Cumulants in rectangular finite free probability and beta-deformed singular values

Cesar Cuenca
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Abstract

Motivated by the $(q,\gamma)$-cumulants, introduced by Xu [arXiv:2303.13812] to study $\beta$-deformed singular values of random matrices, we define the $(n,d)$-rectangular cumulants for polynomials of degree $d$ and prove several moment-cumulant formulas by elementary algebraic manipulations; the proof naturally leads to quantum analogues of the formulas. We further show that the $(n,d)$-rectangular cumulants linearize the $(n,d)$-rectangular convolution from Finite Free Probability and that they converge to the $q$-rectangular free cumulants from Free Probability in the regime where $d\to\infty$, $1+n/d\to q\in[1,\infty)$. As an application, we employ our formulas to study limits of symmetric empirical root distributions of sequences of polynomials with nonnegative roots. One of our results is akin to a theorem of Kabluchko [arXiv:2203.05533] and shows that applying the operator $\exp(-\frac{s^2}{n}x^{-n}D_xx^{n+1}D_x)$, where $s>0$, asymptotically amounts to taking the rectangular free convolution with the rectangular Gaussian distribution of variance $qs^2/(q-1)$.
矩形有限自由概率中的累积量和β变形奇异值
受许[arXiv:2303.13812]为研究随机矩阵的$\beta$变形奇异值而引入的$(q,\gamma)$累积量的启发,我们定义了度数为$d$的多项式的$(n,d)$矩形累积量,并通过基本代数操作证明了几个矩形累积量公式;证明自然地引出了公式的量子类比。我们进一步证明了$(n,d)$-矩形积线性化了来自有限自由概率的$(n,d)$-矩形卷积,并且在$d\to\infty$, $1+n/d\toq\in[1,\infty)$ 的情况下,它们收敛于来自自由概率的$q$-矩形自由积。作为应用,我们用我们的公式来研究具有负根的多项式序列的对称经验根分布的极限。我们的一个结果类似于卡布卢奇科的一个定理[arXiv:2203.05533],表明应用算子$exp(-\frac{s^2}{n}x^{-n}D_xx^{n+1}D_x)$,其中$s>0$,近似等价于取方差为$qs^2/(q-1)$的矩形高斯分布的矩形自由卷积。
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