Two-sided bounds on free energy of directed polymers on strongly recurrent graphs

TL;DR

This paper investigates the free energy behavior of directed polymers on strongly recurrent graphs with sub-Gaussian heat kernel bounds, establishing bounds and relations between quenched and annealed free energies for small inverse temperatures.

Contribution

It proves the existence and equality of quenched and annealed free energies and characterizes their difference in terms of the spectral dimension for such graphs.

Findings

Quenched and annealed free energies coincide.

Difference between free energies scales as 2^{4/(2-d_s)} for small 2.

Results apply to graphs with spectral dimension less than two.

Abstract

We study the directed polymers in random environment on an infinite graph $G=(V,E)$ on which the underlying random walk satisfies sub-Gaussian heat kernel bounds with spectral dimension $d_{s}$ strictly less than two. Our goal in this paper is to show (i) the existence and the coincidence of the quenched and the annealed free energy $F_q(\beta)$, $F_a(\beta)$ and (ii) that $F_a(\beta)-F_q(\beta)$ is comparable to $\beta^{\frac{4}{2-d_{s}}}$ for small inverse temperature $\beta$.