arXiv subjects

Louis Vassaux

Publications and source records attributed to Louis Vassaux.

4 recordsLinked to original sources

arXiv2026

Graph alignment in sparse inhomogeneous models via self-overlap

We develop a general framework for understanding when graph alignment is information-theoretically feasible in sparse inhomogeneous random graph models, by studying the set of vertices on which the underlying matching can be recovered. Our main theorem gives a general lower bound on this set by leveraging the balanced load function introduced by Hajek (1990). The corresponding obstruction is captured by a new graph parameter, the self-overlap, which measures the extent to which a graph can imitate itself under a non-trivial relabelling. We then show that this criterion is sharp in a broad class of sparse inhomogeneous models, recovering known Erd\H{o}s--R\'enyi phenomena and yielding sharp thresholds for Chung--Lu graphs and stochastic block models.

math.PR↗

arXiv2026

Phase Transition in Convex Relaxations for Graph Alignment

We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+\sigma^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $\sigma = o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(\Pi^\star)$, i.e., $\|X^\star-\Pi^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-\Pi^\star\|_F^2$ transitions from $o(n)$ for $\sigma = \tilde{o}(n^{-1/2})$ to $\Omega(n)$ for $\sigma = \tilde{\Omega}(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.

stat.ML↗

arXiv2026

The feasibility of multi-graph alignment: a Bayesian approach

We establish thresholds for the feasibility of random multi-graph alignment in two models. In the Gaussian model, we demonstrate an "all-or-nothing" phenomenon: above a critical threshold, exact alignment is achievable with high probability, while below it, even partial alignment is statistically impossible. In the sparse Erdős-Rényi model, we rigorously identify a threshold below which no meaningful partial alignment is possible and conjecture that above this threshold, partial alignment can be achieved. To prove these results, we develop a general Bayesian estimation framework over metric spaces, which provides insight into a broader class of high-dimensional statistical problems.

math.ST↗

arXiv2025

Brownian behaviour of the Riemann zeta function around the critical line

We establish a Brownian extension to Selberg's central limit theorem for the Riemann zeta function. This implies various limiting distributions for $\zeta$, including an analogue of the reflection principle for the maximum of the Brownian motion: as $T$ diverges, for any $u>0$ we have \[ \frac{1}{T}\cdot {\rm meas}\Big\{0\leq t\leq T:\max_{\sigma\geq \tfrac{1}{2}}\log|\zeta(\sigma+i t)|\geq u \sqrt{\tfrac{1}{2}\log \log T} \Big\}\to 2 \displaystyle\int_u^{\infty} \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}\mathrm{d} x. \]

math.NT↗