Abstract
We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\geq 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ exhibits a phase transition at level $h_{\star }$, which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite $d$-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level $h$ does not contain any connected component of larger than logarithmic size whenever $h>h_{\star }$, and on the contrary, whenever $h<h_{\star }$, a linear fraction of the vertices is contained in connected components of the level set above level $h$ having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase $h<h_{\star }$, as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level $h$. The proofs in this article make use of results from the accompanying paper [2].
Highlights
In this article we study level-set percolation of the zero-average Gaussian free field on a class of large d-regular graphs with d ≥ 3
Through suitable local approximations of the zero-average Gaussian free field by the Gaussian free field on the infinite d-regular tree we are able to establish a phase transition for level-set percolation of the zero-average Gaussian free field which occurs at the critical value for level-set percolation in the infinite model, that is, on the d-regular tree
Our main results concerning the size of the connected components of the level sets of ΨGn on the finite graphs (Gn)n≥1 satisfying (1.1)–(1.3) are the following: we show in essence that in the subcritical phase h > h, with high probability for large n, the level set EΨ≥Ghn of ΨGn only contains microscopic connected components; (1.8)
Summary
In this article we study level-set percolation of the zero-average Gaussian free field on a class of large d-regular graphs with d ≥ 3. In the accompanying paper [2] we make heavy use of this characterisation to obtain new results about φTd on Td. Our main results concerning the size of the connected components of the level sets of ΨGn on the finite graphs (Gn)n≥1 satisfying (1.1)–(1.3) are the following: we show in essence that (see Section 4, Theorem 4.1, for the precise statement) in the subcritical phase h > h , with high probability for large n, the level set EΨ≥Ghn of ΨGn only contains microscopic connected components (i.e. containing at most a logarithmic number of vertices of Gn);. The emergence of a unique giant connected component in the supercritical phase has been shown for Bernoulli bond percolation on d-regular expanders of large girth in [4] (see [15]) and for vacant-set percolation of simple random walk on exactly the same graphs (Gn)n≥1 like here in [8].
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