Publisher Summary This chapter discusses the random geometry of equilibrium phases.
Percolation will come into play here on various levels. Its concepts like clusters, open paths …

We consider upper level sets of the Gaussian free field (GFF) on Z d, for d≥ 3, above a
given real-valued height parameter h. As h varies, this defines a canonical percolation …

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 …

In this paper, we study the critical level-set of Gaussian free field (GFF) on the metric graph
Z~ d, d> 6. We prove that the one-arm probability (ie the probability of the event that the …

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 …

We provide a description and computational investigation of an efficient method to
stochastically generate realistic pore structures. Smolarkiewicz and Winter introduced this …

We consider the Gaussian free field φ on Z d, for d≥ 3, and give sharp bounds on the
probability that the radius of a finite cluster in the excursion set {φ≥ h} exceeds a large value …

We consider the bond percolation problem on a transient weighted graph induced by the
excursion sets of the Gaussian free field on the corresponding cable system. Owing to the …

We establish the sharpness of the phase transition for a wide class of Gaussian percolation
models, on Z d or R d, d≥ 2, with correlations decaying at least algebraically with exponent …

Z d to contain a unique infinite connected component for which the chemical distances are
comparable to the Euclidean distance. In addition, we show that these conditions also imply …