On the radius of Gaussian free field excursion clusters

With Franco Severo (Geneva and IHES)

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, 14:0015:00
  • Venue: Zoom.
  • Series: Probability; organiser: Perla Sousi.

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