American Mathematical Society

Asymptotics of Plancherel measures for symmetric groups

By Alexei Borodin, Andrei Okounkov, Grigori Olshanski

Abstract

We consider the asymptotics of the Plancherel measures on partitions of n as n goes to infinity. We prove that the local structure of a Plancherel typical partition in the middle of the limit shape converges to a determinantal point process with the discrete sine kernel.

On the edges of the limit shape, we prove that the joint distribution of suitably scaled 1st, 2nd, and so on rows of a Plancherel typical diagram converges to the corresponding distribution for eigenvalues of random Hermitian matrices (given by the Airy kernel). This proves a conjecture due to Baik, Deift, and Johansson by methods different from the Riemann-Hilbert techniques used in their original papers and from the combinatorial proof given by the second author.

Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures in terms of a new kernel involving Bessel functions. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.

1. Introduction

1.1. Plancherel measures

Given a finite group upper G , by the corresponding Plancherel measure we mean the probability measure on the set upper G Superscript logical-and of irreducible representations of upper G which assigns to a representation pi element-of upper G Superscript logical-and the weight left-parenthesis dimension pi right-parenthesis squared slash StartAbsoluteValue upper G EndAbsoluteValue . For the symmetric group upper S left-parenthesis n right-parenthesis , the set upper S left-parenthesis n right-parenthesis Superscript logical-and is the set of partitions lamda of the number n , which we shall identify with Young diagrams with n squares throughout this paper. The Plancherel measure on partitions lamda arises naturally in representation–theoretic, combinatorial, and probabilistic problems. For example, the Plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation Reference31.

We denote the Plancherel measure on partitions of n by upper M Subscript n ,

StartLayout 1st Row with Label left-parenthesis 1.1 right-parenthesis EndLabel upper M Subscript n Baseline left-parenthesis lamda right-parenthesis equals StartFraction left-parenthesis dimension lamda right-parenthesis squared Over n factorial EndFraction comma StartAbsoluteValue lamda EndAbsoluteValue equals n comma EndLayout

where dimension lamda is the dimension of the corresponding representation of upper S left-parenthesis n right-parenthesis . The asymptotic properties of these measures as n right-arrow normal infinity have been studied very intensively; see the References and below.

In the seventies, Logan and Shepp Reference23 and, independently, Vershik and Kerov Reference40Reference42 discovered the following measure concentration phenomenon for upper M Subscript n as n right-arrow normal infinity . Let lamda be a partition of n and let i and j be the usual coordinates on the diagrams, namely, the row number and the column number. Introduce new coordinates u and v by

u equals StartFraction j minus i Over StartRoot n EndRoot EndFraction comma v equals StartFraction i plus j Over StartRoot n EndRoot EndFraction comma

that is, we flip the diagram, rotate it 135 Superscript ring as in Figure 1, and scale it by the factor of n Superscript negative 1 slash 2 in both directions.

After this scaling, the Plancherel measures upper M Subscript n converge as n right-arrow normal infinity (see Reference23Reference40Reference42 for precise statements) to the delta measure supported on the following shape:

StartSet StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to 2 comma StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to v less-than-or-equal-to normal upper Omega left-parenthesis u right-parenthesis EndSet comma

where the function normal upper Omega left-parenthesis u right-parenthesis is defined by

normal upper Omega left-parenthesis u right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction 2 Over pi EndFraction left-parenthesis u arc sine left-parenthesis u slash 2 right-parenthesis plus StartRoot 4 minus u squared EndRoot right-parenthesis comma 2nd Column StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to 2 comma 2nd Row 1st Column StartAbsoluteValue u EndAbsoluteValue comma 2nd Column StartAbsoluteValue u EndAbsoluteValue greater-than 2 period EndLayout

The function normal upper Omega is plotted in Figure 1. As explained in detail in Reference22, this limit shape normal upper Omega is closely connected to Wigner’s semicircle law for distribution of eigenvalues of random matrices; see also Reference19Reference20Reference21.

From a different point of view, the connection with random matrices was observed in Reference3Reference4, and also in earlier papers Reference16Reference28Reference29. In Reference3, Baik, Deift, and Johansson made the following conjecture. They conjectured that in the n right-arrow normal infinity limit and after proper scaling the joint distribution of lamda Subscript i , i equals 1 comma 2 comma period period period , becomes identical to the joint distribution of properly scaled largest eigenvalues of a Gaussian random Hermitian matrix (which form the so-called Airy ensemble; see Section 1.4). They proved this for the individual distribution of lamda 1 and lamda 2 in Reference3 and Reference4, respectively. A combinatorial proof of the full conjecture was given by one of us in Reference25. It was based on an interplay between maps on surfaces and ramified coverings of the sphere.

In this paper we study the local structure of a typical Plancherel diagram both in the bulk of the limit shape normal upper Omega and on its edge, where by the study of the edge we mean the study of the behavior of lamda 1 , lamda 2 , and so on.

We employ an analytic approach based on an exact formula in terms of Bessel functions for the correlation functions of the so-called poissonization of the Plancherel measures upper M Subscript n (see Theorem 1 in the following subsection), and the so-called depoissonization techniques (see Section 1.4).

The exact formula in Theorem 1 is a limit case of a formula from Reference8; see also the recent papers Reference26Reference27 for more general results. The use of poissonization and depoissonization is very much in the spirit of Reference3Reference16Reference39 and represents a well–known statistical mechanics principle of the equivalence of canonical and grand canonical ensembles.

Our main results are the following two. In the bulk of the limit shape normal upper Omega , we prove that the local structure of a Plancherel typical partition converges to a determinantal point process with the discrete sine kernel; see Theorem 3. This result is parallel to the corresponding result for random matrices. On the edge of the limit shape, we give an analytic proof of the Baik-Deift-Johansson conjecture; see Theorem 4. These results will be stated in Sections 1.3 and 1.4 of the present Introduction, respectively.

Simultaneously and independently, results equivalent to our Theorems 2 and 4 were obtained by K. Johansson Reference17.

1.2. Poissonization and correlation functions

For theta greater-than 0 , consider the poissonization upper M Superscript theta of the measures upper M Subscript n :

upper M Superscript theta Baseline left-parenthesis lamda right-parenthesis equals e Superscript negative theta Baseline sigma-summation Underscript n Endscripts StartFraction theta Superscript n Baseline Over n factorial EndFraction upper M Subscript n Baseline left-parenthesis lamda right-parenthesis equals e Superscript negative theta Baseline theta Superscript StartAbsoluteValue lamda EndAbsoluteValue Baseline left-parenthesis StartFraction dimension lamda Over StartAbsoluteValue lamda EndAbsoluteValue factorial EndFraction right-parenthesis squared period

This is a probability measure on the set of all partitions. Our first result is the computation of the correlation functions of the measures upper M Superscript theta .

By correlation functions we mean the following. By definition, set

script upper D left-parenthesis lamda right-parenthesis equals StartSet lamda Subscript i Baseline minus i EndSet subset-of double-struck upper Z period

Also, following Reference41, define the modified Frobenius coordinates upper F r left-parenthesis lamda right-parenthesis of a partition lamda by

StartLayout 1st Row with Label left-parenthesis 1.2 right-parenthesis EndLabel StartLayout 1st Row 1st Column upper F r left-parenthesis lamda right-parenthesis 2nd Column equals left-parenthesis script upper D left-parenthesis lamda right-parenthesis plus one-half right-parenthesis white up pointing triangle left-parenthesis double-struck upper Z Subscript 0 Baseline minus one-half right-parenthesis 2nd Row 1st Column Blank 2nd Column equals StartSet p 1 plus one-half comma ellipsis comma p Subscript d Baseline plus one-half comma minus q 1 minus one-half comma ellipsis comma minus q Subscript d Baseline minus one-half EndSet subset-of double-struck upper Z plus one-half comma EndLayout EndLayout

where white up pointing triangle stands for the symmetric difference of two sets, d is the number of squares on the diagonal of lamda , and p Subscript i ’s and q Subscript i ’s are the usual Frobenius coordinates of lamda . Recall that p Subscript i is the number of squares in the i th row to the right of the diagonal, and q Subscript i is number of squares in the i th column below the diagonal. The equality Equation1.2 is a well–known combinatorial fact discovered by Frobenius; see Ex. I.1.15(a) in Reference24. Note that, in contrast to upper F r left-parenthesis lamda right-parenthesis , the set script upper D left-parenthesis lamda right-parenthesis is infinite and, moreover, it contains all but finitely many negative integers.

The sets script upper D left-parenthesis lamda right-parenthesis and upper F r left-parenthesis lamda right-parenthesis have the following nice geometric interpretation. Let the diagram lamda be flipped and rotated 135 Superscript ring as in Figure 1, but not scaled. Denote by omega Subscript lamda a piecewise linear function with omega prime Subscript lamda Baseline equals plus-or-minus 1 whose graph is given by the upper boundary of lamda completed by the lines

v equals StartAbsoluteValue u EndAbsoluteValue comma u not-an-element-of left-bracket minus lamda prime 1 comma lamda 1 right-bracket period

Then

k element-of script upper D left-parenthesis lamda right-parenthesis left right double arrow omega Subscript lamda Superscript prime Baseline vertical-bar Subscript left-bracket k comma k plus 1 right-bracket Baseline equals negative 1 period

In other words, if we consider omega Subscript lamda as a history of a walk on double-struck upper Z , then script upper D left-parenthesis lamda right-parenthesis are those moments when a step is made in the negative direction. It is therefore natural to call script upper D left-parenthesis lamda right-parenthesis the descent set of lamda . As we shall see, the correspondence lamda right-arrow from bar script upper D left-parenthesis lamda right-parenthesis is a very convenient way to encode the local structure of the boundary of lamda .

The halves in the definition of upper F r left-parenthesis lamda right-parenthesis have the following interpretation: one splits the diagonal squares in half and gives half to the rows and half to the columns.

