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Mathematics > Probability
Title: Correlation functions between singular values and eigenvalues
(Submitted on 28 Mar 2024 (v1), last revised 11 Apr 2024 (this version, v2))
Abstract: Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size we aim at finding the induced probability measure on $j$ eigenvalues and $k$ singular values that we coin $j,k$-point correlation measure. We fully derive all $j,k$-point correlation measures in the simplest cases for matrices of size $n=1$ and $n=2$. For $n>2$, we find a general formula for the $1,1$-point correlation measure. This formula reduces drastically when assuming the singular values are drawn from a polynomial ensemble, yielding an explicit formula in terms of the kernel corresponding to the singular value statistics. These expressions simplify even further when the singular values are drawn from a P\'{o}lya ensemble and extend known results between the eigenvalue and singular value statistics of the corresponding bi-unitarily invariant ensemble.
Submission history
From: Matthias Allard [view email][v1] Thu, 28 Mar 2024 05:42:07 GMT (945kb,D)
[v2] Thu, 11 Apr 2024 01:30:47 GMT (946kb,D)
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