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

Abstract

We consider the Anderson polymer partition function u(t):=EX[e∫0tdBsX(s)],where {Btx;t≥0}x∈Zd is a family of independent fractional Brownian motions all with Hurst parameter H∈ (0 , 1) , and {X(t)}t∈R≥0 is a continuous-time simple symmetric random walk on Zd with jump rate κ and started from the origin. EX is the expectation with respect to this random walk. We prove that when H≤ 1 / 2 , the function u(t) almost surely grows asymptotically like eλt, where λ> 0 is a deterministic number. More precisely, we show that as t approaches + ∞, the expression {1tlogu(t)}t∈R>0 converges both almost surely and in the L 1 sense to some positive deterministic number λ. For H> 1 / 2 , we first show that limt→∞1tlogu(t) exists both almost surely and in the L 1 sense and equals a strictly positive deterministic number (possibly + ∞); hence, almost surely u(t) grows asymptotically at least like eαt for some deterministic constant α> 0. On the other hand, we also show that almost surely and in the L 1 sense, lim supt→∞1tlogtlogu(t) is a deterministic finite real number (possibly zero), hence proving that almost surely u(t) grows asymptotically at most like eβtlogt for some deterministic positive constant β. Finally, for H> 1 / 2 when Zd is replaced by a circle endowed with a Hölder continuous covariance function, we show that lim supt→∞1tlogu(t) is a deterministic finite positive real number, hence proving that almost surely u(t) grows asymptotically at most like ect for some deterministic positive constant c.