The tail distribution of the partition function for directed polymer in the weak disorder phase
Stefan Junk, Hubert Lacoin
TL;DR
The paper analyzes the upper tail of the directed polymer partition function in the weak disorder phase, proving a universal power-law decay ${\mathbb P}(W^{\beta}_{\infty}>u) \asymp u^{-p^*(\beta)}$ with $p^*(\beta)\in[1+\tfrac{2}{d},\infty)$. A central technical tool is an overshoot bound that controls $L^p$ norms at the first passage time when the partition function exceeds a large level, enabling tail results without assuming an upper bound on the environment. The authors also show that the same tail behavior extends to the suprema over time and to point-to-point partitions, and they develop a robust framework (including a spine/size-biased approach) to derive corollaries about disorder regimes, moment growth, and fluctuation phenomena. They connect the tail exponent to broader questions about the free energy near criticality and to fluctuation fields, with extensions to general reference walks. Overall, the work broadens the understanding of tail behavior in disordered polymers and provides tools likely applicable to other disordered systems.
Abstract
We investigate the upper tail distribution of the partition function of the directed polymer in a random environment on $\mathbb Z^d$ in the weak disorder phase. We show that the distribution of the infinite volume partition function $W^β_{\infty}$ displays a power-law decay, with an exponent $p^*(β)\in [1+\frac{2}{d},\infty)$. We also prove that the distribution of the suprema of the point-to-point and point-to-line partition functions display the same behavior. On the way to these results, we prove a technical estimate of independent interest: the $L^p$-norm of the partition function at the time when it overshoots a high value $A$ is comparable to $A$. We use this estimate to extend the validity of many recent results that were proved under the assumption that the environment is upper bounded.