Products of random matrices in statistical physics.(English)Zbl 0784.58003
Springer Series in Solid-State Sciences. 104. Berlin: Springer-Verlag. xiii, 166 p. (1993).
This monograph presents a descriptive rather than deductive account of products of random matrices and their applications particularly in statistical physics. The book is divided into three parts. The first part contains a collection of useful formulae and results concerning random matrices and their products with references to works where further details can be found. More than half of the space in this part is devoted to a separate chapter on Lyapunov exponents for the products of random matrices.
The second part is concerned with major applications of random matrices and their products and is divided into three chapters. The first chapter describes the application of random matrices to deterministic chaos and a separate section is devoted to the description of the role of products of independent random matrices in finding approximations to Lyapunov exponents. The second chapter describes how products of random matrices appear as a natural tool for the study of one-dimensional disordered systems with particular attention paid to the following topics: Ising model and transfer matrices, Ising chains with random fields and random coupling, generalized Lyapunov exponents and free energy fluctuations and correlation functions. A brief description of applications to two- and three-dimensional systems is also given. The third chapter describes the role of the product of random matrices to the study of localization.
The third part entitled “miscellany” contains a chapter on certain other applications. These being application to the propagation of light in random media, random magnetic dynamos and image compression. This part also contains seven appendices with more mathematical formulae and some source codes for a computer programme for the computation of the Lyapunov spectrum.
There are just under 200 well selected references in the bibliography taking the reader through important works published in the area until 1990. The book is nicely laid out and printed in the high typographic standards one has come to expect from Springer. All in all this is a nicely presented summary of the state of the art in the subject.
The second part is concerned with major applications of random matrices and their products and is divided into three chapters. The first chapter describes the application of random matrices to deterministic chaos and a separate section is devoted to the description of the role of products of independent random matrices in finding approximations to Lyapunov exponents. The second chapter describes how products of random matrices appear as a natural tool for the study of one-dimensional disordered systems with particular attention paid to the following topics: Ising model and transfer matrices, Ising chains with random fields and random coupling, generalized Lyapunov exponents and free energy fluctuations and correlation functions. A brief description of applications to two- and three-dimensional systems is also given. The third chapter describes the role of the product of random matrices to the study of localization.
The third part entitled “miscellany” contains a chapter on certain other applications. These being application to the propagation of light in random media, random magnetic dynamos and image compression. This part also contains seven appendices with more mathematical formulae and some source codes for a computer programme for the computation of the Lyapunov spectrum.
There are just under 200 well selected references in the bibliography taking the reader through important works published in the area until 1990. The book is nicely laid out and printed in the high typographic standards one has come to expect from Springer. All in all this is a nicely presented summary of the state of the art in the subject.
Reviewer: C.S.Sharma (London)
MSC:
| 58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |
| 82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |
| 15B52 | Random matrices (algebraic aspects) |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 60K40 | Other physical applications of random processes |
| 82B31 | Stochastic methods applied to problems in equilibrium statistical mechanics |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
