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.
Supercritical sharpness for Voronoi percolation
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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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The supercritical phase of the $\varphi^4$ model is well behaved
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.