Level-set percolation for the Gaussian free field on a transient tree

Abstract

We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison with percolation of the vacant set of random interlacements, which is more tractable in the case of trees. If $h_{\ast }$ and $u_{\ast }$ denote the respective (non-negative) critical values of level-set percolation of the Gaussian free field and of the vacant set of random interlacements, we show here that $h_{\ast }<{\sqrt{2u}}_{\ast }$ in a broad enough set-up, but provide an example where $0=h_{\ast }=u_{\ast }$ occurs. We also obtain some sufficient conditions ensuring that $h_{\ast }>0$.

34 pages, this version appears in the Annales Institut Henri Poincaré Probabilités et Statistiques

References in corpus (2)

• From loop clusters and random interlacement to the free field
• An introduction to Galton-Watson trees and their local limits

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