arXiv subjects

Youssef Djellouli

Publications and source records attributed to Youssef Djellouli.

2 recordsLinked to original sources

arXiv2026

Optimal Covariance Estimates for Schrödinger Semigroups with White Noise in $d=1,2$

For $d\in\{1,2\}$, let $H=-\frac{1}{2}Δ+ V +ξ$ be the random Schrödinger operator on $L^2(\mathbb{R}^d)$ where $ξ$ is a standard Gaussian white noise and $V$ is a deterministic potential with power-law growth at infinity. Using a Feynman-Kac formula for the trace of the Schrödinger semigroup, we give optimal asymptotic upper and lower bounds on the covariance of $\mathrm{Tr}[e^{-sH}]$ and $\mathrm{Tr}[e^{-tH}]$ as $s,t\to0$ through estimates on Brownian bridge local times. These estimates are a significant improvement on previous bounds in the case $d=1$ and are the first of their kind for $d=2$. As an application of these new estimates, we prove a quantitative hyperuniformity-type property and decorrelation rate for the trace as $s,t\to0$.

math.PR↗

arXiv2025

On the Rigidity of Projected Perturbed Lattices

We study the occurrence of number rigidity and deletion singularity in a class of point processes that we call {\it projected perturbed lattices}. These are generalizations of processes of the form $Π=\{\|z\|^α+g_z\}_{z\in\mathbb{Z}^d}$ where $(g_z)_{z\in\mathbb{Z}^d}$ are jointly Gaussian, $α>0$, $d\in\mathbb{N}$, and $\|\cdot\|$ is a norm. We develop a new technique to prove sufficient conditions for the deletion singularity of $Π$, which improves significantly on the conditions one can obtain using the standard rigidity toolkit (e.g., the variance of linear statistics). In particular, we obtain the first lower bounds on $α$ for the deletion singularity of $Π$ that are independent of the dimension $d$ and the correlation of the $g_z$'s.

math.PR↗