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

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• 作者：Frederi G. Viens ; Tao Zhang
• 关键词：Polymer ; Random media ; Lyapunov exponent ; Malliavin derivative ; Partition function ; Fractional Brownian motion ; Gaussian field ; Long memory ; Anderson model
• 刊名：Journal of Functional Analysis
• 出版年：2008
• 出版时间：15 November 2008
• 年：2008
• 卷：255
• 期：10
• 页码：2810-2860
• 全文大小：408 K

文摘

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 Here b is a continuous-time nearest neighbor random walk on Z with fixed intensity 2κ, 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 2π. For , we prove existence and positivity of the Lyapunov exponent defined as the almost sure limit lim_t→∞t⁻¹logu(t). For , we prove that the upper and lower almost sure limits lim sup_t→∞t^−2Hlogu(t) and lim inf_t→∞(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 , 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.