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

Following the strategy proposed by Makarov and Smirnov in arXiv:0909.5377, 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 of arXiv:0909.5377 appeared since then, we believe that such a treatment might be of interest for the community. We do not require any regularity of the limiting planar domain $\mathrm{\Omega }$ near its degenerate prime ends $a$ and $b$ except that $({\mathrm{\Omega }}^{\delta },a^{\delta },b^{\delta })$ are assumed to be `close discrete approximations’ to $(\mathrm{\Omega },a,b)$ near $a$ and $b$ in the sense of a recent work arXiv:1810.05608.