Stephen Muirhead: Sharpness of the phase transition for level set percolation of strongly correlated Gaussian fields


A few years ago Duminil-Copin, Goswami, Rodriguez and Severo established the sharpness of the phase transition for level-set percolation of the Gaussian free field on  \mathbb{Z}^d, d \ge 3, in the sense that there is a single regime of subcritical behaviour. We give a new, simpler, proof of this result that (i) extends immediately to a wide class of strongly correlated Gaussian fields, discrete and continuous, whose correlations decay algebraically with exponent \alpha > 0, including the Gaussian membrane model on \mathbb{Z}^d, d \ge 5 (\alpha = d-4), and (ii) yields near-critical information on the percolation density that is new even in the case of the GFF.

Our proof is based on the celebrated OSSS argument of Duminil-Copin, Raoufi and Tassion, with a new twist to handle the strong correlations. The upshot of the approach is to establish a weaker form of the Menshikov differential inequality that is (i) strong enough to prove sharpness, but (ii) weak enough to be consistent with the lack of exponential decay in strongly correlated models.

We also establish some additional new results in the planar case, including the “mean field bound” for models with decay exponent \alpha \ge 1 (which may lie outside the Bernoulli universality class), and the “interface RSW” property valid for all decay exponents \alpha > 0.


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