The localization transition for the directed polymer in a random environment is smooth

Abstract

When $d\ge 3$, the directed polymer a in random environment on ${\mathbb{Z}}^d$ is known to display a phase transition from a diffusive phase, known as \textit{weak disorder} to a localized phase, referred to as \textit{strong disorder}. This transition is encoded by the behavior of the the free energy of the model, defined by $$\mathfrak{f}({\hat{I}}^2):=\underset{N\to \infty }{\lim }(1/n)\log W_n^2$$ where $W_n^2$ is the normalized partition function for the directed polymer of length $n$. More precisely weak disorder corresponds to $\mathfrak{f}({\hat{I}}^2)=0$ and strong disorder to $\mathfrak{f}({\hat{I}}^2)<0$. Monotonicity and continuity of $\mathfrak{f}$ implies that there exists ${\hat{I}}_c^2\in [0,\infty ]$ such that weak disorder is equivalent to ${\hat{I}}^2\in [0,{\hat{I}}_c^2]$. Furthermore ${\hat{I}}_c^2>0$ if and only if $d\ge 3$. We prove that this transition is infinitely smooth in the sense that $\mathfrak{f}$ grows slower than any power function at the vicinity of ${\hat{I}}_c^2$, that is $$\underset{{\hat{I}}^2\downarrow {\hat{I}}_c^2}{\lim }\frac{\log |\mathfrak{f}({\hat{I}}^2)|}{\log ({\hat{I}}^2-{\hat{I}}_c^2)}=\infty .$$

19 pages