Anderson Polymer in a Fractional Brownian Environment: Asymptotic Behavior of the Partition Function

Abstract

We consider the Anderson polymer partition function

$$\begin{aligned} u(t):=\mathbb {E}^X\left[ e^{\int _0^t \mathrm {d}B^{X(s)}_s}\right] \,, \end{aligned}$$

where $\{B^{x}_t\,;\, t\ge 0\}_{x\in \mathbb {Z}^d}$ is a family of independent fractional Brownian motions all with Hurst parameter $H\in (0,1)$, and $\{X(t)\}_{t\in \mathbb {R}^{\ge 0}}$ is a continuous-time simple symmetric random walk on $\mathbb {Z}^d$ with jump rate $\kappa$ and started from the origin. $\mathbb {E}^X$ is the expectation with respect to this random walk. We prove that when $H\le 1/2$, the function u(t) almost surely grows asymptotically like $e^{\lambda t}$, where $\lambda >0$ is a deterministic number. More precisely, we show that as t approaches $+\infty$, the expression $\{\frac{1}{t}\log u(t)\}_{t\in \mathbb {R}^{>0}}$ converges both almost surely and in the $\hbox {L}^1$ sense to some positive deterministic number $\lambda$. For $H>1/2$, we first show that $\lim _{t\rightarrow \infty } \frac{1}{t}\log u(t)$ exists both almost surely and in the $\hbox {L}^1$ sense and equals a strictly positive deterministic number (possibly $+\infty$); hence, almost surely u(t) grows asymptotically at least like $e^{\alpha t}$ for some deterministic constant $\alpha >0$. On the other hand, we also show that almost surely and in the $\hbox {L}^1$ sense, $\limsup _{t\rightarrow \infty } \frac{1}{t\sqrt{\log t}}\log u(t)$ is a deterministic finite real number (possibly zero), hence proving that almost surely u(t) grows asymptotically at most like $e^{\beta t\sqrt{\log t}}$ for some deterministic positive constant $\beta$. Finally, for $H>1/2$ when $\mathbb {Z}^d$ is replaced by a circle endowed with a Hölder continuous covariance function, we show that $\limsup _{t\rightarrow \infty } \frac{1}{t}\log u(t)$ is a deterministic finite positive real number, hence proving that almost surely u(t) grows asymptotically at most like $e^{c t}$ for some deterministic positive constant c.