On the Lyapunov exponent for the random field Ising transfer matrix, in the critical case
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912401630167040 |
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| author | Collin, Orphée Giacomin, Giambattista Greenblatt, Rafael L. Hu, Yueyun |
| author_facet | Collin, Orphée Giacomin, Giambattista Greenblatt, Rafael L. Hu, Yueyun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_08797 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Lyapunov exponent for the random field Ising transfer matrix, in the critical case Collin, Orphée Giacomin, Giambattista Greenblatt, Rafael L. Hu, Yueyun Probability Mathematical Physics 60K37, 82B44, 60K35, 37H15 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. |
| title | On the Lyapunov exponent for the random field Ising transfer matrix, in the critical case |
| topic | Probability Mathematical Physics 60K37, 82B44, 60K35, 37H15 |
| url | https://arxiv.org/abs/2501.08797 |