On the convergence of massive loop-erased random walks to massive SLE(2) curves

Abstract

Following the strategy proposed by Makarov and Smirnov [24] in 2009 (see also [3, 2] for theoretical physics arguments), we provide technical details for the proof of convergence of massive loop-erased random walks to the chordal mSLE(2) process. As no follow-up to [24] appeared since then, we believe that such a treatment might be of interest to the community. We do not require any regularity of the limiting planar domain near its degenerate prime ends a and b except that $({\mathrm{\Omega }}^{\mathit{\delta }},a^{\mathit{\delta }},b^{\mathit{\delta }})$ are assumed to be ‘close discrete approximations’ to $(\mathrm{\Omega },a,b)$ near a and b in the sense of [13].