arXiv subjects

Mohammed Osman

Publications and source records attributed to Mohammed Osman.

13 recordsLinked to original sources

arXiv2026

Precise Delocalisation and Gumbel Laws for Eigenvectors of Wigner Matrices

We prove a delocalisation bound for eigenvectors of Wigner matrices with the precise relationship between the size of the largest entry and the decay exponent of the probability. We also prove that the largest entry of an individual eigenvector and the largest entry of all eigenvectors are both Gumbel distributed. The proof is based on the inclusion-exclusion principle, the small probability comparison of Erd\H{o}s--Xu, and partial diagonalisation of Gaussian divisible matrices.

math.PR↗

arXiv2026

Mesoscopic Linear Statistics for Two Ensembles of Quantum Graphs

We study mesoscopic linear spectral statistics for two ensembles of random quantum graphs. These are defined by a discrete graph $G$ and a unitary-matrix-valued function $U(k)$ indexed by directed edges of $G$. The matrix function $U(k)$ is constructed from unitary matrices $U^{(v)}$ indexed by the neighbours of each vertex $v$. The first ensemble is obtained by sampling the underlying discrete graph uniformly from the set of $d$-regular graphs. The second ensemble is obtained by sampling $U^{(v)}$ uniformly from the Haar measure, independently for each vertex. We prove that the variance of a linear spectral statistic in the large graph limit on polynomial mesoscopic scales coincides with that of the Gaussian Orthogonal/Unitary Ensemble.

math-ph↗

arXiv2026

Mapping Subnational Vulnerability to Inadequate Micronutrient Intake using a Bayesian Small Area Estimation Framework

Inadequate dietary micronutrient intake is a significant risk factor for deficiency and remains a major global health challenge. Nutrition programmes and interventions are most effective when targeted to populations at greatest risk. Household Consumption and Expenditure Surveys (HCES) are a widely available source of dietary data; however, they are often not powered for estimation below the first administrative level, limiting their utility for geographically targeted interventions. To address this, we applied Bayesian Small Area Estimation (SAE) methods to estimate the prevalence of apparent inadequate intake at the second administrative level. Three approaches were considered: a cluster level Beta binomial model and two area level models (mean smoothing and joint smoothing). Models were evaluated using a Rwanda HCES survey that supports inference at this scale. All models were implemented in a fully Bayesian framework to propagate uncertainty. Simulation results in Rwanda showed that the cluster level Beta binomial model achieved the strongest performance, while the area level joint smoothing model was the most reliable alternative among models accounting for survey design. Based on these results, models were applied to Senegal and Nigeria. In Senegal, second administrative level estimates captured meaningful subnational variation, reduced uncertainty relative to direct estimates, and remained consistent with first administrative level benchmarks. In Nigeria, despite smaller sample sizes and survey design constraints, modelled estimates reduced extreme uncertainty and showed good agreement with first administrative level estimates. This study demonstrates that Bayesian SAE methods can be applied to HCES data to generate reliable fine scale estimates of inadequate micronutrient intake, supporting localised nutrition interventions.

stat.AP↗

arXiv2025

Bulk Universality for Sparse Complex non-Hermitian Random Matrices

We prove that the local eigenvalue statistics in the bulk for complex random matrices with independent entries whose $r$-th absolute moment decays as $N^{-1-(r-2)\epsilon}$ for some $\epsilon>0$ are universal. This includes sparse matrices whose entries are the product of a Bernouilli random variable with mean $N^{-1+\epsilon}$ and an independent complex-valued random variable. By a standard truncation argument, we can also conclude universality for complex random matrices with $4+\epsilon$ moments. The main ingredient is a sparse multi-resolvent local law for products involving any finite number of resolvents of the Hermitisation and deterministic $2N\times2N$ matrices whose $N\times N$ blocks are multiples of the identity.

math.PR↗

arXiv2025

Functional renormalization group study of rho condensate at a finite isospin chemical potential in the quark meson model

We investigate the effect of an isospin chemical potential ($\mu_{I}$) within the quark-meson model, which approximates quantum chromodynamics (QCD) by modeling low energy phenomena such as chiral symmetry breaking and phase structure under varying conditions of temperature and chemical potential. Using the functional renormalization group (FRG) flow equations, we calculate the phase diagram in the chiral limit within the two-flavor quark-meson model in a finite $\mu_{I}$ with $\rho$ vector meson interactions. Fluctuation effects significantly decrease the critical chemical potential from the mean-field (MF) value $\mu_{I, MF} > m_\rho$ to lower value, at which point the $\rho$ vector meson condensates alongside the chiral condensate once the isospin chemical potential exceeds the critical value $\mu_{I}^{\text{crit}}$. This $\rho$ condensation is investigated numerically for different meson coupling strengths. The $\rho$ meson dominated region is delineated from other phases by a second-order phase transition at lower $\mu_{I}$ and a first-order transition at slightly higher $\mu_{I}$.

hep-ph↗

arXiv2024

Universality for Diagonal Eigenvector Overlaps of non-Hermitian Random Matrices

We prove the universality of the joint distribution of an eigenvalue and the corresponding diagonal eigenvector overlap, in the bulk and at the edge, for eigenvalues of complex matrices and real eigenvalues of real matrices. As part of the proof we obtain a bound for the least non-zero singular value of $X-z$ when $z$ is an edge eigenvalue and a bound for the inner product between left and right singular vectors of $X-z$ when $|z|=1+O(N^{-1/2})$.

