In this short note, we prove a central limit theorem for a type of replica overlap of the Brownian directed polymer in a Gaussian random environment, in the low temperature regime and in all dimensions. The proof relies on a superconcentration result for the KPZ equation driven by a spatially mollified noise, which is inspired by the recent work of Chatterjee .
Y.G. was partially supported by the NSF through DMS-2203007/2203014. T.K. acknowledges the support of NCN grant 2020/37/B/ST1/00426.
We thank Erik Bates and Sourav Chatterjee for comments on the draft and two anonymous referees for a careful reading of the paper which helped to improve the presentation.
"Gaussian fluctuations of replica overlap in directed polymers." Electron. Commun. Probab. 27 1 - 12, 2022. https://doi.org/10.1214/22-ECP476