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Supercritical sharpness for Voronoi percolation

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arxiv 2311.00555 v2 pith:HSDO7LPV submitted 2023-11-01 math.PR math-phmath.MP

classification math.PRmath-phmath.MP
keywords percolationsupercriticalhappensphasevoronoianaloguebehavedbernoulli
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abstract

We prove that the supercritical phase of Voronoi percolation on $\mathbb{R}^d$, $d\geq 3$, is well behaved in the sense that for every $p>p_c(d)$ local uniqueness of macroscopic clusters happens with high probability. As a consequence, truncated connection probabilities decay exponentially fast and percolation happens on sufficiently thick 2D slabs. This is the analogue of the celebrated result of Grimmett & Marstrand for Bernoulli percolation and serves as the starting point for renormalization techniques used to study several fine properties of the supercritical phase.

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  1. The supercritical phase of the $\varphi^4$ model is well behaved

    math.PR 2025-01 conditional novelty 8.0 of 10

    For every β above the critical inverse temperature, the φ^4 random cluster model on Z^d has a unique macroscopic cluster with high probability, yielding surface-order large deviations and spectral gap decay.

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