Definition 1.1

The correlation functions of upper M Superscript theta are the probabilities that the sets upper F r left-parenthesis lamda right-parenthesis or, similarly, script upper D left-parenthesis lamda right-parenthesis contain a fixed subset upper X . More precisely, we set

StartLayout 1st Row with Label left-parenthesis 1.3 right-parenthesis EndLabel 1st Column rho Superscript theta Baseline left-parenthesis upper X right-parenthesis 2nd Column equals upper M Superscript theta Baseline left-parenthesis StartSet lamda vertical-bar upper X subset-of upper F r left-parenthesis lamda right-parenthesis EndSet right-parenthesis comma 3rd Column Blank 4th Column upper X subset-of double-struck upper Z plus one-half comma 2nd Row with Label left-parenthesis 1.4 right-parenthesis EndLabel 1st Column bold-italic rho Superscript theta Baseline left-parenthesis upper X right-parenthesis 2nd Column equals upper M Superscript theta Baseline left-parenthesis StartSet lamda vertical-bar upper X subset-of script upper D left-parenthesis lamda right-parenthesis EndSet right-parenthesis comma 3rd Column Blank 4th Column upper X subset-of double-struck upper Z period EndLayout

Theorem 1

For any upper X equals StartSet x 1 comma ellipsis comma x Subscript s Baseline EndSet subset-of double-struck upper Z plus one-half we have

rho Superscript theta Baseline left-parenthesis upper X right-parenthesis equals det left-bracket sans-serif upper K left-parenthesis x Subscript i Baseline comma x Subscript j Baseline right-parenthesis right-bracket Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to s Baseline comma

where the kernel sans-serif upper K is given by the following formula:

StartLayout 1st Row with Label left-parenthesis 1.5 right-parenthesis EndLabel sans-serif upper K left-parenthesis x comma y right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartRoot theta EndRoot StartFraction sans-serif k Subscript plus Baseline left-parenthesis StartAbsoluteValue x EndAbsoluteValue comma StartAbsoluteValue y EndAbsoluteValue right-parenthesis Over StartAbsoluteValue x EndAbsoluteValue minus StartAbsoluteValue y EndAbsoluteValue EndFraction comma 2nd Column x y greater-than 0 comma 2nd Row 1st Column StartRoot theta EndRoot StartFraction sans-serif k Subscript minus Baseline left-parenthesis StartAbsoluteValue x EndAbsoluteValue comma StartAbsoluteValue y EndAbsoluteValue right-parenthesis Over x minus y EndFraction comma 2nd Column x y less-than 0 period EndLayout EndLayout

The functions sans-serif k Subscript plus-or-minus are defined by

StartLayout 1st Row with Label left-parenthesis 1.6 right-parenthesis EndLabel 1st Column sans-serif k Subscript plus Baseline left-parenthesis x comma y right-parenthesis 2nd Column equals upper J Subscript x minus one-half Baseline upper J Subscript y plus one-half Baseline minus upper J Subscript x plus one-half Baseline upper J Subscript y minus one-half Baseline comma 2nd Row with Label left-parenthesis 1.7 right-parenthesis EndLabel 1st Column sans-serif k Subscript minus Baseline left-parenthesis x comma y right-parenthesis 2nd Column equals upper J Subscript x minus one-half Baseline upper J Subscript y minus one-half Baseline plus upper J Subscript x plus one-half Baseline upper J Subscript y plus one-half Baseline comma EndLayout

where upper J Subscript x Baseline equals upper J Subscript x Baseline left-parenthesis 2 StartRoot theta EndRoot right-parenthesis is the Bessel function of order x and argument 2 StartRoot theta EndRoot . The diagonal values sans-serif upper K left-parenthesis x comma x right-parenthesis are determined by the l’Hospital rule.

This theorem is established in Section 2.1; see also Remark 1.2 below. By the complementation principle (see Sections A.3 and 2.2), Theorem 1 is equivalent to the following

Theorem 2

For any upper X equals StartSet x 1 comma ellipsis comma x Subscript s Baseline EndSet subset-of double-struck upper Z we have

StartLayout 1st Row with Label left-parenthesis 1.8 right-parenthesis EndLabel bold-italic rho Superscript theta Baseline left-parenthesis upper X right-parenthesis equals det left-bracket sans-serif upper J left-parenthesis x Subscript i Baseline comma x Subscript j Baseline right-parenthesis right-bracket Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to s Baseline period EndLayout

Here the kernel sans-serif upper J is given by the following formula:

StartLayout 1st Row with Label left-parenthesis 1.9 right-parenthesis EndLabel sans-serif upper J left-parenthesis x comma y right-parenthesis equals sans-serif upper J left-parenthesis x comma y semicolon theta right-parenthesis equals StartRoot theta EndRoot StartFraction upper J Subscript x Baseline upper J Subscript y plus 1 Baseline minus upper J Subscript x plus 1 Baseline upper J Subscript y Baseline Over x minus y EndFraction comma EndLayout

where upper J Subscript x Baseline equals upper J Subscript x Baseline left-parenthesis 2 StartRoot theta EndRoot right-parenthesis . The diagonal values sans-serif upper J left-parenthesis x comma x right-parenthesis are determined by the l’Hospital rule.

Remark 1.2

Theorem 1 is a limit case of Theorem 3.3 of Reference8. For the reader’s convenience a direct proof of it is given in Section 2. Various limit cases of the results of Reference8 are discussed in Reference9. By different methods, the formula Equation1.8 was obtained by K. Johansson Reference17.

A representation–theoretic proof of a more general formula than Theorem 3.3 of Reference8 has been subsequently given in Reference27Reference26; see also Reference7.

Remark 1.3

Observe that all Bessel functions involved in the above formulas are of integer order. Also note that the ratios like sans-serif upper J left-parenthesis x comma y right-parenthesis are entire functions of x and y because upper J Subscript x is an entire function of x . In particular, the values sans-serif upper J left-parenthesis x comma x right-parenthesis are well defined. Various denominator–free formulas for the kernel sans-serif upper J are given in Section 2.1.

1.3. Asymptotics in the bulk of the spectrum

Given a sequence of subsets

upper X left-parenthesis n right-parenthesis equals StartSet x 1 left-parenthesis n right-parenthesis less-than dot dot dot less-than x Subscript s Baseline left-parenthesis n right-parenthesis EndSet subset-of double-struck upper Z comma

where s equals StartAbsoluteValue upper X left-parenthesis n right-parenthesis EndAbsoluteValue is some fixed integer, we call this sequence regular if the limits

StartLayout 1st Row with Label left-parenthesis 1.10 right-parenthesis EndLabel 1st Column a Subscript i 2nd Column equals limit Underscript n right-arrow normal infinity Endscripts StartFraction x Subscript i Baseline left-parenthesis n right-parenthesis Over StartRoot n EndRoot EndFraction comma 2nd Row with Label left-parenthesis 1.11 right-parenthesis EndLabel 1st Column d Subscript i j 2nd Column equals limit Underscript n right-arrow normal infinity Endscripts left-parenthesis x Subscript i Baseline left-parenthesis n right-parenthesis minus x Subscript j Baseline left-parenthesis n right-parenthesis right-parenthesis EndLayout

exist, finite or infinite. Here i comma j equals 1 comma ellipsis comma s . Observe that if d Subscript i j is finite, then d Subscript i j Baseline equals x Subscript i Baseline left-parenthesis n right-parenthesis minus x Subscript j Baseline left-parenthesis n right-parenthesis for n much-greater-than 0 .

In the case when upper X left-parenthesis n right-parenthesis can be represented as upper X left-parenthesis n right-parenthesis equals upper X prime left-parenthesis n right-parenthesis union upper X double-prime left-parenthesis n right-parenthesis and the distance between upper X prime left-parenthesis n right-parenthesis and upper X double-prime left-parenthesis n right-parenthesis goes to normal infinity as n right-arrow normal infinity we shall say that the sequence splits; otherwise, we call it nonsplit. Obviously, upper X left-parenthesis n right-parenthesis is nonsplit if and only if all x Subscript i Baseline left-parenthesis n right-parenthesis stay at a finite distance from each other.

Define the correlation functions bold-italic rho left-parenthesis n comma dot right-parenthesis of the measures upper M Subscript n by the same rule as in Equation1.4:

bold-italic rho left-parenthesis n comma upper X right-parenthesis equals upper M Subscript n Baseline left-parenthesis StartSet lamda vertical-bar upper X subset-of script upper D left-parenthesis lamda right-parenthesis EndSet right-parenthesis period

We are interested in the limit of bold-italic rho left-parenthesis n comma upper X left-parenthesis n right-parenthesis right-parenthesis as n right-arrow normal infinity . This limit will be computed in Theorem 3 below. As we shall see, if upper X left-parenthesis n right-parenthesis splits, then the limit correlations factor accordingly.

Introduce the following discrete sine kernel which is a translation invariant kernel on the lattice double-struck upper Z ,

sans-serif upper S left-parenthesis k comma l semicolon a right-parenthesis equals sans-serif upper S left-parenthesis k minus l comma a right-parenthesis comma k comma l element-of double-struck upper Z comma

depending on a real parameter a :

StartLayout 1st Row 1st Column sans-serif upper S left-parenthesis k comma a right-parenthesis 2nd Column equals StartFraction sine left-parenthesis arc cosine left-parenthesis a slash 2 right-parenthesis dot k right-parenthesis Over pi k EndFraction comma k element-of double-struck upper Z period EndLayout

Note that sans-serif upper S left-parenthesis k comma a right-parenthesis equals sans-serif upper S left-parenthesis negative k comma a right-parenthesis and for k greater-than-or-equal-to 1 we have

sans-serif upper S left-parenthesis k comma a right-parenthesis equals StartFraction StartRoot 4 minus a squared EndRoot Over 2 pi EndFraction StartFraction upper U Subscript k minus 1 Baseline left-parenthesis a slash 2 right-parenthesis Over k EndFraction comma

where upper U Subscript k is the Tchebyshev polynomials of the second kind. We agree that

sans-serif upper S left-parenthesis 0 comma a right-parenthesis equals StartFraction arc cosine left-parenthesis a slash 2 right-parenthesis Over pi EndFraction comma sans-serif upper S left-parenthesis normal infinity comma a right-parenthesis equals 0

and also that

sans-serif upper S left-parenthesis k comma a right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 comma 2nd Column a greater-than-or-equal-to 2 or a less-than-or-equal-to negative 2 and k not-equals 0 comma 2nd Row 1st Column 1 comma 2nd Column a less-than-or-equal-to negative 2 and k equals 0 period EndLayout

The following result describes the local structure of a Plancherel typical partition.

Theorem 3

Let upper X left-parenthesis n right-parenthesis subset-of double-struck upper Z be a regular sequence and let the numbers a Subscript i , d Subscript i j be defined by Equation1.10, Equation1.11. If upper X left-parenthesis n right-parenthesis splits, that is, if upper X left-parenthesis n right-parenthesis equals upper X prime left-parenthesis n right-parenthesis union upper X double-prime left-parenthesis n right-parenthesis and the distance between upper X prime left-parenthesis n right-parenthesis and upper X double-prime left-parenthesis n right-parenthesis goes to normal infinity as n right-arrow normal infinity , then

StartLayout 1st Row with Label left-parenthesis 1.12 right-parenthesis EndLabel limit Underscript n right-arrow normal infinity Endscripts bold-italic rho left-parenthesis n comma upper X left-parenthesis n right-parenthesis right-parenthesis equals limit Underscript n right-arrow normal infinity Endscripts bold-italic rho left-parenthesis n comma upper X prime left-parenthesis n right-parenthesis right-parenthesis dot limit Underscript n right-arrow normal infinity Endscripts bold-italic rho left-parenthesis n comma upper X double-prime left-parenthesis n right-parenthesis right-parenthesis period EndLayout

If upper X left-parenthesis n right-parenthesis is nonsplit, then

StartLayout 1st Row with Label left-parenthesis 1.13 right-parenthesis EndLabel limit Underscript n right-arrow normal infinity Endscripts bold-italic rho left-parenthesis n comma upper X left-parenthesis n right-parenthesis right-parenthesis equals det left-bracket sans-serif upper S left-parenthesis d Subscript i j Baseline comma a right-parenthesis right-bracket Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to s Baseline comma EndLayout

where sans-serif upper S is the discrete sine kernel and a equals a 1 equals a 2 equals period period period  .

We prove this theorem in Section 3.

Remark 1.4

Notice that, in particular, Theorem 3 implies that, as n right-arrow normal infinity , the shape of a typical partition lamda near any point of the limit curve normal upper Omega is described by a stationary random process. For distinct points on the curve normal upper Omega these random processes are independent.

Remark 1.5

By complementation (see Sections A.3 and 3.2), one obtains from Theorem 3 an equivalent statement about the asymptotics of the following correlation functions:

rho left-parenthesis n comma upper X right-parenthesis equals upper M Subscript n Baseline left-parenthesis StartSet lamda vertical-bar upper X subset-of upper F r left-parenthesis lamda right-parenthesis EndSet right-parenthesis period

Remark 1.6

The discrete sine kernel was studied before (see Reference44Reference45), mainly as a model case for the continuous sine kernel. In particular, the asymptotics of Toeplitz determinants built from the discrete sine kernel was obtained by H. Widom Reference45 who was answering a question of F. Dyson. We thank S. Kerov for pointing out this reference.

