Sign cluster geometry of the Gaussian free field

Jeudi 4 avril 15:45-16:45 - Pierre-François Rodriguez - IHES

Résumé : We consider the Gaussian free field on a large class of transient weighted graphs G, and show that its sign clusters contain an infinite connected component. In fact, we prove 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. 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.

Sign cluster geometry of the Gaussian free field  Version PDF