2024/09/06 by Cuenca, Cesar · 2 citations
#05A19 (Primary) 60C05 #05A30 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2409.04305
Motivated by the (q,γ)-cumulants, introduced by Xu [arXiv:2303.13812] to study β-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→∞, 1+n/d→ q∈[1,∞). 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(-(s2)/(n)x-nDxxn+1Dx), where s>0, asymptotically amounts to taking the rectangular free convolution with the rectangular Gaussian distribution of variance qs2/(q-1).