Remark 1.7

Note that, in particular, Theorem 3 implies that the limit density (the 1-point correlation function) is given by

StartLayout 1st Row with Label left-parenthesis 1.14 right-parenthesis EndLabel bold-italic rho left-parenthesis normal infinity comma a right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction 1 Over pi EndFraction arc cosine left-parenthesis a slash 2 right-parenthesis comma 2nd Column StartAbsoluteValue a EndAbsoluteValue less-than-or-equal-to 2 comma 2nd Row 1st Column 0 comma 2nd Column a greater-than 2 comma 3rd Row 1st Column 1 comma 2nd Column a less-than negative 2 period EndLayout EndLayout

This is in agreement with the Logan-Shepp-Vershik-Kerov result about the limit shape normal upper Omega . More concretely, the function normal upper Omega is related to the density Equation1.14 by

bold-italic rho left-parenthesis normal infinity comma u right-parenthesis equals StartFraction 1 minus normal upper Omega prime left-parenthesis u right-parenthesis Over 2 EndFraction comma

which can be interpreted as follows. Approximately, we have

number-sign StartSet i vertical-bar StartFraction lamda Subscript i Baseline minus i Over StartRoot n EndRoot EndFraction element-of left-bracket u comma u plus normal upper Delta u right-bracket EndSet almost-equals StartRoot n EndRoot bold-italic rho left-parenthesis normal infinity comma u right-parenthesis normal upper Delta u period

Set w equals StartFraction i Over StartRoot n EndRoot EndFraction . Then the above relation reads normal upper Delta w almost-equals bold-italic rho left-parenthesis normal infinity comma u right-parenthesis normal upper Delta u and it should be satisfied on the boundary v equals normal upper Omega left-parenthesis u right-parenthesis of the limit shape. Since v equals u plus 2 w , we conclude that

bold-italic rho left-parenthesis normal infinity comma u right-parenthesis almost-equals StartFraction d w Over d u EndFraction equals StartFraction 1 minus normal upper Omega Superscript prime Baseline Over 2 EndFraction comma

as was to be shown.

Remark 1.8

The discrete sine kernel sans-serif upper S becomes especially nice near the diagonal, that is, where a equals 0 . Indeed,

sans-serif upper S left-parenthesis x comma 0 right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 slash 2 comma 2nd Column x equals 0 comma 2nd Row 1st Column left-parenthesis negative 1 right-parenthesis Superscript left-parenthesis x minus 1 right-parenthesis slash 2 Baseline slash left-parenthesis pi x right-parenthesis comma 2nd Column x equals plus-or-minus 1 comma plus-or-minus 3 comma period period period comma 3rd Row 1st Column 0 comma 2nd Column x equals plus-or-minus 2 comma plus-or-minus 4 comma period period period period EndLayout

1.4. Behavior near the edge of the spectrum and the Airy ensemble

The discrete sine kernel sans-serif upper S left-parenthesis k comma a right-parenthesis vanishes if a greater-than-or-equal-to 2 . Therefore, it follows from Theorem 3 that the limit correlations limit bold-italic rho left-parenthesis n comma upper X left-parenthesis n right-parenthesis right-parenthesis vanish if a Subscript i Baseline greater-than-or-equal-to 2 for some i . However, as will be shown below in Proposition 4.1, after a suitable scaling near the edge a equals 2 , the correlation functions bold-italic rho Superscript theta converge to the correlation functions given by the Airy kernel Reference12Reference36

sans-serif upper A left-parenthesis x comma y right-parenthesis equals StartFraction upper A left-parenthesis x right-parenthesis upper A prime left-parenthesis y right-parenthesis minus upper A prime left-parenthesis x right-parenthesis upper A left-parenthesis y right-parenthesis Over x minus y EndFraction period

Here upper A left-parenthesis x right-parenthesis is the Airy function:

StartLayout 1st Row with Label left-parenthesis 1.15 right-parenthesis EndLabel upper A left-parenthesis x right-parenthesis equals StartFraction 1 Over pi EndFraction integral Subscript 0 Superscript normal infinity Baseline cosine left-parenthesis StartFraction u cubed Over 3 EndFraction plus x u right-parenthesis d u period EndLayout

In fact, the following more precise statement is true about the behavior of the Plancherel measure near the edge a equals 2 . By symmetry, everything we say about the edge a equals 2 applies to the opposite edge a equals negative 2 .

Consider the random point process on double-struck upper R whose correlation functions are given by the determinants

rho Subscript k Superscript Airy Baseline left-parenthesis x 1 comma ellipsis comma x Subscript k Baseline right-parenthesis equals det left-bracket sans-serif upper A left-parenthesis x Subscript i Baseline comma x Subscript j Baseline right-parenthesis right-bracket Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to k Baseline comma

and let

zeta equals left-parenthesis zeta 1 greater-than zeta 2 greater-than zeta 3 greater-than period period period right-parenthesis element-of double-struck upper R Superscript normal infinity

be its random configuration. We call the random variables zeta Subscript i ’s the Airy ensemble. It is known Reference12Reference36 that the Airy ensemble describes the behavior of the (properly scaled) 1st, 2nd, and so on largest eigenvalues of a Gaussian random Hermitian matrix. The distribution of individual eigenvalues was obtained by Tracy and Widom in Reference36 in terms of certain Painlevé transcendents.

It has been conjectured by Baik, Deift, and Johansson that the random variables

lamda overTilde equals left-parenthesis lamda overTilde Subscript 1 Baseline greater-than-or-equal-to lamda overTilde Subscript 2 Baseline greater-than-or-equal-to period period period right-parenthesis comma lamda overTilde Subscript i Baseline equals n Superscript 1 slash 3 Baseline left-parenthesis StartFraction lamda Subscript i Baseline Over n Superscript 1 slash 2 Baseline EndFraction minus 2 right-parenthesis comma

converge, in distribution and together with all moments, to the Airy ensemble. They verified this conjecture for individual distribution of lamda 1 and lamda 2 in Reference3 and Reference4, respectively. In particular, in the case of lamda 1 , this generalizes the result of Reference40Reference42 that StartFraction lamda 1 Over StartRoot n EndRoot EndFraction right-arrow 2 in probability as n right-arrow normal infinity . The computation of limit StartFraction lamda 1 Over StartRoot n EndRoot EndFraction was known as the Ulam problem; different solutions to this problem were given in Reference1Reference16Reference32; see also the survey Reference2.

Convergence of all expectations of the form

StartLayout 1st Row with Label left-parenthesis 1.16 right-parenthesis EndLabel mathematical left-angle product Underscript k equals 1 Overscript r Endscripts sigma-summation Underscript i equals 1 Overscript normal infinity Endscripts e Superscript t Super Subscript k Superscript lamda overTilde Super Subscript i Superscript Baseline mathematical right-angle comma t 1 comma ellipsis comma t Subscript r Baseline greater-than 0 comma r equals 1 comma 2 comma period period period comma EndLayout

to the corresponding quantities for the Airy ensemble was established in Reference25. The proof in Reference25 was based on a combinatorial interpretation of Equation1.16 as the asymptotics in a certain enumeration problem for random surfaces.

In the present paper we use different ideas to prove the following

Theorem 4

As n right-arrow normal infinity , the random variables lamda overTilde converge, in joint distribution, to the Airy ensemble.

This is done in Section 4 using methods described in the next subsection. The result stated in Theorem 4 was independently obtained by K. Johansson in Reference17. See, for example, Reference13 for an application of Theorem 4.

1.5. Poissonization and depoissonization

We obtain Theorems 3 and 4 from Theorem 1 using the so-called depoissonization techniques. We recall that the fundamental idea of depoissonization is the following.

Given a sequence b 1 comma b 2 comma b 3 comma period period period its poissonization is, by definition, the function

StartLayout 1st Row with Label left-parenthesis 1.17 right-parenthesis EndLabel upper B left-parenthesis theta right-parenthesis equals e Superscript negative theta Baseline sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts StartFraction theta Superscript k Baseline Over k factorial EndFraction b Subscript k Baseline period EndLayout

Provided the b Subscript k ’s grow not too rapidly this is an entire function of theta . In combinatorics, it is usually called the exponential generating function of the sequence StartSet b Subscript k Baseline EndSet . Various methods of extracting asymptotics of sequences from their generating functions are classically known and widely used (see for example Reference39 where such methods are used to obtain the limit shape of a typical partition under various measures on the set of partitions).

A probabilistic way to look at the generating function Equation1.17 is the following. If theta greater-than-or-equal-to 0 , then upper B left-parenthesis theta right-parenthesis is the expectation of b Subscript eta where eta element-of StartSet 0 comma 1 comma 2 comma period period period EndSet is a Poisson random variable with parameter theta . Because eta has mean theta and standard deviation StartRoot theta EndRoot , one expects that

StartLayout 1st Row with Label left-parenthesis 1.18 right-parenthesis EndLabel upper B left-parenthesis n right-parenthesis almost-equals b Subscript n Baseline comma n right-arrow normal infinity comma EndLayout

provided the variations of b Subscript k for StartAbsoluteValue k minus n EndAbsoluteValue less-than-or-equal-to c o n s t StartRoot n EndRoot are small. One possible regularity condition on b Subscript n which implies Equation1.18 is monotonicity. In a very general and very convenient form, a depoissonization lemma for nonincreasing nonnegative b Subscript n was established by K. Johansson in Reference16. We use this lemma in Section 4 to prove Theorem 4.

Another approach to depoissonization is to use a contour integral

StartLayout 1st Row with Label left-parenthesis 1.19 right-parenthesis EndLabel b Subscript n Baseline equals StartFraction n factorial Over 2 pi i EndFraction integral Underscript upper C Endscripts StartFraction upper B left-parenthesis z right-parenthesis e Superscript z Baseline Over z Superscript n Baseline EndFraction StartFraction d z Over z EndFraction comma EndLayout

where upper C is any contour around z equals 0 . Suppose, for a moment, that b Subscript n is constant, b equals b Subscript n Baseline equals upper B left-parenthesis z right-parenthesis . The function e Superscript z Baseline slash z Superscript n Baseline equals e Superscript z minus n ln z has a unique critical point z equals n . If we choose StartAbsoluteValue z EndAbsoluteValue equals n as the contour upper C , then only neighborhoods of size StartAbsoluteValue z minus n EndAbsoluteValue less-than-or-equal-to c o n s t StartRoot n EndRoot contribute to the asymptotics of Equation1.19. Therefore, for general StartSet b Subscript n Baseline EndSet , we still expect that provided the overall growth of upper B left-parenthesis z right-parenthesis is under control and the variations of upper B left-parenthesis z right-parenthesis for StartAbsoluteValue z minus n EndAbsoluteValue less-than-or-equal-to c o n s t StartRoot n EndRoot are small, the asymptotically significant contribution to Equation1.19 will come from z equals n . That is, we still expect Equation1.18 to be valid. See, for example, Reference15 for a comprehensive discussion and survey of this approach.

We use this approach to prove Theorem 3 in Section 3. The growth conditions on upper B left-parenthesis z right-parenthesis which are suitable in our situation are spelled out in Lemma 3.1.

In our case, the functions upper B left-parenthesis theta right-parenthesis are combinations of the Bessel functions. Their asymptotic behavior as theta almost-equals n right-arrow normal infinity can be obtained directly from the classical results on asymptotics of Bessel functions which are discussed, for example, in the fundamental Watson’s treatise Reference43. These asymptotic formulas for Bessel functions are derived using the integral representations of Bessel functions and the steepest descent method. The different behavior of the asymptotics in the bulk left-parenthesis negative 2 comma 2 right-parenthesis of the spectrum, near the edges plus-or-minus 2 of the spectrum, and outside of left-bracket negative 2 comma 2 right-bracket is produced by the different location of the saddle point in these three cases.

