Freud随机矩阵系综中的弱与强约束及间隙概率
Weak and strong confinement in the Freud random matrix ensemble and gap probabilities
AI总结：
本文研究了Freud随机矩阵系综在零点附近特征值的局部行为，发现在临界值$\beta=1$处存在相变，确定了弱约束范围下大间隙概率的前两项及强约束下多重积分渐近展开的乘积常数。
摘要：
Freud随机矩阵系综是对应于实数轴上权重$\exp(-n |x|^\beta)$（$\beta>0$）的酉不变系综。我们考虑特征值在零点附近的局部行为，该行为在$\beta$上表现出相变。若$\beta\ge 1$，其由标准正弦过程描述。在临界值$\beta=1$以下，其由依赖于$\beta$值的过程描述，我们确定了其中大间隙概率的前两项。此即所谓弱约束范围$0<\beta<1$，对应于具有不定矩问题的Freud权重。我们还求出了$\beta\ge 1$时Freud多重积分渐近展开中的乘积常数。
英文摘要：
The Freud ensemble of random matrices is the unitary invariant ensemble corresponding to the weight $\exp(-n |x|^β)$, $β>0$, on the real line. We consider the local behaviour of eigenvalues near zero, which exhibits a transition in $β$. If $β\ge 1$, it is described by the standard sine process. Below the critical value $β=1$, it is described by a process depending on the value of $β$, and we determine the first two terms of the large gap probability in it. This so called weak confinement range $0<β<1$ corresponds to the Freud weight with the indeterminate moment problem. We also find the multiplicative constant in the asymptotic expansion of the Freud multiple integral for $β\ge 1$.
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