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

2025/01/15 by Collin, Orphée, Giacomin, Giambattista, Greenblatt, Rafael L. +1 · 2 citations
#37H15 #60K35 #60K37 #82B44 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2501.08797

Abstract

We study the top Lyapunov exponent of a product of random 2 × 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 R² 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.

Cited by

• Further results on the free energy of the random field Ising chain in the case of centered disorder
• Dimers with layered disorder

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