论文标题
雅各比和库奇随机矩阵合奏之间的关系
Relations between moments for the Jacobi and Cauchy random matrix ensembles
论文作者
论文摘要
我们概述了$β$增强的密度相对于jacobi重量$（1-x）^a（1+x）^a（1+x）^b $在间隔$（ - 1,1）$上支撑的$（1,1,1）$与cauchy stript $（1-- \ m m mathrm {i}）这意味着后者的密度满足了第三级的线性微分方程的$β= 2 $，而第五级的$β= 1 $和$ 4 $的五级，类似物的类似物已经以jacobi重量$ x^a（1-x a（1-x）^b $（$（0,1）$（0,1）$所闻名。我们专注于$ a = b $（jacobi重量$（-1,1）$）和$η$ real（cauchy weight），因为密度是一个均匀函数，并且微分方程简化了。从微分方程中，可以在$（-1,1）$和/或Cauchy重量的时刻支撑的Jacobi重量的时刻获得复发。特别注意$β= 2 $，在对称情况下的$（-1,1）$的雅各比的重量$ a = b $，这与Assiotis等人最近获得的结果保持一致。同样，对于$η=-β（n-1）/2-1-α$的对称cauchy重量，在适当地缩放$α$与$ n $成比例的$α$之后，我们使用微分方程在$ 1/n^2 $（$ 1/n $）中以$β= 2 $（$β= 2 $β= 1,4 $）计算$ 1/n^2 $（$ 1/n $）。
We outline a relation between the densities for the $β$-ensembles with respect to the Jacobi weight $(1-x)^a(1+x)^b$ supported on the interval $(-1,1)$ and the Cauchy weight $(1-\mathrm{i}x)^η(1+\mathrm{i}x)^{\barη}$ by appropriate analytic continuation. This has the consequence of implying that the latter density satisfies a linear differential equation of degree three for $β=2$, and of degree five for $β=1$ and $4$, analogues of which are already known for the Jacobi weight $x^a(1-x)^b$ supported on $(0,1)$. We concentrate on the case $a=b$ (Jacobi weight on $(-1,1)$) and $η$ real (Cauchy weight) since the density is then an even function and the differential equations simplify. From the differential equations, recurrences can be obtained for the moments of the Jacobi weight supported on $(-1,1)$ and/or the moments of the Cauchy weight. Particular attention is paid to the case $β=2$ and the Jacobi weight on $(-1,1)$ in the symmetric case $a=b$, which in keeping with a recent result obtained by Assiotis et al.~for the $β=2$ case of the symmetric Cauchy weight (parameter $η$ real), allows for an explicit solution of the recurrence in terms of particular continuous Hahn polynomials. Also for the symmetric Cauchy weight with $η=-β(N-1)/2-1-α$, after appropriately scaling $α$ proportional to $N$, we use differential equations to compute terms in the $1/N^2$ ($1/N$) expansion of the resolvent for $β=2$ ($β=1,4$).
