Singular behavior of the leading Lyapunov exponent of a product of random $2 \times 2$ matrices
Giuseppe Genovese, Giambattista Giacomin, Rafael Leon Greenblatt
备注：
35 pages, 1 figure, LaTeX. Various revisions including many changes to the introduction, a simpler statement of the results of Section 4, and some simplifications of the calculations in Section 5
发表信息：
Communications in Mathematical Physics vol. 351, pp. 923-958 (2017)
本文考虑了统计力学中某些$1$和$1+1$维无序模型的解中出现的随机$2 \times 2$矩阵的某种无穷乘积，它依赖于一个参数$\varepsilon>0$和一个具有$μ$分布的实随机变量.对于一大类$μ$，我们证明了B.德里达和H. J. Hilhorst（J. Phys. A 16：2641，1983）指出，在极限$\varepsilon \searrow 0$中，李雅普诺夫指数表现得像$C \varepsilon^{2 α}$，其中$α\in (0,1)$和$C>0$由$μ$确定。德里达和希尔霍斯特对与矩阵乘积相关的马尔可夫链的不变分布的积分方程进行了双尺度分析，并获得了一个概率测度，预计它将接近小$\varepsilon$的不变分布。我们引入合适的规范和利用收缩性质表明，这样的概率测度确实是接近不变的，在某种意义上，这意味着一个合适的控制的李雅普诺夫指数。
英文摘要：
We consider a certain infinite product of random $2 \times 2$ matrices appearing in the solution of some $1$ and $1+1$ dimensional disordered models in statistical mechanics, which depends on a parameter $\varepsilon>0$ and on a real random variable with distribution $μ$. For a large class of $μ$, we prove the prediction by B. Derrida and H. J. Hilhorst (J. Phys. A 16:2641, 1983) that the Lyapunov exponent behaves like $C \varepsilon^{2 α}$ in the limit $\varepsilon \searrow 0$, where $α\in (0,1)$ and $C>0$ are determined by $μ$. Derrida and Hilhorst performed a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product and obtained a probability measure that is expected to be close to the invariant one for small $\varepsilon$. We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one in a sense which implies a suitable control of the Lyapunov exponent.
