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

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 $\beta$.