随机矩阵顶特征值的截断线性统计量
Truncated linear statistics associated with the top eigenvalues of random matrices
摘要：
给定一个由特征值$P(λ_1,\ldots,λ_N)$的联合概率分布表征的不变随机矩阵系综，许多重要问题都与特征值$L=\sum_{i=1}^Nf(λ_i)$的线性统计研究有关，其中$f(λ)$是已知函数。我们在这里研究截断线性统计的总和被限制到$N_1<N$最大的特征值：$\tilde{L}=\sum_{i=1}^{N_1}f(λ_i)$。基于对一维起伏界面统计物理的分析，我们考虑了随机矩阵的Laguerre系综与$f(λ)=\sqrtλ$的情形。利用库仑气体技术研究了固定$N_1/N$的$N\to\infty$的极限。我们表明，$\tilde{L}=\sum_{i=1}^{N_1}f(λ_i)$是固定的约束驱动器的一个无限阶相变在底层的库仑气体。这种转变对应于气体密度的变化，从两个不相交区间上定义的密度到单个区间。在后一种情况下，密度在体块内部呈现对数发散。假设$f(λ)$是单调的，我们表明，这些功能出现任何随机矩阵合奏和截断线性统计量，这使得这里描述的情况下，强大的和普遍的。
英文摘要：
Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues $P(λ_1,\ldots,λ_N)$, many important questions have been related to the study of linear statistics of eigenvalues $L=\sum_{i=1}^Nf(λ_i)$, where $f(λ)$ is a known function. We study here truncated linear statistics where the sum is restricted to the $N_1<N$ largest eigenvalues: $\tilde{L}=\sum_{i=1}^{N_1}f(λ_i)$. Motivated by the analysis of the statistical physics of fluctuating one-dimensional interfaces, we consider the case of the Laguerre ensemble of random matrices with $f(λ)=\sqrtλ$. Using the Coulomb gas technique, we study the $N\to\infty$ limit with $N_1/N$ fixed. We show that the constraint that $\tilde{L}=\sum_{i=1}^{N_1}f(λ_i)$ is fixed drives an infinite order phase transition in the underlying Coulomb gas. This transition corresponds to a change in the density of the gas, from a density defined on two disjoint intervals to a single interval. In this latter case the density presents a logarithmic divergence inside the bulk. Assuming that $f(λ)$ is monotonous, we show that these features arise for any random matrix ensemble and truncated linear statitics, which makes the scenario described here robust and universal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
