LA Probability Forum

We consider the number of open paths in a supercritical oriented site percolation. In the finite volume, it is the zero temperature limit of the directed polymer in the Bernoulli environment. Unlike in the positive temperature regime, even the existence of the "free energy" (=growth rate) is a non-trivial problem due to the problem of extinction. It was proved in 2017 by Garet, Gouere and Marchand but fundamental properties of the growth rate, such as the continuity with respect to the percolation parameter, have been unknown. We prove that the free energy of the directed polymer in the 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 the 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 supercritical phase, which implies that the growth rate depends continuously on the percolation parameter.