Cumulants in rectangular finite free probability and beta-deformed singular values
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914493547675648 |
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| author | Cuenca, Cesar |
| author_facet | Cuenca, Cesar |
| contents | 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 $d\to\infty$, $1+n/d\to q\in[1,\infty)$. 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 $qs^2/(q-1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04305 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cumulants in rectangular finite free probability and beta-deformed singular values Cuenca, Cesar Combinatorics Probability 05A19 (Primary) 60C05, 05A30 (Secondary) 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 $d\to\infty$, $1+n/d\to q\in[1,\infty)$. 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 $qs^2/(q-1)$. |
| title | Cumulants in rectangular finite free probability and beta-deformed singular values |
| topic | Combinatorics Probability 05A19 (Primary) 60C05, 05A30 (Secondary) |
| url | https://arxiv.org/abs/2409.04305 |