The stability of large random matrices and their products.

*(English)*Zbl 0543.60098For fixed n, let \(\{\) A(1),A(2),...\(\}\) be an infinite sequence of random independent identically distributed (i.i.d.) real \(n\times n\) matrices. Let x(0) be a non-zero vector in \({\mathbb{R}}^ n\) and define \(x(t)=A(t)A(t- 1)...A(1)x(0)\) for \(t=1,2,..\).. Define \(\lambda\) (x(0)) by \(\lambda(x(0))=\lim_{t\to \infty}\| x(t)\|^{1/t}\) whenever the limit exists almost surely and is non-random. (Due to work by H. Furstenberg, Trans. Am. Math. Soc. 108, 377-428 (1963; Zbl 0203.191) and others it can be shown that in many situations this is the case.) Whenever \(\lambda\) (x(0)) exists for all x(0))\(\neq 0\) one can define \({\bar \lambda}\) and \({\underline \lambda}\) by \({\bar \lambda}=\sup \{\lambda(x(0)):x(0)\neq 0\}\) and \({\underline \lambda}=\inf \{\lambda(x(0)):x(0)\neq 0\}\). (Very often the situation is in fact such that \({\bar \lambda}={\underline \lambda}.)\)

For a fixed n, the system \(\{A(t),t=1,2,...\}\) is called strongly stable if \({\bar \lambda}<1\) and strongly unstable if \({\underline \lambda}>1\). As n tends to infinity the sequence of systems is called asymptotically strongly stable (resp. unstable) if there exists an integer N such that strong stability (resp. instability) is valid for each \(n>N.\)

The purpose of this paper is threefold. In the first part the authors present a case when \({\bar \lambda}={\underline \lambda}\) and when this quantity can be determined explicitly. The case considered is the case when the elements of A(1) are i.i.d. standard symmetric random variables with exponent \(\alpha (0<\alpha \leq 2)\). The authors also give precise conditions when a sequence of systems of random matrices of this type is asymptotically strongly stable (resp. unstable).

In the second part the authors first give a general upper bound for \({\bar \lambda}\) and then also a lower bound for \({\underline \lambda}\) in a fairly general situation. To obtain this latter bound quite a few sharp estimates are needed. The authors then use these bounds when determining criteria for asymptotic strong stability (resp. unstability). In the third part of the paper the authors construct counter-examples to claims by R. M. May [Will a large complex system be stable? Nature 238, 413-414 (1972)] and H. M. Hastings [The May-Wigner stability theorem. J. Theor. Biol. 97, 155-166 (1982)] concerning asymptotic strong stability resp. unstability.

For a fixed n, the system \(\{A(t),t=1,2,...\}\) is called strongly stable if \({\bar \lambda}<1\) and strongly unstable if \({\underline \lambda}>1\). As n tends to infinity the sequence of systems is called asymptotically strongly stable (resp. unstable) if there exists an integer N such that strong stability (resp. instability) is valid for each \(n>N.\)

The purpose of this paper is threefold. In the first part the authors present a case when \({\bar \lambda}={\underline \lambda}\) and when this quantity can be determined explicitly. The case considered is the case when the elements of A(1) are i.i.d. standard symmetric random variables with exponent \(\alpha (0<\alpha \leq 2)\). The authors also give precise conditions when a sequence of systems of random matrices of this type is asymptotically strongly stable (resp. unstable).

In the second part the authors first give a general upper bound for \({\bar \lambda}\) and then also a lower bound for \({\underline \lambda}\) in a fairly general situation. To obtain this latter bound quite a few sharp estimates are needed. The authors then use these bounds when determining criteria for asymptotic strong stability (resp. unstability). In the third part of the paper the authors construct counter-examples to claims by R. M. May [Will a large complex system be stable? Nature 238, 413-414 (1972)] and H. M. Hastings [The May-Wigner stability theorem. J. Theor. Biol. 97, 155-166 (1982)] concerning asymptotic strong stability resp. unstability.

Reviewer: T.Kaijser

##### MSC:

60K99 | Special processes |

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

15B52 | Random matrices (algebraic aspects) |

60F99 | Limit theorems in probability theory |

92D40 | Ecology |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |