An invariance principle for the 2d weakly self-repelling Brownian polymer

Abstract: We investigate the large-scale behaviour of the Self-Repelling Brownian Polymer (SRBP) in the critical dimension $d=2$. The SRBP is a model of self-repelling motion, which is formally given by the solution a stochastic differential equation driven by a standard Brownian motion and with a drift given by the negative gradient of its own local time. As with its discrete counterpart, the "true" self-avoiding walk (TSAW) of [D.J. Amit, G. Parisi, & L. Peliti, Asymptotic behaviour of the "true" self-avoiding walk, Phys. Rev. B, 1983], it is conjectured to be logarithmically superdiffusive, i.e. to be such that its mean-square displacement grows as $t(\log t)^\beta$ for $t$ large and some currently unknown $\beta\in(0,1)$. The main result of the paper is an invariance principle for the SRBP under the weak coupling scaling, which corresponds to scaling the SRBP diffusively and simultaneously tuning down the strength of the self-interaction in a scale-dependent way. The diffusivity for the limiting Brownian motion is explicit and its expression provides compelling evidence that the $\beta$ above should be $1/2$. Further, we derive the scaling limit of the so-called environment seen by the particle process, which formally solves a non-linear singular stochastic PDE of transport-type, and prove this is given by the solution of a stochastic linear transport equation with enhanced diffusivity.