**Contemporary Mathematics**

Volume: 50;
1986;
358 pp;
Softcover

MSC: Primary 60;

Print ISBN: 978-0-8218-5044-2

Product Code: CONM/50

List Price: $54.00

AMS Member Price: $43.20

MAA Member Price: $48.60

**Electronic ISBN: 978-0-8218-7635-0
Product Code: CONM/50.E**

List Price: $54.00

AMS Member Price: $43.20

MAA Member Price: $48.60

# Random Matrices and Their Applications

Share this page *Edited by *
*J. E. Cohen; H. Kesten; C. M. Newman*

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

## Random Matrices and Their Applications

- Contents ix10 free
- Preface xiii14 free
- I. Basic theory of products of random matrices 116 free
- A. Overviews: 318
- B. Perturbation theory: 6580
- Multiplicative ergodic theorems for random diffeomorphisms 6782
- Furstenberg-Kesten results: asymptotic analysis 7994
- Stability of exponential growth rate for randomly perturbed random matrix products via Markov-chain arguments 87102
- Representation, positivity and expansion of Lyapunov exponents for linear stochastic systems 93108

- C. Theory of matrix products: 107122
- D. Connections with spectral theory: 119134

- II. Spectral theory of random matrices 143158
- III. Applications to computer science, probability and statistics of products of random matrices 169184
- A. Applications to computer science and statistics: 171186
- B. Applications to Markov chains in random environments: 197212
- C. Other applications to probability theory: 241256
- A note on random systems with complete connections and their applications to products of random matrices 243258
- Using random matrices to give recurrence and transience criteria for random walk in a random environment 255270
- A contraction principle for certain Markov chains and its applications 263278

- IV. Scientific applications of random matrices and their products 275290
- Lyapounov exponents and spectra for one-dimensional random Schroedinger operators 277292
- The density of states of random Schroedinger operators 287302
- Random matrices in nuclear physics and number theory 295310
- Random matrices and waves in random media 311326
- Demographic applications of random matrix products 319334

- V. Supplements 327342