We extend the spherical-integral approach to $\mathcal{R}$-transforms of non-Hermitian random matrices to random matrices with a fixed $K \times K$ block structure. We introduce a rank-$K$ spherical integral defining a scalar $H$-transform on $2K \times 2K$ overlap matrices and an associated matrix-valued $\mathcal{R}$-transform encoding the joint block structure. Using the replica method, we derive a conjectural subordination relation for sums of independent block matrices. This framework provides a method for determining the spectral boundaries of large non-Hermitian random matrices with a fixed block structure.
We study finite-rank multiplicative deformations of rotationally invariant non-Hermitian random matrices. More precisely, we consider models of the form $\mathbf{A}(\mathbf{I}+\mathbf{T})$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix, $\mathbf{T}$ is a finite-rank normal perturbatio...
In this paper, we study spectral properties of multiplicative deformations of non-Hermitian random matrices. We consider matrices of the form $\mathbf{A}\mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. We show t...
We define a rank-two extension $\mathcal{H}^{(2)}$ of the spherical $\mathcal{H}$-transform, together with its associated transform $\mathcal{R}^{(2)}$. For sums, both transforms are additive. The transform $\mathcal{R}^{(2)}$ yields a $4\times4$ formula for the average product of two hermitized resolvents, and hence f...
We study the large-$N$ asymptotics of a constrained spherical integral for non-Hermitian random matrices, in which the norms and mutual scalar product of two vectors are fixed. In the delocalized regime, the asymptotics are governed by the non-Hermitian transforms $\mathcal R_1$ and $\mathcal R_2$. At saddle-point leve...
In this paper, we introduce and study conformal permutation matrices. In particular, for a permutation $\pi_s\in S_n$, we define the corresponding conformal permutation matrix $H_{\pi_s}$ and show that its $d-$th power, where $d=n/\gcd(s,n)$, is a diagonal block matrix whose blocks are products of certain matrix blocks...
We establish nonasymptotic spectral comparison results for matrices whose entries are non-linear functions of an iid random field. Under an exponential decay assumption on an $\mathbb L^\infty$-coupling coefficient, we derive high-probability bounds for the Hausdorff distance between their spectra and those of Gaussian...
Marwa Banna, Issa-Mbenard Dabo, F. Merlevède· 0 citations
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