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

Abstract

We consider the Anderson polymer partition function u(t) := E X e t 0 dB X(s) s, where {B x t ; t ≥ 0} x∈Z d 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 Z d with jump rate κ and started from the origin. E X 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 e λt , where λ > 0 is a deterministic number. More precisely, we show that as t approaches +∞, the expression { 1 t log u(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→∞ 1 t log u(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 e αt for some deterministic constant α > 0. On the other hand, we also show that almost surely and in the L 1 sense, 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 βt √ log t for some deterministic positive constant β.Finally, for H > 1/2 when Z d is replaced by a circle endowed with a Hölder continuous covariance function, we show that 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 ct for some deterministic positive constant c.