Tag - Random tilings

Random tilings of large domains have long been objects of fascination for mathematicians and physicists. What one witnesses for these systems is that the shapes of their domains have a significant impact on the local geometry of the tiling. Indeed, these shapes influence the probabilities of (microscopic) tiling patterns well inside the domain; can induce phase transitions (arctic curves) at facet edges; and can create "turning points" where the limiting behaviour of the model is discontinuous. At least for certain special shapes, there now exist powerful algebraic frameworks providing exact formulas, which can be analysed to precisely identify the limiting statistics that appear in each of these situations. This gives rise to a universality prediction, that the same limiting statistical behaviours should also arise in random tilings not only of those particular domains, but also of far more general ones. Over the past several years, there has been progress in establishing such universality statements for random tilings of the triangular lattice, or lozenge tilings. In these talks, we review some of these developments, and outline how they are obtained, by intertwining the above mentioned algebraic frameworks with analytic and probabilistic ones.

Random tilings of large domains have long been objects of fascination for mathematicians and physicists. What one witnesses for these systems is that the shapes of their domains have a significant impact on the local geometry of the tiling. Indeed, these shapes influence the probabilities of (microscopic) tiling patterns well inside the domain; can induce phase transitions (arctic curves) at facet edges; and can create "turning points" where the limiting behaviour of the model is discontinuous. At least for certain special shapes, there now exist powerful algebraic frameworks providing exact formulas, which can be analysed to precisely identify the limiting statistics that appear in each of these situations. This gives rise to a universality prediction, that the same limiting statistical behaviours should also arise in random tilings not only of those particular domains, but also of far more general ones. Over the past several years, there has been progress in establishing such universality statements for random tilings of the triangular lattice, or lozenge tilings. In these talks, we review some of these developments, and outline how they are obtained, by intertwining the above mentioned algebraic frameworks with analytic and probabilistic ones.