The Discrete Tacnode Kernel: a Universal and a Master Kernel
Pierre van Moerbeke (Brandeis University; Université Catholique de Louvain)
MSRI: Simons Auditorium, Online/Virtual
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Consider random tilings of nonconvex polygons and skew-Aztec rectangles; tilings by lozenges or dominos. For large sizes of the domain and upon using an appropriate scaling in certain regions, one finds split tacnodes at the boundary of liquid regions with two distinct adjacent frozen phases descending into the tacnode. Zooming about such split tacnodes, one sees filaments appearing between liquid patches evolving in a bricklike sea of dimers of another type. The random fluctuations near this tacnode are governed asymptotically by the discrete tacnode kernel, providing strong evidence that this kernel is a universal discrete-continuous limiting kernel occurring naturally whenever we have doubly interlacing patterns. For nonconvex polygons the problem is intimately related to the number of skew-Young tableaux of a given shape and for skew-Aztec rectangles it involves the inversion of a finite Toeplitz matrix. It is believed that the discrete tacnode kernel also enjoys the status of a master kernel, from which all existing kernels can be derived by appropriate double scaling. This is joint work with Mark Adler and Kurt Johansson.