Let {Xt : t ∈ Z} be a collection of random variables defined on some probability space (Ω,F , P ). Mixing conditions provide one way to formalize the notion that these random variables are only weakly dependent on one another. There are many ways to define mixing; the monographs by Doukhan [8] and Bradley [5] list five classical definitions. The oldest and most general of these is the α-mixing ...

Given a countable graph G and a finite graph H, we consider Hom(G ,H) the set of graph homomorphisms from G to H and we study Gibbs measures supported on Hom(G ,H). We develop some sufficient and other necessary conditions on Hom(G ,H) for the existence of Gibbs specifications satisfying strong spatial mixing (with exponential decay rate). We relate this with previous work of Brightwell and Win...

The so-called STIT tessellations form the class of homogeneous (spatially stationary) tessellations of R which are stable under the nesting/iteration operation. In this paper, we establish the strong mixing property for these tessellations and give the optimal form of the rate of decay for the quantity |P(A ∩ Y = ∅, ThB ∩ Y = ∅) − P(A ∩ Y = ∅)P(B ∩ Y = ∅)| when A and B are two compact sets, h a...

We study the strong spatial mixing (decay of correlation) property of proper q-colorings of random graphG(n, d/n) with a fixed d. The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as G(n, d/n), an easy counterexample shows that colorings do not exh...