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Random Matrices and Their Applications
Edited by: J. E. Cohen, H. Kesten, and C. M. Newman
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Contemporary Mathematics
1986; 358 pp; softcover
Volume: 50
Reprint/Revision History:
reprinted 1988
ISBN-10: 0-8218-5044-X
ISBN-13: 978-0-8218-5044-2
List Price: US$51
Member Price: US$40.80
Order Code: CONM/50
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These twenty-six expository papers on random matrices and products of random matrices survey the major results of the last thirty years. They reflect both theoretical and applied concerns in fields as diverse as computer science, probability theory, mathematical physics, and population biology. Many of the articles are tutorial, consisting of examples, sketches of proofs, and interpretations of results. They address a wide audience of mathematicians and scientists who have an elementary knowledge of probability theory and linear algebra, but not necessarily any prior exposure to this specialized area. More advanced articles, aimed at specialists in allied areas, survey current research with references to the original literature.

The book's major topics include the computation and behavior under perturbation of Lyapunov exponents and the spectral theory of large random matrices. The applications to mathematical and physical sciences under consideration include computer image generation, card shuffling, and other random walks on groups, Markov chains in random environments, the random Schroedinger equations and random waves in random media.

Most of the papers were originally presented at an AMS-IMS-SIAM Joint Summer Research Conference held at Bowdoin College in June, 1984. Of special note are the papers by Kotani on random Schroedinger equations, Yin and Bai on spectra for large random matrices, and Newman on the relations between the Lyapunov and eigenvalue spectra.

Table of Contents

  • J. C. Watkins -- Limit theorems for products of random matrices
  • J. Cohen, H. Kesten, and C. M. Newman -- Oseledec's multiplicative ergodic theorem
  • Y. Guivarc'h and A. Raugi -- Products of random matrices: convergence theorems
  • F. Ledrappier -- Examples of application of Oseledec's theorem
  • Y. Kifer -- Multiplicative ergodic theorems for random diffeomorphisms
  • S. Pincus -- Furstenberg-Kesten results: asymptotic analysis
  • E. V. Slud -- Stability of exponential growth rate for randomly perturbed random matrix products via Markov-chain arguments
  • V. Wihstutz -- Representation, positivity, and expansion of Lyapunov exponents for linear stochastic systems
  • M. Wojtkowski -- On uniform contraction generated by positive matrices
  • C. M. Newman -- Lyapunov exponents for some products of random matrices
  • V.-R. Hwang -- A brief survey on the spectral radius and the spectral distribution of large random matrices with i.i.d. entries
  • J. W. Silverstein -- Eigenvalues and eigenvectors of large dimensional sample covariance matrices
  • Y. Q. Yin and Z. D. Bai -- Spectra for large dimensional random matrices
  • P. Diaconis and M. Shahshahani -- Products of random matrices and computer image generation
  • P. Diaconis and M. Shahshahani -- Products of random matrices as they arise in the study of random walks on groups
  • R. Cogburn -- On products of random stochastic matrices
  • M. Rosenblatt -- Convolution sequences of measures on the semigroup of stochastic matrices
  • T.-C. Sun -- Random walks on semigroups
  • T.Kaijser -- A note on random systems with complete connections and their applications to products of random matrices
  • E. Key -- Using random matrices to give recurrence and transience criteria for random walk in a random environment
  • G. Letac -- A contraction principle for certain Markov chains and its application
  • S. Kotani -- Lyapunov exponents and spectra for one-dimensional random Schroedinger operators
  • R. S. Maier -- The density of states of random Schroedinger operators
  • M. L. Mehta -- Random matrices in nuclear physics and number theory
  • G. Papanicolaou -- Random matrices and waves in random media
  • S. Tuljapurkar -- Demographic applications of random matrix products
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