Limiting Results for the Free Energy of Directed Polymers in Random Environment with Unbounded Jumps
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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 β . Our proof of existence at β=−∞ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at β=−∞ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the socalled 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.
Journal

 Journal of Statistical Physics

Journal of Statistical Physics 161(3), 577597, 201511
Springer US