Products of random matrices with applications to Schrödinger operators.

*(English)*Zbl 0572.60001
Progress in Probability and Statistics, Vol. 8. Boston - Basel - Stuttgart: Birkhäuser. X, 283 p. DM 88.00 (1985).

This is a very readable book on two highly interesting subjects. The first part of the book, written by P. Bougerol, deals with the analysis of the limit behaviour of products of i.i.d. non-singular random matrices. This subject, initiated by R. Bellman [Duke Math. J. 21, 491-500 (1954; Zbl 0057.112)], was fully developed by the work of Furstenberg, Kesten, Oseledec, Tutubalin, Guivarc’h, LePage, Cohen, Ledrappier, Royer, Raugi, Virtser and others.

The purpose of the first part of the book is to prove in detail the analogues of the classical limit theorems (e.g. the law of large numbers, the central limit theorem). The text is primarily based on the works by H. Furstenberg, Trans. Am. Math. Soc. 108, 377-428 (1963; Zbl 0203.191), Y. Guivarc’h and H. Raugi, Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 187-242 (1985; Zbl 0558.60009) and E. LePage, Probability measures on groups. Proc. 6th Conf., Oberwolfach 1981, Lect. Notes Math. 928, 258-303 (1982; Zbl 0506.60019). The level is kept as elementary as possible which makes the text suitable for graduate courses.

The second part of the book, written by J. Lacroix, deals with the following type of stochastic operators. Let \(\{\Psi_ n,n\in {\mathbb{Z}}\}\) be a sequence of complex numbers, let \(\{(a_ n,b_ n),n\in {\mathbb{Z}}\}\) be a stationary sequence of real-valued random vectors, such that \(b_ n>0\), and define the operator H acting on complex sequences by \[ (H\Psi)_ n=b_ n^{-1}(-\Psi_{n+1}-\Psi_{n-1}+a_ n\Psi_ n). \] This type of second order stochastic difference operator arises naturally in the study of nonhomogeneous (disordered) one-dimensional discrete physical systems. Equations such as the Schrödinger equation, the wave equation, the heat equation can all be formulated using an operator of this type, and when analysing the limit behaviour of solutions to such equations, as time tends to infinity, it is well known that the limit behaviour is determined by the spectral properties of the operator H viewed as a self-adjoint operator on a certain Hilbert-space.

Two early and famous papers on random Schrödinger operators are by P. Anderson, Phys. Rev. 109, 1492-1505 (1958) and N. Mott and W. Twose, Adv. Phys. 10, 107-155 (1961). The first to use results on products of random matrices when investigating the operator H were H. Matsuda and K. Ishii, Prog. Theor. Phys. Suppl. 45, 56-86 (1970) and from then on there has been an increasing number of papers on the properties of the operator H almost every year.

The purpose of the second part of the book is to give a rigorous and unified presentation of the spectral analysis of the operator H defined above, also in the case when the operator acts on sequences of complex vectors and the a:s and the b:s in the definition of H are random matrices. (This situation corresponds to discrete physical systems in a strip). The main tools used in the analysis are results from the theory on products of random matrices developed in the first part of the book, but the analysis is of course also very much based on general spectral theory of self-adjoint operators.

As far as the reviewer knows this is the first book on any of these subjects and therefore this well written book fills a gap in the literature. However the reviewer looks forward to the next edition when hopefully each of the two subjects gets its own book. They deserve it.

The purpose of the first part of the book is to prove in detail the analogues of the classical limit theorems (e.g. the law of large numbers, the central limit theorem). The text is primarily based on the works by H. Furstenberg, Trans. Am. Math. Soc. 108, 377-428 (1963; Zbl 0203.191), Y. Guivarc’h and H. Raugi, Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 187-242 (1985; Zbl 0558.60009) and E. LePage, Probability measures on groups. Proc. 6th Conf., Oberwolfach 1981, Lect. Notes Math. 928, 258-303 (1982; Zbl 0506.60019). The level is kept as elementary as possible which makes the text suitable for graduate courses.

The second part of the book, written by J. Lacroix, deals with the following type of stochastic operators. Let \(\{\Psi_ n,n\in {\mathbb{Z}}\}\) be a sequence of complex numbers, let \(\{(a_ n,b_ n),n\in {\mathbb{Z}}\}\) be a stationary sequence of real-valued random vectors, such that \(b_ n>0\), and define the operator H acting on complex sequences by \[ (H\Psi)_ n=b_ n^{-1}(-\Psi_{n+1}-\Psi_{n-1}+a_ n\Psi_ n). \] This type of second order stochastic difference operator arises naturally in the study of nonhomogeneous (disordered) one-dimensional discrete physical systems. Equations such as the Schrödinger equation, the wave equation, the heat equation can all be formulated using an operator of this type, and when analysing the limit behaviour of solutions to such equations, as time tends to infinity, it is well known that the limit behaviour is determined by the spectral properties of the operator H viewed as a self-adjoint operator on a certain Hilbert-space.

Two early and famous papers on random Schrödinger operators are by P. Anderson, Phys. Rev. 109, 1492-1505 (1958) and N. Mott and W. Twose, Adv. Phys. 10, 107-155 (1961). The first to use results on products of random matrices when investigating the operator H were H. Matsuda and K. Ishii, Prog. Theor. Phys. Suppl. 45, 56-86 (1970) and from then on there has been an increasing number of papers on the properties of the operator H almost every year.

The purpose of the second part of the book is to give a rigorous and unified presentation of the spectral analysis of the operator H defined above, also in the case when the operator acts on sequences of complex vectors and the a:s and the b:s in the definition of H are random matrices. (This situation corresponds to discrete physical systems in a strip). The main tools used in the analysis are results from the theory on products of random matrices developed in the first part of the book, but the analysis is of course also very much based on general spectral theory of self-adjoint operators.

As far as the reviewer knows this is the first book on any of these subjects and therefore this well written book fills a gap in the literature. However the reviewer looks forward to the next edition when hopefully each of the two subjects gets its own book. They deserve it.

Reviewer: T.Kaijser

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |