交替符号矩阵顶路最大值的GOE涨落
GOE fluctuations for the maximum of the top path in alternating sign matrices
摘要：
六顶点模型是二维冰统计力学中一个重要的玩具模型，其自然参数为$Δ$。当$Δ= 0$，所谓的自由费米子点，该模型是在自然对应的多米诺骨牌镶嵌的阿兹特克钻石。虽然这个模型对所有$Δ$都是可积的，但在理解它对其他值的标度极限的统计方面进展甚微。在这项工作中，我们专注于在$Δ= 1/2$，它对应于交替符号矩阵（ASM）的区域壁边界条件的六顶点模型。我们认为，在一个高度函数表示的ASM的水平线。我们发现，最高水平线的最大值均匀随机ASMs具有GOE特雷西-Widom分布后适当的重新标度。在我们的证明中的一个关键因素是Zeilberger的ASM猜想的证明。据我们所知，这是畴壁六顶点模型在非自由费米子情形下第一个远离切点的边缘涨落结果。
英文摘要：
The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter $Δ$. When $Δ= 0$, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all $Δ$, there has been very little progress in understanding its statistics in the scaling limit for other values. In this work, we focus on the six-vertex model with domain wall boundary conditions at $Δ= 1/2$, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We show that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy--Widom distribution after appropriate rescaling. A key ingredient in our proof is Zeilberger's proof of the ASM conjecture. As far as we know, this is the first edge fluctuation result away from the tangency points for the domain-wall six-vertex model when we are not in the free fermion case.
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