1.6. Organization of the paper

Section 2 contains the proof of Theorems 1 and Equation2 and also various formulas for the kernels sans-serif upper K and sans-serif upper J . We also discuss a difference operator which commutes with sans-serif upper J and its possible applications.

Section 3 deals with the behavior of the Plancherel measure in the bulk of the spectrum; there we prove Theorem 3. Theorem 4 and a similar result (Theorem 5) for the poissonized measure upper M Superscript theta are established in Section 4.

At the end of the paper there is an Appendix, where we collected some necessary results about Fredholm determinants, point processes, and convergence of trace class operators.

2. Correlation functions of the measures upper M Superscript theta

2.1. Proof of Theorem 1

As noted above, Theorem 1 is a limit case of Theorem 3.3 of Reference8. That theorem concerns a family StartSet upper M Subscript z z prime Superscript left-parenthesis n right-parenthesis Baseline EndSet of probability measures on partitions of n , where z comma z prime are certain parameters. When the parameters go to infinity, upper M Subscript z z prime Superscript left-parenthesis n right-parenthesis tends to the Plancherel measure upper M Subscript n . Theorem 3.3 in Reference8 gives a determinantal formula for the correlation functions of the measure

StartLayout 1st Row with Label left-parenthesis 2.1 right-parenthesis EndLabel upper M Subscript z z prime Superscript xi Baseline equals left-parenthesis 1 minus xi right-parenthesis Superscript t Baseline sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts StartFraction left-parenthesis t right-parenthesis Subscript n Baseline Over n factorial EndFraction xi Superscript n Baseline upper M Subscript z z prime Superscript left-parenthesis n right-parenthesis EndLayout

in terms of a certain hypergeometric kernel. Here t equals z z Superscript prime Baseline greater-than 0 and xi element-of left-parenthesis 0 comma 1 right-parenthesis is an additional parameter. As z comma z prime right-arrow normal infinity and xi equals StartFraction theta Over t EndFraction right-arrow 0 , the negative binomial distribution in Equation2.1 tends to the Poisson distribution with parameter theta . In the same limit, the hypergeometric kernel becomes the kernel sans-serif upper K of Theorem 1. The Bessel functions appear as a suitable degeneration of hypergeometric functions.

Recently, these results of Reference8 were considerably generalized in Reference26, where it was shown how this type of correlation functions can be computed using simple commutation relations in the infinite wedge space.

For the reader’s convenience, we present here a direct and elementary proof of Theorem 1 which uses the same ideas as in Reference8 plus an additional technical trick, namely, differentiation with respect to theta which kills denominators. This trick yields a denominator–free integral formula for the kernel sans-serif upper K ; see Proposition 2.7. Our proof here is a verification, not a derivation. For more conceptual approaches the reader is referred to Reference26Reference27Reference7.

Let x comma y element-of double-struck upper Z plus one-half . Introduce the following kernel sans-serif upper L :

sans-serif upper L left-parenthesis x comma y semicolon theta right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 comma 2nd Column x y greater-than 0 comma 2nd Row 1st Column StartFraction 1 Over x minus y EndFraction StartFraction theta Superscript left-parenthesis StartAbsoluteValue x EndAbsoluteValue plus StartAbsoluteValue y EndAbsoluteValue right-parenthesis slash 2 Baseline Over normal upper Gamma left-parenthesis StartAbsoluteValue x EndAbsoluteValue plus one-half right-parenthesis normal upper Gamma left-parenthesis StartAbsoluteValue y EndAbsoluteValue plus one-half right-parenthesis EndFraction comma 2nd Column x y less-than 0 period EndLayout

We shall consider the kernels sans-serif upper K and sans-serif upper L as operators in the script l squared space on double-struck upper Z plus one-half .

We recall that simple multiplicative formulas (for example, the hook formula) are known for the number dimension lamda in Equation1.1. For our purposes, it is convenient to rewrite the hook formula in the following determinantal form. Let lamda equals left-parenthesis p 1 comma ellipsis comma p Subscript d Baseline vertical-bar q 1 comma ellipsis comma q Subscript d Baseline right-parenthesis be the Frobenius coordinates of lamda ; see Section 1.2. We have

StartLayout 1st Row with Label left-parenthesis 2.2 right-parenthesis EndLabel StartFraction dimension lamda Over StartAbsoluteValue lamda EndAbsoluteValue factorial EndFraction equals det left-bracket StartFraction 1 Over left-parenthesis p Subscript i Baseline plus q Subscript j Baseline plus 1 right-parenthesis p Subscript i Baseline factorial q Subscript i Baseline factorial EndFraction right-bracket Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to d Baseline period EndLayout

The following proposition is a straightforward computation using Equation2.2.

Proposition 2.1

Let lamda be a partition. Then

StartLayout 1st Row with Label left-parenthesis 2.3 right-parenthesis EndLabel upper M Superscript theta Baseline left-parenthesis lamda right-parenthesis equals e Superscript negative theta Baseline det left-bracket sans-serif upper L left-parenthesis x Subscript i Baseline comma x Subscript j Baseline semicolon theta right-parenthesis right-bracket Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to s Baseline comma EndLayout

where upper F r left-parenthesis lamda right-parenthesis equals StartSet x 1 comma ellipsis comma x Subscript s Baseline EndSet subset-of double-struck upper Z plus one-half are the modified Frobenius coordinates of lamda defined in Equation1.2.

Let upper F r Subscript asterisk Baseline left-parenthesis upper M Superscript theta Baseline right-parenthesis be the push-forward of upper M Superscript theta under the map upper F r . Note that the image of upper F r consists of sets upper X subset-of double-struck upper Z plus one-half having equally many positive and negative elements. For other upper X subset-of double-struck upper Z plus one-half , the right-hand side of Equation2.3 can be easily seen to vanish. Therefore upper F r Subscript asterisk Baseline left-parenthesis upper M Superscript theta Baseline right-parenthesis is a determinantal point process (see the Appendix) corresponding to sans-serif upper L , that is, its configuration probabilities are determinants of the form Equation2.3.

Corollary 2.2

det left-parenthesis 1 plus sans-serif upper L right-parenthesis equals e Superscript theta .

This follows from the fact that upper M Superscript theta is a probability measure. This is explained in Propositions A.1 and A.4 in the Appendix. Note that, in general, one needs to check that sans-serif upper L is a trace class operator.Footnote1 Actually, sans-serif upper L is of trace class because the sum of the absolute values of its matrix elements is finite. We are grateful to P. Deift for this remark. However, because of the special form of sans-serif upper L , it suffices to check a weaker claim – that sans-serif upper L is a Hilbert–Schmidt operator, which is immediate.

Theorem 1 now follows from general properties of determinantal point processes (see Proposition A.6 in the Appendix) and the following

Proposition 2.3

sans-serif upper K equals sans-serif upper L left-parenthesis 1 plus sans-serif upper L right-parenthesis Superscript negative 1 .

We shall need three identities for Bessel functions which are degenerations of the identities (3.13–15) in Reference8 for the hypergeometric function. The first identity is due to Lommel (see Reference43, Section 3.2, or Reference14, 7.2.(60)):

StartLayout 1st Row with Label left-parenthesis 2.4 right-parenthesis EndLabel upper J Subscript nu Baseline left-parenthesis 2 z right-parenthesis upper J Subscript 1 minus nu Baseline left-parenthesis 2 z right-parenthesis plus upper J Subscript negative nu Baseline left-parenthesis 2 z right-parenthesis upper J Subscript nu minus 1 Baseline left-parenthesis 2 z right-parenthesis equals StartFraction sine pi nu Over pi z EndFraction period EndLayout

The other two identities are the following.

Lemma 2.4

For any nu not-equals 0 comma negative 1 comma negative 2 comma period period period and any z not-equals 0 we have

StartLayout 1st Row with Label left-parenthesis 2.5 right-parenthesis EndLabel 1st Column Blank 2nd Column sigma-summation Underscript m equals 0 Overscript normal infinity Endscripts StartFraction 1 Over m plus nu EndFraction StartFraction z Superscript m Baseline Over m factorial EndFraction upper J Subscript m Baseline left-parenthesis 2 z right-parenthesis equals StartFraction normal upper Gamma left-parenthesis nu right-parenthesis upper J Subscript nu Baseline left-parenthesis 2 z right-parenthesis Over z Superscript nu Baseline EndFraction comma 2nd Row with Label left-parenthesis 2.6 right-parenthesis EndLabel 1st Column Blank 2nd Column sigma-summation Underscript m equals 0 Overscript normal infinity Endscripts StartFraction 1 Over m plus nu EndFraction StartFraction z Superscript m Baseline Over m factorial EndFraction upper J Subscript m plus 1 Baseline left-parenthesis 2 z right-parenthesis equals StartFraction 1 Over z EndFraction minus StartFraction normal upper Gamma left-parenthesis nu right-parenthesis upper J Subscript nu minus 1 Baseline left-parenthesis 2 z right-parenthesis Over z Superscript nu Baseline EndFraction period EndLayout

Proof.

Another identity due to Lommel (see Reference43, Section 5.23, or Reference14, 7.15.(10)) reads

sigma-summation Underscript m equals 0 Overscript normal infinity Endscripts StartFraction normal upper Gamma left-parenthesis nu minus s plus m right-parenthesis Over normal upper Gamma left-parenthesis nu plus m plus 1 right-parenthesis EndFraction StartFraction z Superscript m Baseline Over m factorial EndFraction upper J Subscript m plus s Baseline left-parenthesis 2 z right-parenthesis equals StartFraction normal upper Gamma left-parenthesis nu minus s right-parenthesis Over normal upper Gamma left-parenthesis s plus 1 right-parenthesis EndFraction StartFraction upper J Subscript nu Baseline left-parenthesis 2 z right-parenthesis Over z Superscript nu minus s Baseline EndFraction period

Substituting s equals 0 we get Equation2.5. Substituting s equals 1 yields

StartLayout 1st Row with Label left-parenthesis 2.7 right-parenthesis EndLabel sigma-summation Underscript m equals 0 Overscript normal infinity Endscripts StartFraction 1 Over left-parenthesis m plus nu right-parenthesis left-parenthesis m plus nu minus 1 right-parenthesis EndFraction StartFraction z Superscript m Baseline Over m factorial EndFraction upper J Subscript m plus 1 Baseline left-parenthesis 2 z right-parenthesis equals StartFraction normal upper Gamma left-parenthesis nu minus 1 right-parenthesis upper J Subscript nu Baseline left-parenthesis 2 z right-parenthesis Over z Superscript nu minus 1 Baseline EndFraction period EndLayout

Let r left-parenthesis nu comma z right-parenthesis be the difference of the left-hand side and the right-hand side in Equation2.6. Using Equation2.7 and the recurrence relation

StartLayout 1st Row with Label left-parenthesis 2.8 right-parenthesis EndLabel upper J Subscript nu plus 1 Baseline left-parenthesis 2 z right-parenthesis minus StartFraction nu Over z EndFraction upper J Subscript nu Baseline left-parenthesis 2 z right-parenthesis plus upper J Subscript nu minus 1 Baseline left-parenthesis 2 z right-parenthesis equals 0 EndLayout

we find that r left-parenthesis nu plus 1 comma z right-parenthesis equals r left-parenthesis nu comma z right-parenthesis . Hence for any z it is a periodic function of nu and it suffices to show that limit Underscript nu right-arrow normal infinity Endscripts r left-parenthesis nu comma z right-parenthesis equals 0 . Clearly, the left-hand side in Equation2.6 goes to 0 as nu right-arrow normal infinity . From the defining series for upper J Subscript nu it is clear that

StartLayout 1st Row with Label left-parenthesis 2.9 right-parenthesis EndLabel upper J Subscript nu Baseline left-parenthesis 2 z right-parenthesis tilde StartFraction z Superscript nu Baseline Over normal upper Gamma left-parenthesis nu plus 1 right-parenthesis EndFraction comma nu right-arrow normal infinity comma EndLayout

which implies that the right-hand side of Equation2.6 also goes to 0 as nu right-arrow normal infinity . This concludes the proof.

