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Isomorphism theorems and the sign cluster geometry of the Gaussian free field

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If you have a question about this talk, please contact Perla Sousi.

We consider the Gaussian free field (GFF) on a large class of transient weighted graphs G, and prove that its sign clusters contain an infinite connected component. In fact, we show that the sign clusters fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs G belonging to this class include cases in which the random walk on G exhibits anomalous diffusive behavior. Among other things, our proof exploits a certain relation (isomorphism theorem) relating the GFF to random interlacements, which form a Poissonian soup of bi-infinite random walk trajectories. Our findings also imply the existence of a nontrivial percolating regime for the vacant set of random interlacements on G.

Based on joint work with A. Prévost and A. Drewitz.

This talk is part of the Probability series.

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