Dynamics of random interfaces and tilings of the plane

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I will start from the following apparently simple question, motivated by non-equilibrium statistical physics. Given an integer L, color the points "x" of Z^d black for |x|How does the set of black sites evolve macroscopically, as L and the time tend to infinity?
I will show that this question is actually quite challenging and it is related to several interesting mathematical objects: (i) to anisotropic curve shortening flows in the d=2 case, (ii) to random tilings of the plane in the d=3 case; and (iii) to the computation of the running time of probabilistic Markov Chain Monte Carlo sampling algorithms on complex combinatorial structures.

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