Proof of Proposition 2.3.

It is convenient to set z equals StartRoot theta EndRoot . Since the operator 1 plus sans-serif upper L is invertible we have to check that

sans-serif upper K plus sans-serif upper K sans-serif upper L minus sans-serif upper L equals 0 period

This is clearly true for z equals 0 ; therefore, it suffices to check that

StartLayout 1st Row with Label left-parenthesis 2.10 right-parenthesis EndLabel ModifyingAbove sans-serif upper K With dot plus ModifyingAbove sans-serif upper K With dot sans-serif upper L plus sans-serif upper K ModifyingAbove sans-serif upper L With dot minus ModifyingAbove sans-serif upper L With dot equals 0 comma EndLayout

where ModifyingAbove sans-serif upper K With dot equals StartFraction partial-differential sans-serif upper K Over partial-differential z EndFraction and ModifyingAbove sans-serif upper L With dot equals StartFraction partial-differential sans-serif upper L Over partial-differential z EndFraction . Using the formulas

StartLayout 1st Row with Label left-parenthesis 2.11 right-parenthesis EndLabel 1st Column StartFraction d Over d z EndFraction upper J Subscript x Baseline left-parenthesis 2 z right-parenthesis 2nd Column equals minus 3rd Column Blank 4th Column 2 upper J Subscript x plus 1 Baseline left-parenthesis 2 z right-parenthesis plus StartFraction x Over z EndFraction upper J Subscript x Baseline left-parenthesis 2 z right-parenthesis 2nd Row 1st Column Blank 2nd Column equals 3rd Column Blank 4th Column 2 upper J Subscript x minus 1 Baseline left-parenthesis 2 z right-parenthesis minus StartFraction x Over z EndFraction upper J Subscript x Baseline left-parenthesis 2 z right-parenthesis EndLayout

one computes

ModifyingAbove sans-serif upper K With dot left-parenthesis x comma y right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column upper J Subscript StartAbsoluteValue x EndAbsoluteValue minus one-half Baseline upper J Subscript StartAbsoluteValue y EndAbsoluteValue plus one-half Baseline plus upper J Subscript StartAbsoluteValue x EndAbsoluteValue plus one-half Baseline upper J Subscript StartAbsoluteValue y EndAbsoluteValue minus one-half Baseline comma 2nd Column x y greater-than 0 comma 2nd Row 1st Column s g n left-parenthesis x right-parenthesis left-parenthesis upper J Subscript StartAbsoluteValue x EndAbsoluteValue minus one-half Baseline upper J Subscript StartAbsoluteValue y EndAbsoluteValue minus one-half Baseline minus upper J Subscript StartAbsoluteValue x EndAbsoluteValue plus one-half Baseline upper J Subscript StartAbsoluteValue y EndAbsoluteValue plus one-half Baseline right-parenthesis comma 2nd Column x y less-than 0 comma EndLayout

where upper J Subscript x Baseline equals upper J Subscript x Baseline left-parenthesis 2 z right-parenthesis . Similarly,

ModifyingAbove sans-serif upper L With dot left-parenthesis x comma y right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 0 comma 2nd Column x y greater-than 0 comma 2nd Row 1st Column s g n left-parenthesis x right-parenthesis StartFraction z Superscript StartAbsoluteValue x EndAbsoluteValue plus StartAbsoluteValue y EndAbsoluteValue minus 1 Baseline Over normal upper Gamma left-parenthesis StartAbsoluteValue x EndAbsoluteValue plus one-half right-parenthesis normal upper Gamma left-parenthesis StartAbsoluteValue y EndAbsoluteValue plus one-half right-parenthesis EndFraction comma 2nd Column x y less-than 0 period EndLayout

Now the verification of Equation2.10 becomes a straightforward application of the formulas Equation2.5 and Equation2.6, except for the occurrence of the singularity nu element-of double-struck upper Z Subscript 0 in those formulas. This singularity is resolved using Equation2.4. This concludes the proof of Proposition 2.3 and Theorem 1.

2.2. Proof of Theorem Equation2

Recall that by construction

upper F r left-parenthesis lamda right-parenthesis equals left-parenthesis script upper D left-parenthesis lamda right-parenthesis plus one-half right-parenthesis white up pointing triangle left-parenthesis double-struck upper Z Subscript 0 Baseline minus one-half right-parenthesis period

Let us check that this and Proposition A.8 imply Theorem Equation2. In Proposition A.8 we substitute

German upper X equals double-struck upper Z plus one-half comma upper Z equals double-struck upper Z Subscript 0 Baseline minus one-half comma upper K equals sans-serif upper K period

By definition, set

epsilon left-parenthesis x right-parenthesis equals s g n left-parenthesis x right-parenthesis Superscript x plus 1 slash 2 Baseline comma x element-of double-struck upper Z plus one-half period

We have the following

Lemma 2.5

sans-serif upper K Superscript white up pointing triangle Baseline left-parenthesis x comma y right-parenthesis equals epsilon left-parenthesis x right-parenthesis epsilon left-parenthesis y right-parenthesis sans-serif upper J left-parenthesis x minus one-half comma y minus one-half right-parenthesis .

It is clear that since the epsilon -factors cancel out of all determinantal formulas, this lemma and Proposition A.8 establish the equivalence of Theorems 1 and Equation2.

Proof.

Using the relation

upper J Subscript negative n Baseline equals left-parenthesis negative 1 right-parenthesis Superscript n Baseline upper J Subscript n

and the definition of sans-serif upper K one computes

StartLayout 1st Row with Label left-parenthesis 2.12 right-parenthesis EndLabel sans-serif upper K left-parenthesis x comma y right-parenthesis equals s g n left-parenthesis x right-parenthesis epsilon left-parenthesis x right-parenthesis epsilon left-parenthesis y right-parenthesis sans-serif upper J left-parenthesis x minus one-half comma y minus one-half right-parenthesis comma x not-equals y period EndLayout

Clearly, the relation Equation2.12 remains valid for x equals y greater-than 0 . It remains to consider the case x equals y less-than 0 . In this case we have to show that

1 minus sans-serif upper K left-parenthesis x comma x right-parenthesis equals sans-serif upper J left-parenthesis x minus one-half comma y minus one-half right-parenthesis comma x element-of double-struck upper Z Subscript 0 Baseline minus one-half period

Rewrite it as

StartLayout 1st Row with Label left-parenthesis 2.13 right-parenthesis EndLabel 1 minus sans-serif upper J left-parenthesis k comma k right-parenthesis equals sans-serif upper J left-parenthesis negative k minus 1 comma negative k minus 1 right-parenthesis comma k equals negative x minus one-half element-of double-struck upper Z Subscript 0 Baseline period EndLayout

By Equation2.14 this is equivalent to

StartLayout 1st Row 1 minus sigma-summation Underscript m equals 0 Overscript normal infinity Endscripts left-parenthesis negative 1 right-parenthesis Superscript m Baseline StartFraction left-parenthesis 2 k plus m plus 2 right-parenthesis Subscript m Baseline Over normal upper Gamma left-parenthesis k plus m plus 2 right-parenthesis normal upper Gamma left-parenthesis k plus m plus 2 right-parenthesis EndFraction StartFraction theta Superscript k plus m plus 1 Baseline Over m factorial EndFraction 2nd Row equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts left-parenthesis negative 1 right-parenthesis Superscript n Baseline StartFraction left-parenthesis minus 2 k plus n right-parenthesis Subscript n Baseline Over normal upper Gamma left-parenthesis negative k plus n plus 1 right-parenthesis normal upper Gamma left-parenthesis negative k plus n plus 1 right-parenthesis EndFraction StartFraction theta Superscript negative k plus n Baseline Over n factorial EndFraction period EndLayout

Examine the right-hand side. The terms with n equals 0 comma ellipsis comma k minus 1 vanish because then 1 slash normal upper Gamma left-parenthesis negative k plus n plus 1 right-parenthesis equals 0 . The term with n equals k is equal to 1, which corresponds to 1 in the left-hand side. Next, the terms with n equals k plus 1 comma ellipsis comma 2 k vanish because for these values of n , the expression left-parenthesis minus 2 k plus n right-parenthesis Subscript n vanishes. Finally, for n greater-than-or-equal-to 2 k plus 1 , set n equals 2 k plus 1 plus m . Then the n th term in the second sum is equal to minus the m th term in the first sum. Indeed, this follows from the trivial relation

minus left-parenthesis negative 1 right-parenthesis Superscript m Baseline StartFraction left-parenthesis 2 k plus m plus 2 right-parenthesis Subscript m Baseline Over m factorial EndFraction equals left-parenthesis negative 1 right-parenthesis Superscript n Baseline StartFraction left-parenthesis minus 2 k plus n right-parenthesis Subscript n Baseline Over n factorial EndFraction comma n equals 2 k plus 1 plus m period

This concludes the proof.

2.3. Various formulas for the kernel sans-serif upper J

Recall that since upper J Subscript x is an entire function of x , the function sans-serif upper J left-parenthesis x comma y right-parenthesis is entire in x and y . We shall now obtain several denominator–free formulas for the kernel sans-serif upper J .

Proposition 2.6

StartLayout 1st Row with Label left-parenthesis 2.14 right-parenthesis EndLabel sans-serif upper J left-parenthesis x comma y semicolon theta right-parenthesis equals sigma-summation Underscript m equals 0 Overscript normal infinity Endscripts left-parenthesis negative 1 right-parenthesis Superscript m Baseline StartFraction left-parenthesis x plus y plus m plus 2 right-parenthesis Subscript m Baseline Over normal upper Gamma left-parenthesis x plus m plus 2 right-parenthesis normal upper Gamma left-parenthesis y plus m plus 2 right-parenthesis EndFraction StartFraction theta Superscript StartFraction x plus y Over 2 EndFraction plus m plus 1 Baseline Over m factorial EndFraction period EndLayout

Proof.

Straightforward computation using a formula due to Nielsen (see Section 5.41 of Reference43 or Reference14, formula 7.2.(48)).

Proposition 2.7

Suppose x plus y greater-than negative 2 . Then

sans-serif upper J left-parenthesis x comma y semicolon theta right-parenthesis equals one-half integral Subscript 0 Superscript 2 StartRoot theta EndRoot Baseline left-parenthesis upper J Subscript x Baseline left-parenthesis z right-parenthesis upper J Subscript y plus 1 Baseline left-parenthesis z right-parenthesis plus upper J Subscript x plus 1 Baseline left-parenthesis z right-parenthesis upper J Subscript y Baseline left-parenthesis z right-parenthesis right-parenthesis d z period

Proof.

