通过多层环路方程研究 $β$-Krawtchouk 角过程的全局渐近性
Global asymptotics for $β$-Krawtchouk corners processes via multi-level loop equations
AI总结：
本文引入 $β$-Krawtchouk 角过程，利用多层环路方程证明其高度函数在 $K$ 趋于无穷时收敛于显式极限形状，且波动渐近于高斯自由场拉回，与 Wigner 矩阵一致。
摘要：
我们引入了一个双参数概率分布族，由 $β/2 = θ> 0$ 和 $K \in \mathbb{Z}_{\geq 0}$ 索引，称为 $β$-Krawtchouk 角过程。这些测度与 Jack 对称函数有关，可被视为随机矩阵理论中 $β$-角过程的可积离散化，或无限区域菱形铺砌上的非行列式测度。我们证明，当 $K$ 趋于无穷大时，这些模型的高度函数围绕一个显式极限形状集中，并证明其波动渐近地由高斯自由场的拉回描述，这与 Wigner 矩阵的情形一致。我们用于确立结果的主要工具是我们早期工作 arXiv:2108.07710 中引入的某些多层环路方程。
英文摘要：
We introduce a two-parameter family of probability distributions, indexed by $β/2 = θ> 0$ and $K \in \mathbb{Z}_{\geq 0}$, that are called $β$-Krawtchouk corners processes. These measures are related to Jack symmetric functions, and can be thought of as integrable discretizations of $β$-corners processes from random matrix theory, or alternatively as non-determinantal measures on lozenge tilings of infinite domains. We show that as $K$ tends to infinity the height function of these models concentrates around an explicit limit shape, and prove that its fluctuations are asymptotically described by a pull-back of the Gaussian free field, which agrees with the one for Wigner matrices. The main tools we use to establish our results are certain multi-level loop equations introduced in our earlier work arXiv:2108.07710.
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