Limiting Results for the Free Energy of Directed Polymers in Random Environment with Unbounded Jumps

Abstract

We study asymptotics of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by Bernoulli variables. We first establish the existence and continuity of the free energy including the negative infinity value of the coupling constant $\beta$. Our proof of existence at $\beta =-\infty$ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at $\beta =-\infty$ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.

Appendix

We provide a proof of (3) for completeness. We consider $d=1$ case first since the other case will reduce to it. Set

$$\begin{aligned} {\mathscr {L}}=\{(m,x)\in \mathbb {N}\times \mathbb {Z}: m+x\in 2\mathbb {Z}\}. \end{aligned}$$

For $R>0$ and $(m,x)\in {\mathscr {L}}$, we say (m, x) is open if there exists a $y_m\in Rx+(-R,R)\cap \mathbb {Z}$ such that $\eta (m,y_m)=0$. It is easy to see that

$$\begin{aligned} Q((m,x)\text { is open}) \rightarrow 1 \end{aligned}$$

as $R\rightarrow \infty$. Thus when R is large, the directed site percolation on ${\mathscr {L}}$ is supercritical and we can find a percolation point $(1,x)\in {\mathscr {L}}$. This implies that there exists a path $\{(k,y_k)\}_{k\ge 1}$ satisfying

$$\begin{aligned} \eta (k,y_k)=0\text { and }|y_{k+1}-y_{k+2}|\le 3R \end{aligned}$$

for all $k\ge 1$. Then it follows that

$$\begin{aligned} \begin{aligned} \liminf _{n\rightarrow \infty }\frac{1}{n}\log Z_n^{\eta ,-\infty }&\ge \liminf _{n\rightarrow \infty }\frac{1}{n}\log P(X_k=y_k\text { for all } k\le n)\\&\ge -c_2 3^\alpha R^\alpha . \end{aligned} \end{aligned}$$

For the case $d\ge 2$, we have

$$\begin{aligned} Z_n^{\eta ,-\infty }\ge P(H_n^\eta =0 \text { and } X_k \in \mathbb {Z}\times \{0\}^{d-1}\text { for all }1\le k\le n) \end{aligned}$$

and the right-hand side can be bounded from below in the same way as for $d=1$.