A SINGULAR TOEPLITZ DETERMINANT AND THE DISCRETE TACNODE KERNEL FOR SKEW-AZTEC RECTANGLES

• 成果类型: Article
• 署名作者: Adler, Mark; Johansson, Kurt; van Moerbeke, Pierre
• 署名单位: Brandeis University; Royal Institute of Technology; Universite Catholique Louvain; Brandeis University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1708
• 发表日期: 2022
• 页码: 1234-1294
• 关键词: tilings; fluctuations; asymptotics; pearcey; dimers

摘要

Random tilings of geometrical shapes with dominos or lozenges have been a rich source of universal statistical distributions. This paper deals with domino tilings of checker board rectangular shapes such that the top two and bottom two adjacent squares have the same orientation and the two most left and two most right ones as well. It forces these so-called skew-Aztec rectangles to have cuts on either side. For large sizes of the domain and upon an appropriate scaling of the location of the cuts, one finds split tacnodes between liquid regions with two distinct adjacent frozen phases descending into the tacnode. Zooming about such split tacnodes, filaments appear between the liquid patches evolving in a bricklike sea of dimers of another type. This work shows that the random fluctuations in a neighborhood of the split 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. The analysis involves the inversion of a singular Toeplitz matrix which leads to considerable difficulties.