Directed polymer in space-correlated disorder

Directed polymers in random environments describe a perturbation of the simple random walk by a random disorder (environment). The partition function of such models has been extensively studied in recent years, also due to its connection with the solution of the Stochastic Heat Equation. While classical results focus on space-time independent disorder, we investigate the case of a Gaussian environment with (critically) space-correlated disorder. We show that, similarly to the independent case, a phase transition occurs: when the disorder strength is below a critical threshold, the log-partition function satisfies a central limit theorem; above this threshold, it converges to zero in distribution. Remarkably, the appropriate scaling constant for the disorder, as well as the correct limiting variance, emerge from a non-trivial induction scheme that reflects the critical nature of the correlation.
(Joint work with Clément Cosco and Francesca Cottini)