The Tail Distribution of the Partition Function for Directed Polymers in the Weak Disorder Phase

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 \(W^{\beta }_{\infty }\) displays a power-law decay, with an exponent \(p^*(\beta )\in [1+\frac{2}{d},\infty )\). 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.