The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2021
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| _version_ | 1866908469631647744 |
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| author | Mingo, James A. Vazquez-Becerra, Josue |
| author_facet | Mingo, James A. Vazquez-Becerra, Josue |
| contents | Let $X_N$ be a $N \times N$ real Wishart random matrix with aspect ratio $M/N$. The limit eigenvalue distribution of $X_N$ is the Marchenko-Pastur law with parameter $c = \lim_N M/N$. The limit moments $\{m_n\}_n$ are given by $m_n = \sum_π c^{\#(π)}$ where the sum runs over $NC(n)$. Let $m_n'$ be the limit of $N( \mathrm{E}(\mathrm {tr}(X_N^n)) - m_n)$. These are the asymptotic infinitesimal moments of a real Wishart matrix. We show that $m'_n$ can be written as a sum over planar diagrams with two terms, $\sum_π c'(\#(π) -1) c^{\#(π)-1}$, and $\sum_{π\in S_{NC}^δ(n,-n)} c^{\#(π)/2}$, where $S_{NC}^δ(n,-n)$ is a set of non-crossing annular permutations satisfying a symmetry condition. Moreover we present a recursion formula for the second term which is related to one for higher order freeness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_15231 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix Mingo, James A. Vazquez-Becerra, Josue Probability Combinatorics Operator Algebras 60B20, 15B52, 46L54 Let $X_N$ be a $N \times N$ real Wishart random matrix with aspect ratio $M/N$. The limit eigenvalue distribution of $X_N$ is the Marchenko-Pastur law with parameter $c = \lim_N M/N$. The limit moments $\{m_n\}_n$ are given by $m_n = \sum_π c^{\#(π)}$ where the sum runs over $NC(n)$. Let $m_n'$ be the limit of $N( \mathrm{E}(\mathrm {tr}(X_N^n)) - m_n)$. These are the asymptotic infinitesimal moments of a real Wishart matrix. We show that $m'_n$ can be written as a sum over planar diagrams with two terms, $\sum_π c'(\#(π) -1) c^{\#(π)-1}$, and $\sum_{π\in S_{NC}^δ(n,-n)} c^{\#(π)/2}$, where $S_{NC}^δ(n,-n)$ is a set of non-crossing annular permutations satisfying a symmetry condition. Moreover we present a recursion formula for the second term which is related to one for higher order freeness. |
| title | The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix |
| topic | Probability Combinatorics Operator Algebras 60B20, 15B52, 46L54 |
| url | https://arxiv.org/abs/2112.15231 |