math.PR↗

arXiv2024

Least Non-Zero Singular Value and the Distribution of Eigenvectors of non-Hermitian Random Matrices

We obtain a tail bound for the least non-zero singular value of $A-z$ when $A$ is a random matrix and $z$ is an eigenvalue of $A$ in a neighbourhood of a given point $z_0$ in the bulk of the spectrum. The argument relies on a resolvent comparison and a tail bound for Gauss-divisible matrices. The latter can be obtained by the method of partial Schur decomposition. Using this bound we prove that any finite collection of components of a right eigenvector corresponding to an eigenvalue uniformly sampled from a neighbourhood of a point in the bulk is Gaussian. A byproduct of the calculation is an asymptotic formula for the odd moments of the absolute value of the characteristic polynomial of real Gauss-divisible matrices.

math.PR↗

arXiv2024

Functional renormalization group study of the quark-meson model with omega and rho vector mesons

We employ the functional renormalization group flow equations to investigate the phase structure of the two-flavor quark-meson model in the presence of a finite isospin chemical potential, incorporating interactions with omega and rho vector mesons. For comparison, we also compute the phase diagram in the chiral limit using the mean-field approximation. Our findings demonstrate that omega and rho mesons affect the phase structure in markedly distinct ways, and the introduction of an isospin chemical potential leads to significant modifications in the phase boundaries and critical region. Increasing the isospin chemical potential lowers the tricritical points temperature, and tends to suppress the unphysical ``back-bending" of the FRG phase boundary at low temperature.

hep-ph↗

arXiv2024

Bulk Universality for Real Matrices with Independent and Identically Distributed Entries

We consider real, Gauss-divisible matrices $A_{t}=A+\sqrt{t}B$, where $B$ is from the real Ginibre ensemble. We prove that the bulk correlation functions converge to a universal limit for $t=O(N^{-1/3+\epsilon})$ if $A$ satisfies certain local laws. If $A=\frac{1}{\sqrt{N}}(\xi_{jk})_{j,k=1}^{N}$ with $\xi_{jk}$ independent and identically distributed real random variables having zero mean, unit variance and finite moments, the Gaussian component can be removed using local laws proven by Bourgade--Yau--Yin, Alt--Erd\H{o}s--Kr\"{u}ger and Cipolloni--Erd\H{o}s--Schr\"{o}der and the four moment theorem of Tao--Vu.

math.PR↗

arXiv2023

Universality for Weakly Non-Hermitian Matrices: Bulk Limit

We consider complex, weakly non-Hermitian matrices $A = W_1 +i\sqrt{τ_N}W_2$ , where $W_1$ and $W_2$ are Hermitian matrices and $τ_N = O(N^{-1})$. We first show that for pairs of Hermitian matrices $(W_1 , W_2)$ such that $W_1$ satisfies a multi-resolvent local law and $W_2$ is bounded in norm, the bulk correlation functions of the weakly non-Hermitian Gauss-divisible matrix $A + \sqrt{t}B$ converge pointwise to a universal limit for $t = O(N^{-1+ε})$. Using this and the reverse heat flow we deduce bulk universality in the case when $W_1$ and $W_2$ are independent Wigner matrices with sufficiently smooth density.

math.PR↗

arXiv2023

Bulk Universality for Complex non-Hermitian Matrices with Independent and Identically Distributed Entries

We consider N x N matrices with complex entries that are perturbed by a complex Gaussian matrix with small variance. We prove that if the unperturbed matrix satisfies certain local laws then the bulk correlation functions are universal in the large N limit. Assuming the entries are independent and identically distributed with a common distribution that has finite moments, the Gaussian component is removed by the four moment theorem of Tao and Vu.

math.PR↗

arXiv2021

Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities

Motivated by the phenomenon of Coherent Perfect Absorption, we study the shape of the deepest minima in the frequency-dependent single-channel reflection of waves from a cavity with spatially uniform losses. We show that it is largely determined by non-orthogonality factors $O_{nn}$ of the eigenmodes associated with the non-selfadjoint effective Hamiltonian. For cavities supporting chaotic ray dynamics we then use random matrix theory to derive, fully non-perturbatively, the explicit probability density ${\cal P}(O_{nn})$ of the non-orthogonality factors for systems with both broken and preserved time reversal symmetry. The results imply that $O_{nn}$ are heavy-tail distributed, with the universal tail ${\cal P}(O_{nn}\gg 1)\sim O_{nn}^{-3}$.

cond-mat.dis-nn↗

arXiv2020

Chaotic Scattering with Localized Losses: S-Matrix Zeros and Reflection Time Difference for Systems with Broken Time Reversal Invariance

Motivated by recent studies of the phenomenon of Coherent Perfect Absorption, we develop the random matrix theory framework for understanding statistics of the zeros of the (subunitary) scattering matrices in the complex energy plane, as well as of the recently introduced Refection Time Difference (RTD). The latter plays the same role for S-matrix zeros as the Wigner time delay does for its poles. For systems with broken time-reversal invariance, we derive the n -point correlation functions of the zeros in a closed determinantal form, and study various asymptotics and special cases of the associated kernel. The time-correlation function of the RTD is then evaluated and compared with numerical simulations. This allows to identify a cubic tail in the distribution of RTD, which we conjecture to be a superuniversal characteristic valid for all symmetry classes. We also discuss two methods for possible extraction of S-matrix zeroes from scattering data by harmonic inversion.

cond-mat.dis-nn↗