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

TL;DR

The paper proves that the localization transition for the directed polymer in a random environment in $d\ge 3$ is infinitely smooth at the critical point $β_c$. It achieves this by establishing sharp asymptotics for fractional moments $\mathbb E[(W^{β}_n)^p]$ for $p\in(1,1+2/d)$ and relating them to critical moments via conditional Jensen and convex comparisons, ultimately bounding the near-critical growth of the free energy $\mathfrak f(β)$. The work also derives near-critical behavior for endpoint localization metrics and provides a framework (via convexity, concentration, and pinning-type arguments) that yields an infinite-order phase transition and informs the structure of trajectories in both weak and strong disorder regimes. These results improve the understanding of the critical regime, offering tools to analyze near-critical dynamics and suggesting open directions for finer asymptotics and intermediate-disorder scenarios. The findings have implications for the mathematical characterization of phase transitions in disordered systems and contribute rigorous techniques for analyzing fractional moments and their impact on macroscopic observables.

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

Theorems & Definitions (22)

• Theorem A: B89CH02CSY03CV06CY06IS88L10
• Theorem B: B89JL24JL25_2JL25_1
• Theorem C: CH02CSY03
• Theorem 2.1
• Corollary 2.2
• Theorem 2.3
• Proposition 2.4
• Lemma 3.1
• Lemma 3.2
• proof