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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.18550 |
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| _version_ | 1866908532406747136 |
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| author | Bothner, Thomas Shepherd, Toby |
| author_facet | Bothner, Thomas Shepherd, Toby |
| contents | We study invariant random matrix ensembles \begin{equation*}
\mathbb{P}_n(d M)=Z_n^{-1}\exp(-n\,tr(V(M)))\,d M \end{equation*} defined on complex Hermitian matrices $M$ of size $n\times n$, where $V$ is real analytic such that the underlying density of states is one-cut regular. Considering the average \begin{equation*}
E_n[ϕ;λ,α,β]:=\mathbb{E}_n\bigg(\prod_{\ell=1}^n\big(1-ϕ(λ_{\ell}(M))\big)ω_{αβ}(λ_{\ell}(M)-λ)\bigg),\ \ \ \ \ ω_{αβ}(x):=|x|^α\begin{cases}1,&x<0\\ β,&x\geq 0\end{cases}, \end{equation*} taken with respect to the above law and where $ϕ$ is a suitable test function, we evaluate its large-$n$ asymptotic assuming that $λ$ lies within the soft edge boundary layer, and $(α,β)\in\mathbb{R}\times\mathbb{C}$ satisfy $α>-1,β\notin(-\infty,0)$. Our results are obtained by using Riemann-Hilbert problems for orthogonal polynomials and integrable operators and they extend previous results of Forrester and Witte \cite{FW} that were obtained by an application of Okamoto's $τ$-function theory. A key role throughout is played by distinguished solutions to the Painlevé-XXXIV equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18550 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universality for random matrices with an edge spectrum singularity Bothner, Thomas Shepherd, Toby Mathematical Physics Classical Analysis and ODEs Exactly Solvable and Integrable Systems Primary 47B35, Secondary 45B05, 30E25, 34E05 We study invariant random matrix ensembles \begin{equation*} \mathbb{P}_n(d M)=Z_n^{-1}\exp(-n\,tr(V(M)))\,d M \end{equation*} defined on complex Hermitian matrices $M$ of size $n\times n$, where $V$ is real analytic such that the underlying density of states is one-cut regular. Considering the average \begin{equation*} E_n[ϕ;λ,α,β]:=\mathbb{E}_n\bigg(\prod_{\ell=1}^n\big(1-ϕ(λ_{\ell}(M))\big)ω_{αβ}(λ_{\ell}(M)-λ)\bigg),\ \ \ \ \ ω_{αβ}(x):=|x|^α\begin{cases}1,&x<0\\ β,&x\geq 0\end{cases}, \end{equation*} taken with respect to the above law and where $ϕ$ is a suitable test function, we evaluate its large-$n$ asymptotic assuming that $λ$ lies within the soft edge boundary layer, and $(α,β)\in\mathbb{R}\times\mathbb{C}$ satisfy $α>-1,β\notin(-\infty,0)$. Our results are obtained by using Riemann-Hilbert problems for orthogonal polynomials and integrable operators and they extend previous results of Forrester and Witte \cite{FW} that were obtained by an application of Okamoto's $τ$-function theory. A key role throughout is played by distinguished solutions to the Painlevé-XXXIV equation. |
| title | Universality for random matrices with an edge spectrum singularity |
| topic | Mathematical Physics Classical Analysis and ODEs Exactly Solvable and Integrable Systems Primary 47B35, Secondary 45B05, 30E25, 34E05 |
| url | https://arxiv.org/abs/2411.18550 |