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One-arm exponent of critical level-set for metric graph Gaussian free field in high dimensions

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arxiv 2307.04434 v1 pith:IUJRBGZJ submitted 2023-07-10 math.PR

Zhenhao Cai , Jian Ding This is my paper · ORCID
classification math.PR
keywords criticallevel-setbondfieldfreegaussiangraphmetric
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abstract

In this paper, we study the critical level-set of Gaussian free field (GFF) on the metric graph $\widetilde{\mathbb{Z}}^d,d>6$. We prove that the one-arm probability (i.e. the probability of the event that the origin is connected to the boundary of the box $B(N)$) is proportional to $N^{-2}$, where $B(N)$ is centered at the origin and has side length $2\lfloor N \rfloor$. Our proof is hugely inspired by Kozma and Nachmias [29] which proves the analogous result of the critical bond percolation for $d\geq 11$, and by Werner [51] which conjectures the similarity between the GFF level-set and the bond percolation in general and proves this connection for various geometric aspects.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A switching identity for cable-graph loop soups and Gaussian free fields

    math.PR 2025-02 accept novelty 8.0 of 10

    Conditioning two cable-graph points to lie in the same Brownian loop-soup cluster adds an odd-numbered Poisson cloud of Brownian excursions between them, yielding an exact law for the conditional cluster and its GFF analogue.

  2. Cluster volumes for the Gaussian free field on metric graphs

    math.PR 2024-12 accept novelty 8.0 of 10

    Critical clusters of the Gaussian free field on Z^3, Z^4 and Z^5 have tail exponent delta=(d+2)/(d-2) and largest-cluster dimension (d+2)/2, confirming Werner's conjectures.

  3. On the intersection of critical percolation clusters and other tree-like random graphs

    math.PR 2024-11 conditional novelty 7.0 of 10

    Stretched-exponential tail bounds for intersections of independent critical percolation clusters, incipient infinite clusters, and branching random walk ranges are proved with explicit exponents.

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