标题： 随机矩阵轨道的结晶化 显示英文标题

标题： Crystallization of random matrix orbits

摘要： 实数/复数/四元数（对应于$\beta=1,2,4$）矩阵的特征值的三种运算，通过截断主角、相加和相乘得到，可以通过相关的特殊函数推广到一般值$\beta>0$。 我们证明这些运算的$\beta\to\infty$极限分别导致有限自由投影、加法卷积和乘法卷积。 对于截断角来说，极限是最透明的，其中具有固定特征值的均匀随机一般$\beta$自伴矩阵的主角特征值的联合分布已知为$\beta$-角过程。 我们证明当$\beta\to\infty$时，这些特征值会在单个多项式的导数的所有根的不规则格子上结晶。 在二阶情况下，我们观察到一种离散高斯自由场（dGFF）放在该格子之上，这为（连续）高斯自由场为何支配随机矩阵系综的全局渐近行为提供了新的解释。 显示英文摘要

摘要： Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta=1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices can be extrapolated to general values of $\beta>0$ through associated special functions. We show that $\beta\to\infty$ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta$ self-adjoint matrix with fixed eigenvalues is known as $\beta$-corners process. We show that as $\beta\to\infty$ these eigenvalues crystallize on the irregular lattice of all the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field (dGFF) put on top of this lattice, which provides a new explanation of why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.