Sharp Asymptotics for the Partition Function of Some Continuous-time Directed Polymers

Abstract

This paper is concerned with two related types of directed polymers in a random medium. The first one is a d-dimensional Brownian motion living in a random environment which is white-noise in time and homogeneous in space. The second is a continuous-time discrete-space Markov process on Z d , in a random environment with similar properties as in continuous space, albeit defined only on R + × Z d . The case of a space-time white noise environment can be achieved in this second setting. By means of some Gaussian tools, we estimate the free energy of these models at low temperature, and give some further information on the strong disorder regime of the objects under consideration.Keywords Polymer model · Random medium · Gaussian field · Free energy · Lyapunov exponent Mathematics Subject Classifications (2000) 82D60 · 60K37 · 60G15 Frederi Viens' research partially supported by NSF grants no.: DMS 0204999 and DMS 0606615.A. Cadel · S. Tindel (B)

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“…These improved results are made possible by borrowing some tools from [12,18], and the recent preprint [5], using them more efficiently herein, and also introducing new tools.…”

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