论文标题
弗洛伊德随机矩阵合奏和差距概率的弱和强限制
Weak and strong confinement in the Freud random matrix ensemble and gap probabilities
论文作者
论文摘要
随机矩阵的弗洛伊德合奏是对应于重量$ \ exp（-n | x |^β）$，$β> 0 $的单位不变的合奏。我们考虑零附近特征值的局部行为，该行为在$β$中表现出过渡。如果$β\ ge 1 $，则由标准正弦过程描述。低于临界值$β= 1 $，它取决于一个过程，具体取决于$β$的值，我们确定其中大间隙概率的前两个术语。所谓的弱限制范围$ 0 <β<1 $ $对应于弗洛伊德的重量，而不确定力矩问题。我们还发现弗洛伊德多个积分的渐近扩展中的乘法常数以$β\ ge 1 $。
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 $β\ge 1$.
