F. Severo: Sharp phase transition for Gaussian percolation in all dimensions

Abstract: Level-set percolation of continuous Gaussian fields on \mathbb{R}^d has attracted a lot of attention in recent years. While great progress has been made in the planar case d=2, higher dimensions remain rather poorly understood. In this talk, we shall prove that fields with positive and sufficiently fast decaying correlations (along with other mild assumptions) undergo a sharp phase transition in arbitrary dimensions. More precisely, we show that connection probabilities decay exponentially for \ell<\ell_c and percolation occurs in sufficiently thick 2D slabs for \ell>\ell_c, where \ell_c stands for the critical level of the model. In particular, our results apply to the Bargmann-Fock field, for which sharpness was known only in dimension d=2.

The result follows from a global comparison with a truncated and discretized version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a small sprinkling. This approach is inspired by a recent work with Duminil-Copin, Goswami and Rodriguez on sharpness for the (discrete) Gaussian free field.


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