Mercoledì 18 aprile, ore 16:30, aula di Consiglio, Dipartimento di Matematica, La Sapienza.

Speaker: Fabio Toninelli (Université Lyon 1)

*Title: Lozenge tiling dynamics and hydrodynamic equation*

Abstract: We study a reversible continuous-time Markov dynamics on lozenge tilings of the torus, introduced by Luby et al. Single updates consist in concatenations of n elementary lozenge rotations at adjacent vertices, with rate 1/n. The dynamics can also be seen as a reversible stochastic evolution of a 2+1-dimensional interface. The dynamics is known to enjoy especially nice features: a certain Hamming distance between configurations contracts with time on average and the relaxation time of the Markov chain is diffusive. We present another remarkable feature of this dynamics, namely we derive, in the diffusive time scale, a fully explicit hydrodynamic limit equation for the height function (in the form of a non-linear parabolic PDE). The mobility coefficient μ in the equation has non-trivial but explicit dependence on the interface slope and, interestingly, is directly related to the system’s surface free energy. The derivation of the hydrodynamic limit is based on the so-called H^{-1} method due to Yau and Funaki-Spohn. Based on joint work with Benoit Laslier (Paris 7).