Limiting results for the free energy of directed polymers in random environment with unbounded jumps

Abstract

We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first establish the existence and continuity of the free energy including the negative infinity value of the coupling constant $\beta$. Our proof of existence at $\beta =-\infty$ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at $\beta =-\infty$ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.

24 pages, Corrected typos

References in corpus (3)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Localization Transition for Polymers in Poissonian Medium
• On exponential growth for a certain class of linear systems

Cited by in corpus (8)

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• Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation
• Zero temperature limit for the Brownian directed polymer among Poissonian disasters
• Number of paths in oriented percolation as zero temperature limit of directed polymer
• Localization and free energy asymptotics in disordered statistical mechanics and random growth models
• Concentration results for directed polymer with unbounded jumps
• Continuity result for the rate function of the simple random walk on supercritical percolation clusters
• Note on the maximal jump size in a continuum model of directed first passage percolation