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SUMMARY:Percolation phase transition is nontrivial for graphs with isoperi
metric dimension higher than 3
DTSTART;VALUE=DATE-TIME:20180531T123000Z
DTEND;VALUE=DATE-TIME:20180531T133000Z
DTSTAMP;VALUE=DATE-TIME:20200926T125341Z
UID:indico-event-3496@indico.math.cnrs.fr
DESCRIPTION:Let $G$ be a bounded-degree infinite graph. The first step to
study percolation on $G$ is to prove the nontriviality of the phase transi
tion. Let $p_c$ be the critical parameter of bond percolation on $G$ to be
the infimum value of $p$ that you have an infinite cluster almost surely.
We prove that if the isoperimetric dimension of $G$ is higher than 3\, th
en $p_c(G)<1$.\n\nThe theorem settles affirmatively two conjectures of Ben
jamini and Schramm. Notably\, if $G$ is a transitive graph with super-line
ar growth\, then $p_c(G) <1$. In particular\, it implies that if $G$ is a
Cayley graph of a finitely generated group without a finite index cyclic s
ubgroup\, then $p_c(G)<1$.\n\nThe proof of the theorem starts with the exi
stence of an infinite cluster for percolation in a certain in-homogeneous
random environment governed by the Gaussian free field. Then\, by proving
a differential inequality\, we relate the existence of an infinite cluster
in percolation in the random environment to that of percolation with a pa
rameter $p<1$.\n\nThis talk is based on a joint work with H. Duminil-Copin
\, S. Goswami\, F. Severo\, and A. Yadin.\n\nhttps://indico.math.cnrs.fr/e
vent/3496/
LOCATION:UMPA 435
URL:https://indico.math.cnrs.fr/event/3496/
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