The Lyapunov exponent for products of infinite-dimensional random matrices

Abstract

Consider a non-negative random matrix indexed by ℤ^d×ℤ^d, with independent rows, and such that the distribution is invariant under translation down the diagonal. Multiply together independent random matrices with this same law, and define the Lyapunov exponent λ as the exponential growth rate of the sum of the entries in the zero row. For some examples derived from Oriented Percolation, there is positive probability that λ equals the log of the expected value of the sum of entries in the zero row of the original random matrix. The proofs, which are not new, use random walk arguments. Some unsolved problems are described.