Abstract

We consider the Gaussian Free Field (GFF) on $\mathbb{Z}^{d$, for $d\geq 3$, 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 $h\neq h_$. In this talk we shall discuss sharp bounds on the probability that a cluster has radius larger than $N$. For $d\geq 4$, this probability decays exponentially in $N$, similarly to Bernoulli percolation; while for $d=3$ it decays as $\exp(-\frac{\pi}{6}(h-h_*)}2\frac{N}{\log N})$ 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.