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<p>We study the top Lyapunov exponent of a product of random <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2 \times 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> matrices appearing in the analysis of several statistical mechanical models with disorder, extending a previous treatment of the critical case (Giacomin and Greenblatt, ALEA <b>19</b> (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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {R}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> 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.</p>

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On the Lyapunov Exponent for the Random Field Ising Transfer Matrix, in the Critical Case

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月28日

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  • Orphée Collin,
  • Giambattista Giacomin,
  • Rafael L. Greenblatt,
  • Yueyun Hu

摘要

We study the top Lyapunov exponent of a product of random \(2 \times 2\) 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 \(\mathbb {R}^2\) 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.