Follows from a computation done in the proof of Proposition 2.3,

StartFraction partial-differential Over partial-differential theta EndFraction sans-serif upper J left-parenthesis x comma y semicolon theta right-parenthesis equals StartFraction 1 Over 2 StartRoot theta EndRoot EndFraction left-parenthesis upper J Subscript x Baseline upper J Subscript y plus 1 Baseline plus upper J Subscript x plus 1 Baseline upper J Subscript y Baseline right-parenthesis comma upper J Subscript x Baseline equals upper J Subscript x Baseline left-parenthesis 2 StartRoot theta EndRoot right-parenthesis comma

and the following corollary of Equation2.14:

sans-serif upper J left-parenthesis x comma y semicolon 0 right-parenthesis equals 0 comma x plus y greater-than negative 2 period

Remark 2.8

Observe that by Proposition 2.7 the operator StartFraction partial-differential sans-serif upper J Over partial-differential theta EndFraction is a sum of two operators of rank 1.

Proposition 2.9

StartLayout 1st Row with Label left-parenthesis 2.15 right-parenthesis EndLabel sans-serif upper J left-parenthesis x comma y semicolon theta right-parenthesis equals sigma-summation Underscript s equals 1 Overscript normal infinity Endscripts upper J Subscript x plus s Baseline upper J Subscript y plus s Baseline comma upper J Subscript x Baseline equals upper J Subscript x Baseline left-parenthesis 2 StartRoot theta EndRoot right-parenthesis period EndLayout

Proof.

Our argument is similar to an argument due to Tracy and Widom; see the proof of the formula (4.6) in Reference36. The recurrence relation Equation2.8 implies that

StartLayout 1st Row with Label left-parenthesis 2.16 right-parenthesis EndLabel sans-serif upper J left-parenthesis x plus 1 comma y plus 1 right-parenthesis minus sans-serif upper J left-parenthesis x comma y right-parenthesis equals minus upper J Subscript x plus 1 Baseline upper J Subscript y plus 1 Baseline period EndLayout

Consequently, the difference between the left-hand side and the right-hand side of Equation2.15 is a function which depends only on x minus y . Let x and y go to infinity in such a way that x minus y remains fixed. Because of the asymptotics Equation2.9 both sides in Equation2.15 tend to zero and, hence, the difference actually is 0.

In the same way as in Reference36 this results in the following

Corollary 2.10

For any a element-of double-struck upper Z , the restriction of the kernel sans-serif upper J to the subset StartSet a comma a plus 1 comma a plus 2 comma period period period EndSet subset-of double-struck upper Z defines a nonnegative trace class operator in the script l squared space on that subset.

Proof.

By Proposition Equation2.9, the restriction of sans-serif upper J on StartSet a comma a plus 1 comma a plus 2 comma period period period EndSet is the square of the kernel left-parenthesis x comma y right-parenthesis right-arrow from bar upper J Subscript x plus y plus 1 minus a Baseline left-parenthesis 2 StartRoot theta EndRoot right-parenthesis . Since the latter kernel is real and symmetric, the kernel sans-serif upper J is nonnegative. Hence, it remains to prove that its trace is finite. Again, by Proposition Equation2.9, this trace is equal to

sigma-summation Underscript s equals 1 Overscript normal infinity Endscripts s left-parenthesis upper J Subscript a plus s plus 1 Baseline left-parenthesis 2 StartRoot theta EndRoot right-parenthesis right-parenthesis squared period

This sum is clearly finite by Equation2.9.

Remark 2.11

The kernel sans-serif upper J resembles a Christoffel–Darboux kernel and, in fact, the operator in script l squared left-parenthesis double-struck upper Z right-parenthesis defined by the kernel sans-serif upper J is an Hermitian projection operator. Recall that sans-serif upper K equals sans-serif upper L left-parenthesis 1 plus sans-serif upper L right-parenthesis Superscript negative 1 , where sans-serif upper L is of the form

sans-serif upper L equals Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column upper A 2nd Row 1st Column minus upper A Superscript asterisk Baseline 2nd Column 0 EndMatrix period

One can prove that this together with Lemma 2.5 imply that sans-serif upper J is an Hermitian projection kernel. However, in contrast to a Christoffel–Darboux kernel, it projects to an infinite–dimensional subspace.

Note that in Reference17 the restriction of the kernel upper J to double-struck upper Z Subscript plus was obtained as a limit of Christoffel–Darboux kernels for Charlier polynomials.

2.4. Commuting difference operator

Consider the difference operators normal upper Delta and nabla on the lattice double-struck upper Z ,

left-parenthesis normal upper Delta f right-parenthesis left-parenthesis k right-parenthesis equals f left-parenthesis k plus 1 right-parenthesis minus f left-parenthesis k right-parenthesis comma left-parenthesis nabla f right-parenthesis left-parenthesis k right-parenthesis equals f left-parenthesis k right-parenthesis minus f left-parenthesis k minus 1 right-parenthesis period

Note that nabla equals minus normal upper Delta Superscript asterisk as operators on script l squared left-parenthesis double-struck upper Z right-parenthesis . Consider the following second order difference Sturm–Liouville operator:

StartLayout 1st Row with Label left-parenthesis 2.17 right-parenthesis EndLabel upper D equals normal upper Delta ring alpha ring nabla plus beta comma EndLayout

where alpha and beta are operators of multiplication by certain functions alpha left-parenthesis k right-parenthesis , beta left-parenthesis k right-parenthesis . The operator Equation2.17 is self–adjoint in script l squared left-parenthesis double-struck upper Z right-parenthesis . A straightforward computation shows that

StartLayout 1st Row with Label left-parenthesis 2.18 right-parenthesis EndLabel left-bracket upper D f right-bracket left-parenthesis k right-parenthesis equals left-parenthesis minus alpha left-parenthesis k plus 1 right-parenthesis minus alpha left-parenthesis k right-parenthesis plus beta left-parenthesis k right-parenthesis right-parenthesis f left-parenthesis k right-parenthesis plus alpha left-parenthesis k right-parenthesis f left-parenthesis k minus 1 right-parenthesis plus alpha left-parenthesis k plus 1 right-parenthesis f left-parenthesis k plus 1 right-parenthesis period EndLayout

It follows that if alpha left-parenthesis s right-parenthesis equals 0 for a certain s element-of double-struck upper Z , then the space of functions f left-parenthesis k right-parenthesis vanishing for k less-than s is invariant under upper D .

Proposition 2.12

Let left-bracket sans-serif upper J right-bracket Subscript s denote the operator in script l squared left-parenthesis StartSet s comma s plus 1 comma period period period EndSet right-parenthesis obtained by restricting the kernel sans-serif upper J to StartSet s comma s plus 1 comma period period period EndSet . Then the difference Sturm–Liouville operator Equation2.17 commutes with left-bracket sans-serif upper J right-bracket Subscript s provided

alpha left-parenthesis k right-parenthesis equals k minus s comma beta left-parenthesis k right-parenthesis equals minus StartFraction k left-parenthesis k plus 1 minus s minus 2 StartRoot theta EndRoot right-parenthesis Over StartRoot theta EndRoot EndFraction plus const period

Proof.

Since left-bracket sans-serif upper J right-bracket Subscript s is the square of the operator with the kernel upper J Subscript k plus l plus 1 minus s , it suffices to check that the latter operator commutes with upper D , with the above choice of alpha and beta . But this is readily checked using Equation2.18.

This proposition is a counterpart of a known fact about the Airy kernel; see Reference36. Moreover, in the scaling limit when theta right-arrow normal infinity and

k equals 2 StartRoot theta EndRoot plus x theta Superscript 1 slash 6 Baseline comma s equals 2 StartRoot theta EndRoot plus final sigma theta Superscript 1 slash 6 Baseline comma

the difference operator upper D becomes, for a suitable choice of the constant, the differential operator

StartFraction d Over d x EndFraction ring left-parenthesis x minus final sigma right-parenthesis ring StartFraction d Over d x EndFraction minus x left-parenthesis x minus final sigma right-parenthesis comma

which commutes with the Airy operator restricted to left-parenthesis final sigma comma plus normal infinity right-parenthesis . The above differential operator is exactly that of Tracy and Widom Reference36.

Remark 2.13

Presumably, this commuting difference operator can be used to obtain, as was done in Reference36 for the Airy kernel, asymptotic formulas for the eigenvalues of left-bracket sans-serif upper J right-bracket Subscript s , where s equals 2 StartRoot theta EndRoot plus final sigma theta Superscript 1 slash 6 and final sigma much-less-than 0 . Such asymptotic formulas may be very useful if one wishes to refine Theorem 4 and to establish convergence of moments in addition to convergence of distribution functions. For individual distributions of lamda 1 and lamda 2 the convergence of moments was obtained, by other methods, in Reference3Reference4.

3. Correlation functions in the bulk of the spectrum

3.1. Proof of Theorem 3

We refer the reader to Section 1.3 of the Introduction for the definition of a regular sequence upper X left-parenthesis n right-parenthesis subset-of double-struck upper Z and the statement of Theorem 3. Also, in this section, we shall be working in the bulk of the spectrum, that is, we shall assume that all numbers a Subscript i defined in Equation1.10 lie inside left-parenthesis negative 2 comma 2 right-parenthesis . The edges plus-or-minus 2 of the spectrum and its exterior will be treated in the next section.

In our proof, we shall follow the strategy explained in Section 1.5. Namely, in order to compute the limit of bold-italic rho left-parenthesis n comma upper X left-parenthesis n right-parenthesis right-parenthesis we shall use the contour integral

bold-italic rho left-parenthesis n comma upper X left-parenthesis n right-parenthesis right-parenthesis equals StartFraction n factorial Over 2 pi i EndFraction integral Underscript StartAbsoluteValue theta EndAbsoluteValue equals n Endscripts bold-italic rho Superscript theta Baseline left-parenthesis upper X left-parenthesis n right-parenthesis right-parenthesis StartFraction e Superscript theta Baseline Over theta Superscript n plus 1 Baseline EndFraction d theta comma

compute the asymptotics of bold-italic rho Superscript theta for theta almost-equals n , and estimate StartAbsoluteValue bold-italic rho Superscript theta Baseline EndAbsoluteValue away from theta equals n . Both tasks will be accomplished using classical results about the Bessel functions.

We start our proof with the following lemma which formalizes the above informal depoissonization argument. The hypothesis of this lemma is very far from optimal, but it is sufficient for our purposes. For the rest of this section, we fix a number 0 less-than alpha less-than 1 slash 4 which shall play an auxiliary role.

Lemma 3.1

Let StartSet f Subscript n Baseline EndSet be a sequence of entire functions

f Subscript n Baseline left-parenthesis z right-parenthesis equals e Superscript negative z Baseline sigma-summation Underscript k greater-than-or-equal-to 0 Endscripts StartFraction f Subscript n k Baseline Over k factorial EndFraction z Superscript k Baseline comma n equals 1 comma 2 comma period period period comma

and suppose that there exist constants f Subscript normal infinity and gamma such that

StartLayout 1st Row with Label left-parenthesis 3.1 right-parenthesis EndLabel 1st Column Blank 2nd Column max Underscript StartAbsoluteValue z EndAbsoluteValue equals n Endscripts StartAbsoluteValue f Subscript n Baseline left-parenthesis z right-parenthesis EndAbsoluteValue equals upper O left-parenthesis e Superscript gamma StartRoot n EndRoot Baseline right-parenthesis comma 2nd Row with Label left-parenthesis 3.2 right-parenthesis EndLabel 1st Column Blank 2nd Column max Underscript StartAbsoluteValue z slash n minus 1 EndAbsoluteValue less-than-or-equal-to n Superscript negative alpha Baseline Endscripts StartAbsoluteValue f Subscript n Baseline left-parenthesis z right-parenthesis minus f Subscript normal infinity Baseline EndAbsoluteValue e Superscript minus gamma StartAbsoluteValue z minus n EndAbsoluteValue slash StartRoot n EndRoot Baseline equals o left-parenthesis 1 right-parenthesis comma EndLayout

as n right-arrow normal infinity . Then

limit Underscript n right-arrow normal infinity Endscripts f Subscript n n Baseline equals f Subscript normal infinity Baseline period

Proof.

