On the radius of Gaussian free field excursion clusters

We consider the Gaussian Free Field (GFF) on $mathbb{Z}^{d$, for $dgeq 3$, and its excursions above a given real height $h$. As $h$ varies, this defines a natural percolation model with slow decay of correlations and a critical parameter $h_$. Sharpness of phase transition has been recently established for this model. This result directly implies, through classical renormalization techniques, that the radius distribution of a finite excursion cluster decays stretched exponentially fast for any $hneq h_$. In this talk we shall discuss sharp bounds on the probability that a cluster has radius larger than $N$. For $dgeq 4$, this probability decays exponentially in $N$, similarly to Bernoulli percolation; while for $d=3$ it decays as $exp(-frac{pi}{6}(h-h_*)}2frac{N}{log N})$ to principal exponential order. We will explain how the so-called “entropic repulsion phenomenon” allows us to prove such precise estimates for $d=3$. This is a joint work with Subhajit Goswami and Piere-François Rodriguez.

• Speaker: Franco Severo (Geneva and IHES)
• Tuesday 04 May 2021, 14:00–15:00
• Venue: Zoom.
• Series: Probability; organiser: Perla Sousi.