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| Format: | Preprint |
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2021
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| Online Access: | https://arxiv.org/abs/2111.00421 |
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| _version_ | 1866908903485210624 |
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| author | Narayanan, Hariharan Sheffield, Scott |
| author_facet | Narayanan, Hariharan Sheffield, Scott |
| contents | Suppose $α, β$ are Lipschitz strongly concave functions from $[0, 1]$ to $\mathbb{R}$ and $γ$ is a concave function from $[0, 1]$ to $\mathbb{R}$, such that $α(0) = γ(0) = 0$, and $α(1) = β(0) = 0$ and $β(1) = γ(1) = 0.$ For an $n \times n$ Hermitian matrix $W$, let $spec(W)$ denote the vector in $\mathbb{R}^n$ whose coordinates are the eigenvalues of $W$ listed in non-increasing order. Let $λ= \partial^- α$, $μ= \partial^- β$ on $(0, 1]$ and $ν= \partial^- γ,$ at all points of $(0, 1]$, where $\partial^-$ is the left derivative, which is monotonically decreasing. Let $λ_n(i) := n^2(α(\frac{i}{n})-α(\frac{i-1}{n}))$, for $i \in [n]$, and similarly, $μ_n(i) := n^2(β(\frac{i}{n})-β(\frac{i-1}{n}))$, and $ν_n(i) := n^2(γ(\frac{i}{n})-γ(\frac{i-1}{n}))$.
Let $X_n, Y_n$ be independent random Hermitian matrices from unitarily invariant distributions with spectra $λ_n$, $μ_n$ respectively. We define norm $\|\cdot\|_\mathcal{I}$ to correspond in a certain way to the sup norm of an antiderivative. For suitable $λ$ and $μ$, we prove that the following limit exists. \begin{equation} \lim\limits_{n \rightarrow \infty}\frac{\ln \mathbb{P}\left[\|spec(X_n + Y_n) - ν_n\|_{\mathcal{I}} < n^2 ε\right]}{n^2}.\end{equation} We interpret this limit in terms of the surface tension $σ$ of continuum limits of the discrete hives defined by Knutson and Tao. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_00421 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Large deviations for random hives and the spectrum of the sum of two random matrices Narayanan, Hariharan Sheffield, Scott Probability Suppose $α, β$ are Lipschitz strongly concave functions from $[0, 1]$ to $\mathbb{R}$ and $γ$ is a concave function from $[0, 1]$ to $\mathbb{R}$, such that $α(0) = γ(0) = 0$, and $α(1) = β(0) = 0$ and $β(1) = γ(1) = 0.$ For an $n \times n$ Hermitian matrix $W$, let $spec(W)$ denote the vector in $\mathbb{R}^n$ whose coordinates are the eigenvalues of $W$ listed in non-increasing order. Let $λ= \partial^- α$, $μ= \partial^- β$ on $(0, 1]$ and $ν= \partial^- γ,$ at all points of $(0, 1]$, where $\partial^-$ is the left derivative, which is monotonically decreasing. Let $λ_n(i) := n^2(α(\frac{i}{n})-α(\frac{i-1}{n}))$, for $i \in [n]$, and similarly, $μ_n(i) := n^2(β(\frac{i}{n})-β(\frac{i-1}{n}))$, and $ν_n(i) := n^2(γ(\frac{i}{n})-γ(\frac{i-1}{n}))$. Let $X_n, Y_n$ be independent random Hermitian matrices from unitarily invariant distributions with spectra $λ_n$, $μ_n$ respectively. We define norm $\|\cdot\|_\mathcal{I}$ to correspond in a certain way to the sup norm of an antiderivative. For suitable $λ$ and $μ$, we prove that the following limit exists. \begin{equation} \lim\limits_{n \rightarrow \infty}\frac{\ln \mathbb{P}\left[\|spec(X_n + Y_n) - ν_n\|_{\mathcal{I}} < n^2 ε\right]}{n^2}.\end{equation} We interpret this limit in terms of the surface tension $σ$ of continuum limits of the discrete hives defined by Knutson and Tao. |
| title | Large deviations for random hives and the spectrum of the sum of two random matrices |
| topic | Probability |
| url | https://arxiv.org/abs/2111.00421 |