扰动β-角过程
Perturbed Beta Corners Process
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中文总结 AI 辅助
该研究推广经典β-角过程为扰动β-角过程，通过多元贝塞尔函数拓展至所有β>0，分析其结晶行为，在两种 regime 中证明极限定理，发现固定扰动与线性增长扰动下的不同格点与涨落特性。
中文摘要 AI 辅助
我们引入并研究扰动β-角过程，它是经典β-角过程的一种变形。后者是实数交错数组上的概率测度，对于经典值β=1,2,4，描述具有对应GOE/GUE/GSE对称性的高斯随机矩阵主子矩阵的特征值联合分布。对于经典β，该扰动过程由向GOE/GUE/GSE矩阵添加确定性对角矩阵A=diag(a₁,...,a_N)产生，所得随机矩阵的特征值对称依赖于a₁,...,a_N，但该对称性并不延伸至整个角过程。我们通过多元贝塞尔函数将该构造推广至所有β>0，并分析所得交错数组的结晶（β→∞），在两种 regime 中证明了大数定律和中心极限定理。对于固定扰动，特征值排斥占主导，数组结晶于与Gorin和Marcus（arXiv:1706.07393）研究的未扰动情形相同的多项式导数根格点，具有相同的离散高斯自由场涨落。第二种 regime 中出现新现象，此时a_i随β线性增长，外部源在主导阶与排斥力竞争，数组冻结于由耦合最优性方程组表征的变形格点，涨落由附着于该变形格点的同一离散高斯自由场支配。在最后坐标为单个尖峰的特殊情形下，变形格点方程解耦，此时变形格点可通过应用一个平移导数D_c f = f' + cf，再经迭代普通导数显式得到。
英文摘要
We introduce and study the perturbed $β$-corners process, a deformation of the classical $β$-corners process. The latter is a probability measure on interlacing arrays of real numbers that, for the classical values $β=1,2,4$, describes the joint distribution of eigenvalues of principal submatrices of a Gaussian random matrix with the corresponding GOE/GUE/GSE symmetry. For classical $β$, the perturbed process arises from adding a deterministic diagonal matrix $A=diag(a_1,...,a_N)$ to a GOE/GUE/GSE matrix. The eigenvalues of the resulting random matrix depend symmetrically on $a_1,...,a_N$, but this symmetry does not extend to the whole corners process. We extend the construction to all $β>0$ via multivariate Bessel functions, and analyze the crystallization ($β\to\infty$) of the resulting interlacing array, proving a law of large numbers and a central limit theorem in two regimes. For fixed perturbation, the eigenvalue repulsion dominates, and the array crystallizes on the same lattice of polynomial-derivative roots, with the same discrete Gaussian free field fluctuations, as in the unperturbed case treated by Gorin and Marcus (arXiv:1706.07393). New phenomena appear in the second regime, where the $a_i$'s grow linearly in $β$. Then the external source competes with the repulsion at leading order. Here the array freezes on a deformed lattice characterized by a coupled system of optimality equations. The fluctuations are governed by the same discrete Gaussian free field, now attached to the deformed lattice. The deformed lattice equations decouple in the special case of a single spike in the last coordinate. Then the deformed lattice is obtained explicitly by applying one shifted derivative $D_c f = f' + cf$ followed by iterated ordinary derivatives.
