Random lozenge tiling at cusps and the Pearcey process
Random lozenge tiling at cusps and the Pearcey process
It has been known since Cohn-Kenyon-Propp (2000) that uniformlyÌýrandom tiling by lozenges exhibits frozen and disordered regions, which areÌýseparated by the 'arctic curve'. For a generic simply connected polygonalÌýdomain, the microscopic statistics are widely predicted to be universal,Ìýbeing one of (1) discrete sine process inside the disordered region (2) AiryÌýline ensemble around a smooth point of the curve (3) Pearcey process aroundÌýa cusp of the curve (4) GUE corner process around a tangent point of theÌýcurve. These statistics were proved years ago for special domains, usingÌýexact formulas; as for universality, much progress was made more recently.ÌýIn this talk, I will present a proof of the universality of (3), theÌýremaining open case. Our approach is via a refined comparison between tilingÌýand non-intersecting random walks, for which a new universality result ofÌýthe Pearcey process is also proved. This is joint work with Jiaoyang HuangÌýand Fan Yang.