Graph homomorphisms from the Z d lattice to Z are functions on Z d whose gradients equal one in absolute value. These functions are the height functions corresponding to proper 3-colorings of Z d and, in two dimensions, corresponding to the 6-vertex model (square ice). We consider the uniform model, obtained by sampling uniformly such a graph homomorphism subject to boundary conditions. Our main result is that the model delocalizes in two dimensions, having no translation-invariant Gibbs measures. Additional results are obtained in higher dimensions and include the fact that every Gibbs measure which is ergodic under even translations is extremal and that these Gibbs measures are stochastically ordered.

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Abstract. Domino tileability is a classical problem in Discrete Geometry, famously solved by Thurston for simply connected regions in nearly linear time in the area. In this paper, we improve upon Thurston's height function approach to a nearly linear time in the perimeter.

Abstract. We study the critical parameter u * of random interlacements percolation (introduced by A.S Sznitman in [Szn10]) on a Galton-Watson tree conditioned on the non-extinction event. Starting from the previous work of A. Teixeira in [Tei09], we show that, for a given law of a Galton-Watson tree, the value of this parameter is a.s. constant and non-trivial. We also characterize this value as the solution of a certain equation.

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We develop a new robust technique to deduce variational principles for non-integrable discrete systems. To illustrate this technique, we show the existence of a variational principle for graph homomorphisms from Z m to a d-regular tree. This seems to be the first non-trivial example of a variational principle in a non-integrable model. Instead of relying on integrability, the technique is based on a discrete Kirszbraun theorem and a concentration inequality obtained through the dynamic of the model. Using those two results, we also obtain the existence of a continuum of translation-invariant, ergodic, gradient Gibbs measures for graph homomorphisms from Z m to a regular tree.

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