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One-arm probabilities for the two-dimensional metric-graph and discrete Gaussian free field

Yijie Bi, Yifan Gao, Xinyi Li

Abstract

We study the one-arm probability in the level-set percolation of the discrete and metric-graph Gaussian free field (GFF) defined on a box with Dirichlet boundary conditions. For the metric-graph case, we establish asymptotic estimates on two one-arm probabilities of interest. For the discrete case, we show up-to-constants bounds on the point-to-bulk probability and demonstrate its difference from the metric-graph case.

One-arm probabilities for the two-dimensional metric-graph and discrete Gaussian free field

Abstract

We study the one-arm probability in the level-set percolation of the discrete and metric-graph Gaussian free field (GFF) defined on a box with Dirichlet boundary conditions. For the metric-graph case, we establish asymptotic estimates on two one-arm probabilities of interest. For the discrete case, we show up-to-constants bounds on the point-to-bulk probability and demonstrate its difference from the metric-graph case.

Paper Structure

This paper contains 10 sections, 20 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic results on Brownian motions and martingales
  4. The Gaussian free field (GFF)
  5. Exploration martingales
  6. Isomorphism theorem and circuit estimates
  7. The Metric-Graph Case
  8. The Discrete Case
  9. Discrete one-arm probability
  10. Comparison with the metric-graph case: proof of Theorem \ref{['thm:comparison']}

Key Result

Proposition 1.1

The following estimates hold: Above, $a_1,a_2>0$ are some universal constants.

Theorems & Definitions (34)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 24 more