Number of paths in oriented percolation as zero temperature limit of directed polymer

TL;DR

This paper establishes that the free energy of directed polymers in Bernoulli environments converges to the growth rate of open paths in super-critical oriented percolation as temperature approaches zero, with uniform convergence results.

Contribution

It proves the zero-temperature limit of directed polymer free energy converges to the percolation growth rate, showing continuity and uniform convergence in the super-critical phase.

Findings

Convergence of free energy to the percolation growth rate as temperature tends to zero.

Uniform convergence rate in the super-critical phase.

Continuity of the growth rate with respect to the percolation parameter.

Abstract

We prove that the free energy of directed polymer in Bernoulli environment converges to the growth rate for the number of open paths in super-critical oriented percolation as the temperature tends to zero. Our proof is based on rate of convergence results which hold uniformly in the temperature. We also prove that the convergence rate is locally uniform in the percolation parameter inside the super-critical phase, which implies that the growth rate depends continuously on the percolation parameter.