We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved by Aggarwal‐Huang and Aggarwal‐Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with nonintersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formulas.

中文翻译：

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多边形菱形拼贴尖点处的皮尔西普适性

我们研究一般简单连接多边形的均匀随机菱形平铺。在对于多边形形状可能通用的技术假设下，我们表明北极曲线尖点周围的局部统计数据收敛于皮尔西过程。这验证了尖点情况下边缘统计的广泛预测的普遍性。与 Aggarwal-Huang 和 Aggarwal-Gorin 证明的平滑和相切情况一起，这些被认为是通用多边形中可能出现的三种类型的边统计。我们的证明是通过随机平铺与非相交伯努利随机游走 (NBRW) 的局部耦合。为了利用这种耦合，我们为尖点周围的平铺高度函数建立了最佳浓度估计。作为另一个步骤，也是潜在独立兴趣的结果，我们通过对行列式公式的仔细渐近分析，表明当初始配置由具有适当密度增长的两部分组成时，尖点周围的 NBRW 局部统计数据收敛于 Pearcey 过程。