Abstract: The Gaussian free field (GFF) on the metric graph, introduced by Titus Lupu, is a natural extension of the discrete GFF. Its level-set (i.e., the collection of points where the GFF exceeds a given threshold) is a percolation model with favorable properties and strong connections to numerous models in statistical physics (including loop soups, random interlacements, etc). In particular, on the metric graph, the isomorphism theorem states that the sign clusters of the GFF share the same distribution as the clusters of the loop soup with intensity 1/2.
This talk will introduce our recent progress in establishing its critical one-arm exponents, volume exponents, quasi-multiplicativity and incipient infinite clusters. Some of these results lead to new conjectures in Bernoulli percolation. In addition to presenting the results, this talk also aims to demonstrate how the isomorphism theorem enables us to seamlessly convert between the GFF and the loop soup, thereby deepening our understanding of both models.
This is a joint work with Jian Ding (Peking University).
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