Correlations in the multispecies TASEP and a conjecture by Lam

Ayyer, Arvind; Linusson, Svante

doi:10.1090/tran/6806

Correlations in the multispecies TASEP and a conjecture by Lam
HTML articles powered by AMS MathViewer

by Arvind Ayyer and Svante Linusson PDF

Trans. Amer. Math. Soc. 369 (2017), 1097-1125 Request permission

Abstract:

We study correlations in the multispecies TASEP on a ring. Results on the correlation of two adjacent points prove two conjectures by Thomas Lam on

(a) the limiting direction of a reduced random walk in $\tilde A_{n-1}$ and

(b) the asymptotic shape of a random integer partition with no hooks of length $n$, a so called $n$-core.

We further investigate two-point correlations far apart and three-point nearest neighbour correlations and prove explicit formulas in almost all cases. These results can be seen as a finite strengthening of correlations in the TASEP speed process by Amir, Angel and Valkó. We also give conjectures for certain higher order nearest neighbour correlations. We find an unexplained independence property (provably for two points, conjecturally for more points) between points that are closer in position than in value that deserves more study.

References

Erik Aas, A Markov Process on Cyclic Words, PhD-thesis at KTH, Sweden (Dec. 2014), urn:nbn:se:kth:diva-156862
Erik Aas and Svante Linusson, Continuous Multi-line Queues and the TASEP, arXiv:1501.04417.
Erik Aas and Jonas Sjöstrand, A product formula for the TASEP on a ring, Random Struct. Alg. 48 (2015), no. 2, 247–259, arXiv.org/abs/1312.2493.
Gideon Amir, Omer Angel, and Benedek Valkó, The TASEP speed process, Ann. Probab. 39 (2011), no. 4, 1205–1242. MR 2857238, DOI 10.1214/10-AOP561
Arvind Ayyer and Svante Linusson, An inhomogeneous multispecies TASEP on a ring, Adv. in Appl. Math. 57 (2014), 21–43. MR 3206520, DOI 10.1016/j.aam.2014.02.001
Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
Pablo A. Ferrari, Patricia Gonçalves, and James B. Martin, Collision probabilities in the rarefaction fan of asymmetric exclusion processes, Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 4, 1048–1064 (English, with English and French summaries). MR 2572163, DOI 10.1214/08-AIHP303
Pablo A. Ferrari and J. B. Martin, Multi-class processes, dual points and $M/M/1$ queues, Markov Process. Related Fields 12 (2006), no. 2, 175–201. MR 2249628
Pablo A. Ferrari and James B. Martin, Stationary distributions of multi-type totally asymmetric exclusion processes, Ann. Probab. 35 (2007), no. 3, 807–832. MR 2319708, DOI 10.1214/009117906000000944
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Kurt Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2, 437–476. MR 1737991, DOI 10.1007/s002200050027
Thomas Lam, The shape of a random affine Weyl group element and random core partitions, Ann. Probab. 43 (2015), no. 4, 1643–1662. MR 3353811, DOI 10.1214/14-AOP915
Thomas Lam, Luc Lapointe, Jennifer Morse, and Mark Shimozono, Affine insertion and Pieri rules for the affine Grassmannian, Mem. Amer. Math. Soc. 208 (2010), no. 977, xii+82. MR 2741963, DOI 10.1090/S0065-9266-10-00576-4
David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson. MR 2466937, DOI 10.1090/mbk/058
Svante Linusson and James Martin, Stationary probabilities for an inhomogeneous multi-type TASEP, in preparation.
Thomas Mountford and Hervé Guiol, The motion of a second class particle for the TASEP starting from a decreasing shock profile, Ann. Appl. Probab. 15 (2005), no. 2, 1227–1259. MR 2134103, DOI 10.1214/105051605000000151
H. Rost, Nonequilibrium behaviour of a many particle process: density profile and local equilibria, Z. Wahrsch. Verw. Gebiete 58 (1981), no. 1, 41–53. MR 635270, DOI 10.1007/BF00536194
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589