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

TL;DR

This work analyzes the top Lyapunov exponent for products of random $2\times 2$ matrices arising from disordered Ising-type transfer matrices in the critical regime where $\mathbb E[\log Z]=0$. By recasting the problem in Furstenberg theory with the parameter $\Gamma=-\log \varepsilon$ and the centered disorder $\mathtt z=\log Z$, the authors construct a Derrida–Hilhorst probability $\gamma_\Gamma$ that closely approximates the Furstenberg invariant measure $\nu_\Gamma$ via renewal (ladder) analysis of a centered random walk. They prove a sharp leading-order expansion for the Lyapunov exponent: ${\mathcal L}(\Gamma)=\kappa_1/(\Gamma+\kappa_2)+R(\Gamma)$ with $\kappa_1=\mathrm{var}(\mathtt z)/4$ and error terms $R(\Gamma)$ that decay either polynomially or exponentially depending on tail assumptions (H-2 vs H-2′). The work combines a refined edge-chain analysis, detailed stationary-measure asymptotics for the $Y$ process, and a contraction framework (via $T_\Gamma$) to rigorously connect the Derrida–Hilhorst construction to the true Furstenberg measure. The results extend previous critical-case analyses by weakening disorder assumptions and sharpening control of the remainder, providing a universal leading term tied to the disorder variance and illuminating subleading corrections through the parameter $\kappa_2$.

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.

Figures (2)

• Figure 1: For $\Gamma =10$: on the left the plot of $h_\Gamma (\cdot)$ (solid line) and of $x\mapsto {\bf h}_\Gamma(x)= h(x+\Gamma )-\Gamma$ (dotted line); on the right the derivative of the same functions.
• Figure 2: For $\Gamma =10$ we have drawn in black (respectively, in grey) the density of the invariant measure of the $Y$ process translated of $-\Gamma$ (respectively, of the $Y$ process driven by $-\mathtt{z}$, reflected with respect to the origin and then translated of $+ \Gamma$): one can readily see that the densities of these two measure exist if $\mathtt{z}$ has a density. Note also that, in particular,the cumulative function associated to the black density function is $F_\vartriangleleft(\cdot+\Gamma)$, Therefore, up to a multiplicative factor, the density of the probability $\gamma_\Gamma$ is given by the black line on the left of the origin and by the grey one on the right.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• Proposition 2.4
• proof : Proof of Prop. \ref{['th:Y']}
• Theorem 3.1
• Remark 3.2
• Theorem 3.3