Random Lozenge tiling at cusp and the Pearcey process 

It has been known (since Cohn-Kenyon-Propp, 00) that in a uniform random Lozenge tiling, there are frozen regions and disordered regions, separated by the ‘arctic curve’. For a generic simply connected polygonal domain, the microscopic statistics are widely predicted (back to Okounkov-Reshetikhin, 07) to be universal, being one of (1) discrete sine kernel process inside the disordered region (2) Airy line ensemble around a smooth point of the arctic curve (3) Pearcey process around a cusp point of the arctic curve (4) GUE corner process around a tangent point of the arctic curve. For special domains these were proved years ago; and recently, much progress has been made in establishing universality, with (3) being the remaining open case. I will present a proof of it via a refined comparison between tiling and non-intersecting random walks, for which a new universality of the Pearcey process around a bifurcation point is also shown.

This is based on joint work with Jiaoyang Huang and Fan Yang.