Rectangular Matrix Additions in Low and High Temperatures
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916056416649216 |
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| author | Xu, Jiaming |
| author_facet | Xu, Jiaming |
| contents | We study the addition of two independent random $N\times M$ rectangular matrices with invariant distributions in two limiting regimes, where the parameter $β$ (inverse temperature) tends to infinity and $0$. In the low temperature regime the random singular values of the sum concentrate at deterministic points, while in the high temperature regime, we obtain a law of large numbers for the empirical measures. As a consequence, we obtain a duality between low and high temperatures. Our proof uses the type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in the high temperature limit, and degenerate to the classical and free cumulants in special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_13812 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rectangular Matrix Additions in Low and High Temperatures Xu, Jiaming Probability Mathematical Physics Combinatorics Operator Algebras We study the addition of two independent random $N\times M$ rectangular matrices with invariant distributions in two limiting regimes, where the parameter $β$ (inverse temperature) tends to infinity and $0$. In the low temperature regime the random singular values of the sum concentrate at deterministic points, while in the high temperature regime, we obtain a law of large numbers for the empirical measures. As a consequence, we obtain a duality between low and high temperatures. Our proof uses the type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in the high temperature limit, and degenerate to the classical and free cumulants in special cases. |
| title | Rectangular Matrix Additions in Low and High Temperatures |
| topic | Probability Mathematical Physics Combinatorics Operator Algebras |
| url | https://arxiv.org/abs/2303.13812 |