Strong disorder and very strong disorder are equivalent for directed polymers

TL;DR

This work resolves a long-standing conjecture on the phase structure of directed polymers in $\mathbb{Z}^d$ by proving that, for $d\ge 3$ and under an environment bounded above, the weak-strong disorder threshold and the very-strong disorder threshold coincide, i.e., $\beta_c=\bar{\beta}_c$, and that weak disorder persists at criticality. The authors reduce the problem to endpoint localization via a spine/size-biasing framework and establish exponential decay of the normalized partition function in the very-strong regime through a sophisticated combination of fractional moments, tail estimates, and stochastic-calculus techniques in both discrete and continuum settings. A key corollary describes the tail behavior of the critical partition function and shows that weak disorder at $\beta_c$ yields precise moment-dimension control, with implications for the regularity of the free energy near criticality. The results provide a sharp, dichotomous phase picture without intermediate phases, and they open avenues for refined analyses of critical behavior and near-critical decay rates in higher dimensions. Overall, the paper advances the mathematical understanding of disorder-induced localization and sharp phase transitions in polymer models.

Abstract

We show that if the normalized partition function $W^β_n$ of the directed polymer model on $\mathbb Z^d$ converges to zero, then it does so exponentially fast. This implies that there exists a critical value $β_c$ for the inverse temperature such that the normalized partition function has a non-degenerate limit for all $β\in [0,β_c]$ -- weak disorder holds -- while for $β\in (β_c,\infty)$ it converges exponentially fast to zero -- very strong disorder holds. This solves a twenty-years-old conjecture formulated by Comets, Yoshida, Carmona and Hu. Our proof requires a technical assumption on the environment, namely, that it is bounded from above.

Figures (1)

• Figure 1: The environment $\omega^{(j)}$ is simply the environment restricted to the $j$-th slice of width $s$, which has been shifted so that $(js,X_{js})$ (the solid circles on the pictures) plays the role of the origin. Since the slices are disjoint and $X$ is independent of $\omega$, the environments $(\omega^{(j)})_{j\ge 1}$ are independent and distributed like $\omega^{(0)}$. The environments $\widetilde{\omega}^{(j)}$ are obtained by replacing the environment by $\widehat{\omega}$ along the spine. The Markov property applied to $X$ ensures that $(X^{(j)})_{j\ge 0}$ (which described the spine portions after shift) are i.i.d. and the same thing can be said about the restriction of $\widehat{\omega}$ to each time interval $\llbracket js+1,(j+1)s\rrbracket$.

Theorems & Definitions (53)

• Theorem A: CY06
• Remark 2.1
• Theorem 2.2
• Corollary 2.3
• proof
• Theorem B
• Proposition 3.1
• proof
• Lemma 3.2
• proof