We introduce a model of random interlacements made of a countable collection of doubly
infinite paths on ℤ d, d≥ 3. A nonnegative parameter u measures how many trajectories …

It was shown by Le Jan that the occupation field of a Poisson ensemble of Markov loops
(“loop soup”) of parameter 12 associated to a transient symmetric Markov jump process on a …

We consider level-set percolation for the Gaussian free field on Z^ d, d≥ 3, and prove that,
as h varies, there is a non-trivial percolation phase transition of the excursion set above level …

The classical spin O (n) model is a model on ad-dimensional lattice in which a vector on the
(n-1)-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites …

We consider a percolation model, the vacant set V u V^u of random interlacements on Z d
Z^d, d⩾ 3 d\geqslant3, in the regime of parameters u> 0 u>0 in which it is strongly …

Graph homomorphisms from the Z^ d Z d lattice to ZZ are functions on Z^ d Z d whose
gradients equal one in absolute value. These functions are the height functions …

We consider a percolation model, the vacant set $\mathcal {V}^ u $ of random
interlacements on $\mathbb {Z}^ d $, $ d\geq 3$, in the regime of parameters $ u> 0$ in …

We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasi-
transitive graph has a phase in which there are infinitely many infinite clusters, verifying a …

We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more
generally graphs whose automorphism group has a nonunimodular quasi-transitive …

We continue the study of infinite geodesics in planar first-passage percolation, pioneered by
Newman in the mid 1990s. Building on more recent work of Hoffman, and Damron and …