Coincidence of critical points for directed polymers for general environments and random walks

TL;DR

This paper investigates the critical points of directed polymers in random environments, extending known results to more general environments and reference walks, and characterizing when the critical point is zero.

Contribution

It generalizes the equality of critical points for directed polymers to broader environments and reference walks, and characterizes the triviality of the critical point.

Findings

Proves $eta_c = ar{eta}_c$ for general environments and arbitrary reference walks.

Shows $eta_c=0$ if and only if the $L^2$-critical point is trivial.

Extends previous results from upper-bounded environments to more general settings.

Abstract

For the directed polymer in a random environment (DPRE), two critical inverse-temperatures can be defined. The first one, $\beta_c$, separates the strong disorder regime (in which the normalized partition function $W^{\beta}_n$ tends to zero) from the weak disorder regime (in which $W^{\beta}_n$ converges to a nontrivial limit). The other, $\bar \beta_c$, delimits the very strong disorder regime (in which $W^{\beta}_n$ converges to zero exponentially fast). It was proved previously that $\beta_c=\bar \beta_c$ when the random environment is upper-bounded for the DPRE based on the simple random walk. We extend this result to general environment and arbitrary reference walk. We also prove that $\beta_c=0$ if and only the $L^2$-critical point is trivial.