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

AbstractThis paper studies the asymptotic behavior of a one-dimensional directed polymer in a random medium. The latter is represented by a Gaussian field BH 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 isu(t)=Eb[exp∫0tBH(dr,br)]. Here b is a continuous-time nearest neighbor random walk on Z with fixed intensity 2κ, defined on a complete probability space Pb independent of BH. The spatial covariance structure of BH is assumed to be homogeneous and periodic with period 2π. For H<12, we prove existence and positivity of the Lyapunov exponent defined as the almost sure limit limt→∞t−1logu(t). For H>12, we prove that the upper and lower almost sure limits lim supt→∞t−2Hlogu(t) and lim inft→∞(t−2Hlogt)logu(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 12, 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