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

TL;DR

This paper proves that the phase transition in the directed polymer model in random environments for dimensions three and higher is infinitely smooth, with the free energy approaching zero more slowly than any power near the critical point.

Contribution

It demonstrates that the localization transition for the directed polymer in random environments is infinitely smooth, refining understanding of the phase transition's nature.

Findings

The free energy growth near the critical point is slower than any power function.

The transition from weak to strong disorder is infinitely smooth.

The result applies specifically for dimensions three and higher.

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 $$\mathfrak f(\beta):=\lim_{N\to \infty} (1/n)\log W^{\beta}_n$$ where $W^{\beta}_n$ is the normalized partition function for the directed polymer of length $n$. More precisely weak disorder corresponds to $\mathfrak f(\beta)=0$ and strong disorder to $\mathfrak f(\beta)<0$. Monotonicity and continuity of $\mathfrak f$ implies that there exists $\beta_c\in [0,\infty]$ such that weak disorder is equivalent to $\beta\in [0,\beta_c]$. Furthermore $\beta_c>0$ 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 $\beta_c$, that is $$ \lim_{\beta \downarrow \beta_c }\frac{\log |\mathfrak f(\beta)|}{\log (\beta-\beta_c)}=\infty.$$