By replacing f Subscript n Baseline left-parenthesis z right-parenthesis by f Subscript n Baseline left-parenthesis z right-parenthesis minus f Subscript normal infinity , we may assume that f Subscript normal infinity Baseline equals 0 . By Cauchy and Stirling formulas, we have

f Subscript n n Baseline equals left-parenthesis 1 plus o left-parenthesis 1 right-parenthesis right-parenthesis StartRoot StartFraction n Over 2 pi EndFraction EndRoot integral Underscript StartAbsoluteValue zeta EndAbsoluteValue equals 1 Endscripts StartFraction f Subscript n Baseline left-parenthesis n zeta right-parenthesis e Superscript n left-parenthesis zeta minus 1 right-parenthesis Baseline Over zeta Superscript n Baseline EndFraction StartFraction d zeta Over i zeta EndFraction period

Choose some large upper C greater-than 0 and split the circle StartAbsoluteValue zeta EndAbsoluteValue equals 1 into two parts as follows:

upper S 1 equals StartSet StartFraction upper C Over n Superscript 1 slash 4 Baseline EndFraction less-than-or-equal-to StartAbsoluteValue zeta minus 1 EndAbsoluteValue EndSet comma upper S 2 equals StartSet StartFraction upper C Over n Superscript 1 slash 4 Baseline EndFraction greater-than-or-equal-to StartAbsoluteValue zeta minus 1 EndAbsoluteValue EndSet period

The inequality Equation3.1 and the equality

StartAbsoluteValue e Superscript n left-parenthesis zeta minus 1 right-parenthesis Baseline EndAbsoluteValue equals e Superscript minus n StartAbsoluteValue zeta minus 1 EndAbsoluteValue squared slash 2

imply that the integral integral Underscript upper S 1 Endscripts decays exponentially provided upper C is large enough. On upper S 2 , the inequality Equation3.2 applies for sufficiently large n and gives

max Underscript z element-of upper S 2 Endscripts StartAbsoluteValue f Subscript n Baseline left-parenthesis n zeta right-parenthesis EndAbsoluteValue e Superscript minus gamma StartRoot n EndRoot StartAbsoluteValue zeta minus 1 EndAbsoluteValue Baseline equals o left-parenthesis 1 right-parenthesis period

Therefore, the integral integral Underscript upper S 2 Endscripts is o left-parenthesis right-parenthesis of the following integral:

StartRoot n EndRoot integral Underscript StartAbsoluteValue zeta EndAbsoluteValue equals 1 Endscripts StartFraction d zeta Over i zeta EndFraction exp left-parenthesis minus n StartFraction StartAbsoluteValue zeta minus 1 EndAbsoluteValue squared Over 2 EndFraction plus gamma StartRoot n EndRoot StartAbsoluteValue zeta minus 1 EndAbsoluteValue right-parenthesis tilde integral Subscript negative normal infinity Superscript normal infinity Baseline e Superscript minus s squared slash 2 plus gamma StartAbsoluteValue s EndAbsoluteValue Baseline d s period

Hence, integral Underscript upper S 2 Endscripts equals o left-parenthesis 1 right-parenthesis and the lemma follows.

Definition 3.2

Denote by script upper F the algebra (with respect to term-wise addition and multiplication) of sequences StartSet f Subscript n Baseline left-parenthesis z right-parenthesis EndSet which satisfy the properties Equation3.1 and Equation3.2 for some, depending on the sequence, constants f Subscript normal infinity and gamma . Introduce the map

script upper L normal i normal m colon script upper F right-arrow double-struck upper C comma StartSet f Subscript n Baseline left-parenthesis z right-parenthesis EndSet right-arrow from bar f Subscript normal infinity Baseline comma

which is clearly a homomorphism.

Remark 3.3

Note that we do not require f Subscript n Baseline left-parenthesis z right-parenthesis to be entire. Indeed, the kernel sans-serif upper J may have a square root branching; see the formula Equation2.14.

By Theorem Equation2, the correlation functions bold-italic rho Superscript theta belong to the algebra generated by sequences of the form

StartSet f Subscript n Baseline left-parenthesis z right-parenthesis EndSet equals StartSet sans-serif upper J left-parenthesis x Subscript n Baseline comma y Subscript n Baseline semicolon z right-parenthesis EndSet comma

where the sequence upper X equals upper X left-parenthesis n right-parenthesis equals StartSet x Subscript n Baseline comma y Subscript n Baseline EndSet subset-of double-struck upper Z is regular which, we recall, means that the limits

a equals limit Underscript n right-arrow normal infinity Endscripts StartFraction x Subscript n Baseline Over StartRoot n EndRoot EndFraction comma d equals limit Underscript n right-arrow normal infinity Endscripts left-parenthesis x Subscript n Baseline minus y Subscript n Baseline right-parenthesis

exist, finite or infinite. Therefore, we first consider such sequences.

Proposition 3.4

If upper X equals StartSet x Subscript n Baseline comma y Subscript n Baseline EndSet subset-of double-struck upper Z is regular, then

StartSet sans-serif upper J left-parenthesis x Subscript n Baseline comma y Subscript n Baseline semicolon z right-parenthesis EndSet element-of script upper F comma script upper L times normal i times normal m left-parenthesis StartSet sans-serif upper J left-parenthesis x Subscript n Baseline comma y Subscript n Baseline semicolon z right-parenthesis EndSet right-parenthesis equals sans-serif upper S left-parenthesis d comma a right-parenthesis period

In the proof of this proposition it will be convenient to allow upper X subset-of double-struck upper C . For complex sequences upper X we shall require a element-of double-struck upper R ; the number d element-of double-struck upper C may be arbitrary.

Lemma 3.5

Suppose that a sequence upper X subset-of double-struck upper C is as above and, additionally, suppose that normal black letter upper I x Subscript n , normal black letter upper I y Subscript n are bounded and d not-equals 0 . Then the sequence StartSet sans-serif upper J left-parenthesis x Subscript n Baseline comma y Subscript n Baseline semicolon z right-parenthesis EndSet satisfies Equation3.2 with f Subscript normal infinity Baseline equals sans-serif upper S left-parenthesis d comma a right-parenthesis and certain gamma .

Proof of Lemma 3.5.

We shall use Debye’s asymptotic formulas for Bessel functions of complex order and large complex argument; see, for example, Section 8.6 in Reference43. Introduce the function

upper F left-parenthesis x comma z right-parenthesis equals z Superscript 1 slash 4 Baseline upper J Subscript x Baseline left-parenthesis 2 StartRoot z EndRoot right-parenthesis period

The formula Equation1.9 can be rewritten as

StartLayout 1st Row with Label left-parenthesis 3.3 right-parenthesis EndLabel sans-serif upper J left-parenthesis x comma y semicolon z right-parenthesis equals StartFraction upper F left-parenthesis x comma z right-parenthesis upper F left-parenthesis y plus 1 comma z right-parenthesis minus upper F left-parenthesis x plus 1 comma z right-parenthesis upper F left-parenthesis y comma z right-parenthesis Over x minus y EndFraction period EndLayout

The asymptotic formulas for Bessel functions imply that

StartLayout 1st Row with Label left-parenthesis 3.4 right-parenthesis EndLabel upper F left-parenthesis x comma z right-parenthesis equals StartStartFraction cosine left-parenthesis StartRoot z EndRoot upper G left-parenthesis u right-parenthesis plus StartFraction pi Over 4 EndFraction right-parenthesis OverOver upper H left-parenthesis u right-parenthesis Superscript 1 slash 2 Baseline EndEndFraction left-parenthesis 1 plus upper O left-parenthesis z Superscript negative 1 slash 2 Baseline right-parenthesis right-parenthesis comma u equals StartFraction x Over StartRoot z EndRoot EndFraction comma EndLayout

where

upper G left-parenthesis u right-parenthesis equals StartFraction pi Over 2 EndFraction left-parenthesis u minus normal upper Omega left-parenthesis u right-parenthesis right-parenthesis comma upper H left-parenthesis u right-parenthesis equals StartFraction pi Over 2 EndFraction StartRoot 4 minus u squared EndRoot comma

provided that z right-arrow normal infinity in such a way that u stays in some neighborhood of left-parenthesis negative 2 comma 2 right-parenthesis ; the precise form of this neighborhood can be seen in Figure 22 in Section 8.61 of Reference43. Because we assume that

limit Underscript n right-arrow normal infinity Endscripts StartFraction x Subscript n Baseline Over StartRoot n EndRoot EndFraction comma limit Underscript n right-arrow normal infinity Endscripts StartFraction y Subscript n Baseline Over StartRoot n EndRoot EndFraction element-of left-parenthesis negative 2 comma 2 right-parenthesis comma

and because StartAbsoluteValue z slash n minus 1 EndAbsoluteValue less-than n Superscript negative alpha , the ratios x Subscript n Baseline slash StartRoot z EndRoot , y Subscript n Baseline slash StartRoot z EndRoot stay close to left-parenthesis negative 2 comma 2 right-parenthesis . For future reference, we also point out that the constant in upper O left-parenthesis z Superscript negative 1 slash 2 Baseline right-parenthesis in Equation3.4 is uniform in u provided u is bounded away from the endpoints plus-or-minus 2 .

First we estimate normal black letter upper I left-parenthesis StartRoot z EndRoot upper G left-parenthesis u right-parenthesis right-parenthesis . The function upper G clearly takes real values on the real line. From the obvious estimate

StartAbsoluteValue normal black letter upper I left-parenthesis StartRoot z EndRoot upper G left-parenthesis u right-parenthesis right-parenthesis EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue normal black letter upper I left-parenthesis StartRoot n EndRoot upper G left-parenthesis x slash StartRoot n EndRoot right-parenthesis right-parenthesis EndAbsoluteValue plus StartAbsoluteValue StartRoot z EndRoot upper G left-parenthesis x slash StartRoot z EndRoot right-parenthesis minus StartRoot n EndRoot upper G left-parenthesis x slash StartRoot n EndRoot right-parenthesis EndAbsoluteValue

and the boundedness of upper G , upper G prime , and StartAbsoluteValue normal black letter upper I x EndAbsoluteValue we obtain an estimate of the form

StartLayout 1st Row with Label left-parenthesis 3.5 right-parenthesis EndLabel max Underscript StartAbsoluteValue z slash n minus 1 EndAbsoluteValue less-than-or-equal-to n Superscript negative alpha Baseline Endscripts StartAbsoluteValue upper F left-parenthesis x semicolon z right-parenthesis EndAbsoluteValue e Superscript minus c o n s t StartAbsoluteValue z minus n EndAbsoluteValue slash StartRoot n EndRoot Baseline equals upper O left-parenthesis 1 right-parenthesis period EndLayout

If d equals normal infinity , then because of the denominator in Equation3.3 the estimate Equation3.5 implies that

sans-serif upper J left-parenthesis x Subscript n Baseline comma y Subscript n Baseline semicolon z right-parenthesis equals o left-parenthesis e Superscript c o n s t StartAbsoluteValue z minus n EndAbsoluteValue slash StartRoot n EndRoot Baseline right-parenthesis period

Since sans-serif upper S left-parenthesis normal infinity comma a right-parenthesis equals 0 , it follows that in this case the lemma is established.

