On the Lyapunov exponent for the random field Ising transfer matrix, in the critical case

TL;DR

This paper analyzes the top Lyapunov exponent for a product of random 2x2 matrices in the critical case, improving previous methods by weakening assumptions and connecting the invariant measure to ladder times of a related random walk.

Contribution

It introduces a new approach to approximate the Furstenberg measure using ladder times, enhancing the analysis of the Lyapunov exponent under weaker disorder assumptions.

Findings

Improved bounds on the Lyapunov exponent in the critical case.

Weakened assumptions on disorder distribution compared to previous work.

Established a link between invariant measure approximation and ladder times of a random walk.

Abstract

We study the top Lyapunov exponent of a product of random $2 \times 2$ matrices appearing in the analysis of several statistical mechanical models with disorder, extending a previous treatment of the critical case (Giacomin and Greenblatt, ALEA 19 (2022), 701-728) by significantly weakening the assumptions on the disorder distribution. The argument we give completely revisits and improves the previous proof. As a key novelty we build a probability that is close to the Furstenberg probability, i.e. the invariant probability of the Markov chain corresponding to the evolution of the direction of a vector in ${\mathbb R}^2$ under the action of the random matrices, in terms of the ladder times of a centered random walk which is directly related to the random matrix sequence. We then show that sharp estimates on the ladder times (renewal) process lead to a sharp control on the probability measure we build and, in turn, to the control of its distance from the Furstenberg probability.