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Sharpness of the phase transition for level set percolation of long-range correlated Gaussian fields

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We study the phase transition in the connectivity of the excursion sets of long-range correlated Gaussian fields. Our main result establishes `sharpness’ of the transition for a wide class of fields, discrete and continuous, whose correlations decay algebraically with exponent \alpha \in (0,d), including the Gaussian free field on Zd, d \ge 3 (\alpha = d-2), the Gaussian membrane model on Zd, d \ge 5 (\alpha = d – 4), among other examples. This result is new for all models in dimension d \ge 3 except the Gaussian free field, for which sharpness was proven in a recent breakthrough by Duminil-Copin, Goswami, Rodriguez and Severo; even then, our proof is simpler and yields new near-critical information on the percolation density.

This talk is part of the Probability series.

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