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

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 Z^{d, d ge 3 (alpha = d-2), the Gaussian membrane model on Z}d, 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.

• Speaker: Stephen Muirhead (Melbourne)
• Tuesday 22 February 2022, 14:00–15:00
• Venue: MR12.
• Series: Probability; organiser: Perla Sousi.