Almost sure exponential behavior of a directed polymer in a fractional Brownian environment

Abstract

This paper studies the asymptotic behavior of a one-dimensional directed polymer in a random medium. The latter is represented by a Gaussian field B H on R + × R with fractional Brownian behavior in time ( Hurst parameter H) and arbitrary function-valued behavior in space. The partition function of such a polymer is u ( t ) = E b [ exp ∫ 0 t B H ( d r , b r ) ] . Here b is a continuous-time nearest neighbor random walk on Z with fixed intensity 2 κ, defined on a complete probability space P b independent of B H . The spatial covariance structure of B H is assumed to be homogeneous and periodic with period 2 π. For H < 1 2 , we prove existence and positivity of the Lyapunov exponent defined as the almost sure limit lim t → ∞ t −1 log u ( t ) . For H > 1 2 , we prove that the upper and lower almost sure limits lim sup t → ∞ t − 2 H log u ( t ) and lim inf t → ∞ ( t − 2 H log t ) log u ( t ) are non-trivial in the sense that they are bounded respectively above and below by finite, strictly positive constants. Thus, as H passes through 1 2 , the exponential behavior of u ( t ) changes abruptly. This can be considered as a phase transition phenomenon. Novel tools used in this paper include sub-Gaussian concentration theory via the Malliavin calculus, detailed analyses of the long-range memory of fractional Brownian motion, and an almost-superadditivity property.