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

## TL;DR

This paper provides detailed proof of the convergence of massive loop-erased random walks to massive SLE(2) curves, extending previous strategies and removing regularity assumptions on the domain.

## Contribution

It offers a rigorous proof of convergence to massive SLE(2) without regularity constraints, filling a gap in the existing literature.

## Key findings

- Proof of convergence of massive LERW to mSLE(2)
- No regularity assumptions on domain near prime ends
- Applicable to general planar domains

## Abstract

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 $\Omega$ near its degenerate prime ends $a$ and $b$ except that $(\Omega^\delta,a^\delta,b^\delta)$ are assumed to be `close discrete approximations' to $(\Omega,a,b)$ near $a$ and $b$ in the sense of a recent work arXiv:1810.05608.