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We prove that for Bernoulli percolation on $mathbb{Z}^d$, $dgeq 2$, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that $p_c^{bond} <1/2$ for certain families of triangulations for which Benjamini & Schramm conjectured that $p_c^{site} leq 1/2$.
We prove that for Bernoulli bond percolation on $mathbb{Z}^d$, $dgeq 2$ the percolation density is an analytic function of the parameter in the supercritical interval $(p_c,1]$. This answers a question of Kesten from 1981.
We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model
The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $beta>0$ per edge. It arises as the $qto 0$ limit with $p=beta q$ of the $q$-state random cluster model. We prove that
We consider instances of long-range percolation on $mathbb Z^d$ and $mathbb R^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $sin (d,2d)$, and study the asymptotic of the graph-theoretical (a.
A well-known question in the planar first-passage percolation model concerns the convergence of the empirical distribution along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on $mathbb{Z}^2$ with