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

We consider the Anderson polymer partition function $$\begin{aligned} u(t):=\mathbb {E}^X\left[ e^{\int _0^t \mathrm {d}B^{X(s)}_s}\right] \,, \end{aligned}$$ u ( t ) : = E X e ∫ 0 t d B s X ( s ) , where $$\{B^{x}_t\,;\, t\ge 0\}_{x\in \mathbb {Z}^d}$$ { B t x ; t ≥ 0 } x ∈ Z d is a family of independent fractional Brownian motions all with Hurst parameter $$H\in (0,1)$$ H ∈ ( 0 , 1 ) , and $$\{X(t)\}_{t\in \mathbb {R}^{\ge 0}}$$ { X ( t ) } t ∈ R ≥ 0 is a continuous-time simple symmetric random walk on $$\mathbb {Z}^d$$ Z d with jump rate $$\kappa $$ κ and started from the origin. $$\mathbb {E}^X$$ E X is the expectation with respect to this random walk. We prove that when $$H\le 1/2$$ H ≤ 1 / 2 , the function u(t) almost surely grows asymptotically like $$e^{\lambda t}$$ e λ t , where $$\lambda >0$$ λ > 0 is a deterministic number. More precisely, we show that as t approaches $$+\infty $$ + ∞ , the expression $$\{\frac{1}{t}\log u(t)\}_{t\in \mathbb {R}^{>0}}$$ { 1 t log u ( t ) } t ∈ R > 0 converges both almost surely and in the $$\hbox {L}^1$$ L 1 sense to some positive deterministic number $$\lambda $$ λ . For $$H>1/2$$ H > 1 / 2 , we first show that $$\lim _{t\rightarrow \infty } \frac{1}{t}\log u(t)$$ lim t → ∞ 1 t log u ( t ) exists both almost surely and in the $$\hbox {L}^1$$ L 1 sense and equals a strictly positive deterministic number (possibly $$+\infty $$ + ∞ ); hence, almost surely u(t) grows asymptotically at least like $$e^{\alpha t}$$ e α t for some deterministic constant $$\alpha >0$$ α > 0 . On the other hand, we also show that almost surely and in the $$\hbox {L}^1$$ L 1 sense, $$\limsup _{t\rightarrow \infty } \frac{1}{t\sqrt{\log t}}\log u(t)$$ lim sup t → ∞ 1 t 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 $$e^{\beta t\sqrt{\log t}}$$ e β t log t for some deterministic positive constant $$\beta $$ β . Finally, for $$H>1/2$$ H > 1 / 2 when $$\mathbb {Z}^d$$ Z d is replaced by a circle endowed with a Hölder continuous covariance function, we show that $$\limsup _{t\rightarrow \infty } \frac{1}{t}\log u(t)$$ lim sup t → ∞ 1 t log u ( t ) is a deterministic finite positive real number, hence proving that almost surely u(t) grows asymptotically at most like $$e^{c t}$$ e c t for some deterministic positive constant c.