arXiv subjects

Sarah Timhadjelt

Publications and source records attributed to Sarah Timhadjelt.

3 recordsLinked to original sources

arXiv2026

Critical Erd{\H o}s-R\'enyi digraph: all eigenvectors away from zero are delocalized

We consider the adjacency matrix of the directed Erd{\H o}s-R\'enyi graph. As long as the expected degree is larger than the logarithm of the number of vertices, the graph is connected, we show that all eigenvectors are completely delocalized. Below this critical scale, we prove eigenvector delocalization if the corresponding eigenvalue is away from zero. This contrasts the \emph{undirected} or Hermitian setting, where large eigenvalues have localized eigenvectors [arXiv:2005.14180]. Our results also hold for sparse random matrices with independent entries, which can be viewed as weighted Erd{\H o}s-R\'enyi digraphs.

math.PR↗

arXiv2024

Non-Hermitian expander obtained with Haar distributed unitaries

We consider a random quantum channel obtained by taking a selection of $d$ independent and Haar distributed $N$ dimensional unitaries. We follow the argument of Hastings to bound the spectral gap in terms of eigenvalues and adapt it to give an exact estimate of the spectral gap in terms of singular values \cite{hastings2007random,harrow2007quantum}. This shows that we have constructed a random quantum expander in terms of both singular values and eigenvalues. The lower bound is an analog of the Alon-Boppana bound for $d$-regular graphs. The upper bound is obtained using Schwinger-Dyson equations.

math.PR↗

arXiv2023

Spectral gap of convex combination of a random permutation and a bistochastic matrix

We study the spectral gap behavior of an operator obtained by summing a random permutation $M$ and a deterministic bistochastic matrix $Q$. We are interested in the asymptotic in terms of dimension. In the case where $(M,Q)$ are asymptotically free with amalgamation over the diagonal, we can compute limit operators $(u,q)$ which give the weak limit spectral distribution. Therefore we introduce free with amalgamation operators that are suitable for computing the spectral gap limit of our operator in high dimensions. We then approximate the spectral radius of the corresponding limit operator and finally give an upper bound for the spectral radius of the finite-dimensional operator. In particular, we show that if the deterministic matrix underlying graph is an expander, then the underlying graph associated to the sum with a random permutation is again an expander.

math.PR↗