Derrida–Hilhorst regular-expansion conjecture for random matrix characteristic exponents

Let $M_{n,\epsilon}$ be the random $2\times2$ matrix product model considered in the paper, let $Z$ be its positive random parameter, and write $\mathcal{L}(\epsilon)$ for the corresponding characteristic (Lyapunov) exponent. A real-valued random variable $\xi$ is arithmetic if there is a constant $c>0$ such that $c\xi\in\mathbf{Z}\cup\\{\pm\infty\\}$ almost surely. Assume that $\log Z$ is nonarithmetic.

Derrida–Hilhorst conjecture. Suppose first that there exists $\alpha\in(0,+\infty)$ such that $\mathrm{E}[Z^\alpha]=1$. If $\alpha\notin\\{1,2,\ldots\\}$, then, as $\epsilon\to0$,

$$\mathcal{L}(\epsilon)=\sum_{k=1}^{\lfloor\alpha\rfloor}(-1)^{k+1}\ell_k\epsilon^{2k}+(-1)^{\lceil\alpha\rceil+1}C_Z\epsilon^{2\alpha}+o(\epsilon^{2\alpha}),$$

where, for $k\leq\lfloor\alpha\rfloor$, $\ell_k$ is a positive rational function of $\mathrm{E}[Z],\ldots,\mathrm{E}[Z^k]$, and $C_Z>0$. If $\alpha\in\\{1,2,\ldots\\}$, then

$$\mathcal{L}(\epsilon)=\sum_{k=1}^{\alpha-1}(-1)^{k+1}\ell_k\epsilon^{2k}+(-1)^{\alpha+1}C_Z\epsilon^{2\alpha}\log(1/\epsilon)+o\left(\epsilon^{2\alpha}\log\epsilon\right),$$

where the coefficients $\ell_k$ are the same positive rational functions of the moments of $Z$, and $C_Z>0$. Finally, if $\mathrm{E}[\log Z]=0$, the “$\alpha=0$” case, then

$$\mathcal{L}(\epsilon)=\frac{C_Z}{\log(1/\epsilon)}+o\left((\log 1/\epsilon)^{-1}\right).$$

These predictions describe the singular small-$\epsilon$ behaviour of the characteristic exponent when the diagonal limiting matrix remains random. The paper presents them as physical predictions and discusses which cases have been proved; the conjectural regular expansion is expected to hold only up to the order determined by the distribution of $Z$.