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Journal Articles Journal of Theoretical Probability Year : 2022

## Non-Hermitian random matrices with a variance profile (II): properties and examples

Nicholas Cook
• Function : Author
Walid Hachem
Jamal Najim
• Function : Author
• PersonId : 942088
David Renfrew
• Function : Author
• PersonId : 1059400

#### Abstract

For each $n$, let $A_n=(\sigma_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d.\ centered entries of unit variance. In the companion article \cite{cook2018non}, we considered the empirical spectral distribution $\mu_n^Y$ of the rescaled entry-wise product $Y_n = \frac 1{\sqrt{n}} A_n\odot X_n = \left(\frac1{\sqrt{n}} \sigma_{ij}X_{ij}\right)$ and provided a deterministic sequence of probability measures $\mu_n$ such that the difference $\mu^Y_n - \mu_n$ converges weakly in probability to the zero measure. A key feature in \cite{cook2018non} was to allow some of the entries $\sigma_{ij}$ to vanish, provided that the standard deviation profiles $A_n$ satisfy a certain quantitative irreducibility property. In the present article, we provide more information on the sequence $(\mu_n)$, described by a family of \emph{Master Equations}. We consider these equations in important special cases %such as separable variance profiles $\sigma^2_{ij}=d_i \widetilde d_j$ and such as sampled variance profiles $\sigma^2_{ij} = \sigma^2\left(\frac in, \frac jn \right)$ where $(x,y)\mapsto \sigma^2(x,y)$ is a given function on $[0,1]^2$. Associated examples are provided where $\mu_n^Y$ converges to a genuine limit. We study $\mu_n$'s behavior at zero. % and provide examples where $\mu_n$'s density is bounded, blows up, or vanishes while an atom appears. As a consequence, we identify the profiles that yield the circular law. Finally, building upon recent results from Alt et al. \cite{alt2018local,alt2019location}, we prove that, except possibly at the origin, $\mu_n$ admits a positive density on the centered disc of radius \$\sqrt{\rho(V_

#### Domains

Probability [math.PR]

### Dates and versions

hal-03095169 , version 1 (04-01-2021)

### Identifiers

• HAL Id : hal-03095169 , version 1
• DOI :

### Cite

Nicholas Cook, Walid Hachem, Jamal Najim, David Renfrew. Non-Hermitian random matrices with a variance profile (II): properties and examples. Journal of Theoretical Probability, 2022, 35 (4), pp.2343-2382. ⟨10.1007/s10959-021-01140-2⟩. ⟨hal-03095169⟩

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