Anderson polymer in a fractional Brownian environment: asymptotic behavior of the partition function

We consider the Anderson polymer partition function $$ u(t):=\mathbb{E}^X\Bigl[e^{\int_0^t \mathrm{d}B^{X(s)}_s}\Bigr]\,, $$ where $\{B^{x}_t\,;\, t\geq0\}_{x\in\mathbb{Z}^d}$ is a family of independent fractional Brownian motions all with Hurst parameter $H\in(0,1)$, and $\{X(t)\}_{t\in \mathbb{R}^{\geq 0}}$ 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 $H\leq 1/2$, the function $u(t)$ almost surely grows asymptotically like $e^{l t}$, where $l>0$ 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 $\mathcal{L}^1$ sense to some deterministic number $l>0$. 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 $\mathcal{L}^1$ sense, and equals a strictly positive deterministic number (possibly $+\infty$); hence almost surely $u(t)$ grows asymptotically at least like $e^{a t}$ for some deterministic constant $a>0$. On the other hand, we also show that almost surely and in the $\mathcal{L}^1$ 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 $e^{b t\sqrt{\log t}}$ for some deterministic positive constant $b$. Finally, for $H>1/2$ when $\mathbb{Z}^d$ is replaced by a circle endowed with a H\"older continuous covariance function, we show that $\limsup_{t\rightarrow \infty} \frac{1}{t}\log u(t)$ is a finite deterministic positive number, hence proving that almost surely $u(t)$ grows asymptotically at most like $e^{c t}$ for some deterministic positive constant $c$.