Assume, therefore, that d is finite. Observe that for any bounded increment normal upper Delta x we have

StartLayout 1st Row with Label left-parenthesis 3.6 right-parenthesis EndLabel upper F left-parenthesis x plus normal upper Delta x comma z right-parenthesis equals StartStartFraction cosine left-parenthesis StartRoot z EndRoot upper G left-parenthesis u right-parenthesis plus upper G prime left-parenthesis u right-parenthesis normal upper Delta x plus StartFraction pi Over 4 EndFraction right-parenthesis OverOver upper H left-parenthesis u right-parenthesis Superscript 1 slash 2 Baseline EndEndFraction plus upper O left-parenthesis StartFraction left-parenthesis normal upper Delta x right-parenthesis squared Over StartRoot z EndRoot EndFraction e Superscript c o n s t StartAbsoluteValue z minus n EndAbsoluteValue slash StartRoot n EndRoot Baseline right-parenthesis comma EndLayout

and, in particular, the last term is o left-parenthesis e Superscript c o n s t StartAbsoluteValue z minus n EndAbsoluteValue slash StartRoot n EndRoot Baseline right-parenthesis . Using the trigonometric identity

cosine left-parenthesis upper A right-parenthesis cosine left-parenthesis upper B plus upper C right-parenthesis minus cosine left-parenthesis upper A plus upper C right-parenthesis cosine left-parenthesis upper B right-parenthesis equals sine left-parenthesis upper C right-parenthesis sine left-parenthesis upper A minus upper B right-parenthesis comma

and observing that

upper G prime left-parenthesis u right-parenthesis equals arc cosine left-parenthesis u slash 2 right-parenthesis comma sine left-parenthesis upper G prime left-parenthesis u right-parenthesis right-parenthesis equals StartFraction StartRoot 4 minus u squared EndRoot Over 2 EndFraction equals StartFraction upper H left-parenthesis u right-parenthesis Over pi EndFraction comma

we compute

StartLayout 1st Row upper F left-parenthesis x Subscript n Baseline semicolon z right-parenthesis upper F left-parenthesis y Subscript n Baseline plus 1 semicolon z right-parenthesis minus upper F left-parenthesis x Subscript n Baseline plus 1 semicolon z right-parenthesis upper F left-parenthesis y Subscript n Baseline semicolon z right-parenthesis 2nd Row equals StartFraction 1 Over pi EndFraction sine left-parenthesis arc cosine left-parenthesis StartFraction x Subscript n Baseline Over 2 StartRoot z EndRoot EndFraction right-parenthesis left-parenthesis x Subscript n Baseline minus y Subscript n Baseline right-parenthesis right-parenthesis plus o left-parenthesis e Superscript c o n s t StartAbsoluteValue z minus n EndAbsoluteValue slash StartRoot n EndRoot Baseline right-parenthesis period EndLayout

Since, by hypothesis,

StartFraction x Subscript n Baseline Over StartRoot z EndRoot EndFraction right-arrow a comma left-parenthesis x Subscript n Baseline minus y Subscript n Baseline right-parenthesis right-arrow d comma

and d not-equals 0 , the lemma follows.

Remark 3.6

Below we shall need this lemma for a variable sequence upper X equals StartSet x Subscript n Baseline comma y Subscript n Baseline EndSet . Therefore, let us spell out explicitly under what conditions on upper X the estimates in Lemma 3.5 remain uniform. We need the sequences StartFraction x Subscript n Baseline Over StartRoot n EndRoot EndFraction and StartFraction y Subscript n Baseline Over StartRoot n EndRoot EndFraction to converge uniformly; then, in particular, the ratios StartFraction x Subscript n Baseline Over StartRoot n EndRoot EndFraction and StartFraction y Subscript n Baseline Over StartRoot n EndRoot EndFraction are uniformly bounded away from plus-or-minus 2 . Also, we need normal black letter upper I x Subscript n and normal black letter upper I y Subscript n to be uniformly bounded. Finally, we need StartAbsoluteValue d EndAbsoluteValue to be uniformly bounded from below.

Proof of Proposition 3.4.

First, we check the condition Equation3.2. In the case d not-equals 0 this was done in the previous lemma. Suppose, therefore, that StartSet x Subscript n Baseline EndSet is a regular sequence in double-struck upper Z Subscript 0 and consider the asymptotics of sans-serif upper J left-parenthesis x Subscript n Baseline comma x Subscript n Baseline semicolon z right-parenthesis .

Because the function sans-serif upper J left-parenthesis x comma y semicolon z right-parenthesis is an entire function of x and y we have

StartLayout 1st Row with Label left-parenthesis 3.7 right-parenthesis EndLabel sans-serif upper J left-parenthesis x comma x semicolon z right-parenthesis equals StartFraction 1 Over 2 pi EndFraction integral Subscript 0 Superscript 2 pi Baseline sans-serif upper J left-parenthesis x comma x plus r e Superscript i t Baseline semicolon z right-parenthesis d t comma EndLayout

where r is arbitrary; we shall take r to be some small but fixed number. From the previous lemma we know that

sans-serif upper J left-parenthesis x comma x plus r e Superscript i t Baseline semicolon z right-parenthesis equals StartFraction 1 Over pi r e Superscript i t Baseline EndFraction sine left-parenthesis omega left-parenthesis StartFraction x Over StartRoot z EndRoot EndFraction right-parenthesis r e Superscript i t Baseline right-parenthesis plus o left-parenthesis e Superscript c o n s t StartAbsoluteValue z minus n EndAbsoluteValue slash StartRoot n EndRoot Baseline right-parenthesis period

From the above remark it follows that this estimate is uniform in t . This implies the property Equation3.2 for sans-serif upper J left-parenthesis x Subscript n Baseline comma x Subscript n Baseline semicolon z right-parenthesis .

To prove the estimate Equation3.1 we use Schläfli’s integral representation (see Section 6.21 in Reference43)

StartLayout 1st Row with Label left-parenthesis 3.8 right-parenthesis EndLabel upper J Subscript x Baseline left-parenthesis 2 StartRoot z EndRoot right-parenthesis equals StartFraction 1 Over pi EndFraction integral Subscript 0 Superscript pi Baseline cosine left-parenthesis x t minus 2 StartRoot z EndRoot sine t right-parenthesis d t minus StartFraction sine pi x Over pi EndFraction integral Subscript 0 Superscript normal infinity Baseline e Superscript minus x t minus 2 StartRoot z EndRoot hyperbolic sine t Baseline d t comma EndLayout

which is valid for StartAbsoluteValue arg z EndAbsoluteValue less-than pi and even for arg z equals plus-or-minus pi provided normal black letter upper R x greater-than 0 or x element-of double-struck upper Z .

If x element-of double-struck upper Z , then the second summand in Equation3.8 vanishes and the first summand is upper O left-parenthesis e Superscript c o n s t StartAbsoluteValue z EndAbsoluteValue Super Superscript 1 slash 2 Superscript Baseline right-parenthesis uniformly in x element-of double-struck upper Z . This implies the estimate Equation3.1 provided d not-equals 0 .

It remains, therefore, to check Equation3.1 for sans-serif upper J left-parenthesis x Subscript n Baseline comma x Subscript n Baseline semicolon z right-parenthesis where StartSet x Subscript n Baseline EndSet element-of double-struck upper Z is a regular sequence. Again, we use Equation3.7. Observe that since normal black letter upper R StartRoot z EndRoot greater-than-or-equal-to 0 , the second summand in Equation3.8 is uniformly small provided normal black letter upper I x is bounded from above and normal black letter upper R x is bounded from below. Therefore, Equation3.7 produces the Equation3.1 estimate for x Subscript n Baseline greater-than-or-equal-to 1 . For x Subscript n Baseline less-than-or-equal-to 0 we use the relation Equation2.13 and the reccurence Equation2.16 to obtain the estimate.

Proof of Theorem 3.

Let upper X left-parenthesis n right-parenthesis be a regular sequence and let the numbers a Subscript i and d Subscript i j be defined by Equation1.10, Equation1.11. We shall assume that StartAbsoluteValue a Subscript i Baseline EndAbsoluteValue less-than 2 for all i . The validity of the theorem in the case when StartAbsoluteValue a Subscript i Baseline EndAbsoluteValue greater-than-or-equal-to 2 for some i will be obvious from the results of the next section.

We have

StartLayout 1st Row with Label left-parenthesis 3.9 right-parenthesis EndLabel 1st Column bold-italic rho Superscript theta Baseline left-parenthesis upper X left-parenthesis n right-parenthesis right-parenthesis 2nd Column equals e Superscript negative theta Baseline sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts bold-italic rho left-parenthesis k comma upper X left-parenthesis n right-parenthesis right-parenthesis StartFraction theta Superscript k Baseline Over k factorial EndFraction 2nd Row with Label left-parenthesis 3.10 right-parenthesis EndLabel 1st Column Blank 2nd Column equals det left-bracket sans-serif upper J left-parenthesis x Subscript i Baseline left-parenthesis n right-parenthesis comma x Subscript j Baseline left-parenthesis n right-parenthesis right-parenthesis right-bracket Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to s Baseline comma EndLayout

where the first line is the definition of bold-italic rho Superscript theta and the second is Theorem Equation2. From Equation3.9 it is obvious that bold-italic rho Superscript theta is entire. Therefore, we can apply Lemma Equation3.1 to it. It is clear that Lemma Equation3.1, together with Proposition 3.4, implies Theorem 3. The factorization Equation1.12 follows from the vanishing sans-serif upper S left-parenthesis normal infinity comma a right-parenthesis equals 0 .

3.2. Asymptotics of rho left-parenthesis n comma upper X right-parenthesis

Recall that the correlation functions rho left-parenthesis n comma upper X right-parenthesis were defined by

rho left-parenthesis n comma upper X right-parenthesis equals upper M Subscript n Baseline left-parenthesis StartSet lamda vertical-bar upper X subset-of upper F r left-parenthesis lamda right-parenthesis EndSet right-parenthesis comma upper X subset-of double-struck upper Z plus one-half period

The asymptotics of these correlation functions can be easily obtained from Theorem Equation3 by complementation (see Sections A.3 and 2.2), and the result is the following.

Let upper X left-parenthesis n right-parenthesis subset-of double-struck upper Z plus one-half be a regular sequence. If it splits, then limit Underscript n right-arrow normal infinity Endscripts rho left-parenthesis n comma upper X left-parenthesis n right-parenthesis right-parenthesis factors as in Equation1.12. Suppose therefore, that upper X left-parenthesis n right-parenthesis is nonsplit. Here one has to distinguish two cases. If upper X left-parenthesis n right-parenthesis subset-of double-struck upper Z Subscript 0 plus one-half or upper X left-parenthesis n right-parenthesis subset-of double-struck upper Z Subscript 0 minus one-half , then we shall say that this sequence is off-diagonal. Geometrically, it means that upper X left-parenthesis n right-parenthesis corresponds to modified Frobenius coordinates of only one kind: either the row ones or the column ones. For off-diagonal sequences we obtain from Theorem Equation3 by complementation that

limit Underscript n right-arrow normal infinity Endscripts rho left-parenthesis n comma upper X left-parenthesis n right-parenthesis right-parenthesis equals det left-bracket sans-serif upper S left-parenthesis d Subscript i j Baseline comma StartAbsoluteValue a EndAbsoluteValue right-parenthesis right-bracket Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to s Baseline comma

where sans-serif upper S is the discrete sine kernel and a equals a 1 equals a 2 equals period period period .

If upper X left-parenthesis n right-parenthesis