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

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.