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Universality for the 2D Random Walk Loop Soup

Yihao Pang

Abstract

We show that the scaling limit of the random walk loop soup on suitable planar graphs is the Brownian loop soup, under a topology on multisets of unrooted, unparameterized, and macroscopic loops. The result holds assuming only convergence of simple random walk to Brownian motion, a Russo-Seymour-Welsh type crossing estimate, and the bounded density of the graphs. The proof relies on Wilson's algorithm and Schramm's finiteness theorem. Precisely, we approximate the random walk loop soup by the set of loops erased in a greedy variant of Wilson's algorithm, thereby establishing convergence. The resulting limit is identified using the result of Lawler and Ferreras arXiv:math/0409291.

Universality for the 2D Random Walk Loop Soup

Abstract

We show that the scaling limit of the random walk loop soup on suitable planar graphs is the Brownian loop soup, under a topology on multisets of unrooted, unparameterized, and macroscopic loops. The result holds assuming only convergence of simple random walk to Brownian motion, a Russo-Seymour-Welsh type crossing estimate, and the bounded density of the graphs. The proof relies on Wilson's algorithm and Schramm's finiteness theorem. Precisely, we approximate the random walk loop soup by the set of loops erased in a greedy variant of Wilson's algorithm, thereby establishing convergence. The resulting limit is identified using the result of Lawler and Ferreras arXiv:math/0409291.
Paper Structure (24 sections, 20 theorems, 71 equations, 1 figure)

This paper contains 24 sections, 20 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and setup
  3. Graph discretization and assumptions
  4. Loop measures and loop soup
  5. Space of curves.
  6. Space of loops.
  7. Brownian loop soup.
  8. Random Walk Loop Soup.
  9. Markovian Loop Soup.
  10. Space of Loop Soups.
  11. Loop-erased random walk
  12. Convergence of LERW
  13. Wilson's algorithm
  14. Relationship between random walk loop soup and Wilson's algorithm:
  15. Greedy Wilson's algorithm
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $D$ be a simply connected and bounded domain. Let $\{\Gamma ^{\#\delta}\}_{\delta>0}$ be a sequence of graphs, all of which satisfy the assumptions of Section graph. Let $\{\mathcal{L}_D^{\#\delta}\}_{\delta>0}$ be the RWLS on $\{\Gamma^{\#\delta}\}_{\delta>0}$ restricted to $D$. Let $\mathcal{L

Figures (1)

  • Figure 1: An illustration of the $(k+1)$-th step in the sampling of the first branch of the greedy Wilson's algorithm. Left: The red curve $Y^1_k$ is the loop-erased curve obtained from the first $k$ steps. We run a random walk from $X^1(\tau_k^1)=Y^1_k(t_k)$ until it exits $B(X^1(\tau_k^1), r(\varepsilon))$ at the point $X^1(\tau_{k+1}^1)$, shown as the blue curve. The last visit point of the random walk to $Y^1_k$ is $Y^1_k(s^1_{k+1})=X^1(\theta^1_{k+1}-)$. Right: We perform a loop erasure on the random walk path $X^1[\theta^1_{k+1},\tau_{k+1}^1]$ to obtain the simple curve $A^1_{k+1}$(green curve). The updated simple curve $Y_{k+1}^1$ is obtained by concatenating $Y^1_k[0,s^1_{k+1}]$ (solid red line) and $A^1_{k+1}$ (solid green line). The dashed blue curve represents $X^1_{k+1}$.

Theorems & Definitions (45)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Lemma 2.2: Beurling type estimate
  • proof
  • Lemma 2.3: The size of edges is uniformly small
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 35 more