Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4592604 | Journal of Functional Analysis | 2008 | 51 Pages |
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 BH on R+ÃR 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[expâ«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 2Ï. For H<12, we prove existence and positivity of the Lyapunov exponent defined as the almost sure limit limtââtâ1logu(t). For H>12, we prove that the upper and lower almost sure limits limâsuptââtâ2Hlogu(t) and limâinftââ(tâ2Hlogt)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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Frederi G. Viens, Tao Zhang,