Susceptibility to disorder of the optimal resetting rate in the Larkin model of directed polymers

TL;DR

This paper analyzes how disorder affects the optimal resetting rate in the Larkin model of directed polymers, revealing a positive susceptibility and providing a detailed mathematical characterization.

Contribution

It introduces a novel analytical approach to quantify the impact of disorder on the optimal resetting rate in the Larkin model, including a closed-form susceptibility expression.

Findings

Optimal resetting rate has a positive susceptibility to disorder.

The mean time to absorption has a single minimum at the optimal resetting rate.

The susceptibility can be expressed in closed form.

Abstract

We consider the Larkin model of a directed polymer with Gaussian-distributed random forces, with the addition of a resetting process whereby the transverse position of the end-point of the polymer is reset to zero with constant rate $r$. We express the average over disorder of the mean time to absorption by an absorbing target at a fixed value of the transverse position. Thanks to the independence properties of the distribution of the random forces, this expression is analogous to the mean time to absorption for a diffusive particle under resetting, which possesses a single minimum at an optimal value $r^\ast$ of the resetting rate . Moreover, the mean time to absorption can be expanded as a power series of the amplitude of the disorder, around the value $r^\ast$ of the resetting rate. We obtain the susceptibility of the optimal resetting rate to disorder in closed form, and find it to be positive.