arXiv subjects

Omar Hurtado

Publications and source records attributed to Omar Hurtado.

5 recordsLinked to original sources

arXiv2026

Localization for non-stationary Anderson models in three dimensions

We prove localization (near the bottom of the spectrum) for certain non-stationary variants of the Anderson model in three dimensions. More specifically, we prove a Wegner estimate, which implies localization by existing work. Two key inputs are a deterministic quantitative unique continuation theorem by Li and Zhang [Duke Math. J. 171(2): 327-415, 2022] and some combinatorial decompositions/bounds for non-stationary random potentials proved by the author [Commun. Math. Phys. 407:64, 2026].

math-ph↗

arXiv2025

Uniform estimates for random matrix products and applications

For certain natural families of topologies, we study continuity and stability of statistical properties of random walks on linear groups over local fields. We extend large deviation results known in the Archimedean case to non-Archimedean local fields and also demonstrate certain large deviation estimates for heavy tailed distributions unknown even in the Archimedean case. A key technical result, which may be of independent interest, establishes lower semi-continuity for the gap between the first and second Lyapunov exponents. As applications, we are able to obtain a key technical step towards a localization proof for heavy tailed Anderson models (the full proof appearing in a companion article), and show continuity/stability (taking the geometric data as input) of various statistical data associated to hyperbolic surfaces.

math.PR↗

arXiv2025

Localization for heavy-tailed Anderson models

Using recent results on uniform large deviation estimates for random matrix products obtained by S. Raman and the author, we prove localization for one dimensional Anderson models with heavy tails.

math-ph↗

arXiv2024

A "lifting" method for exponential large deviation estimates and an application to certain non-stationary 1D lattice Anderson models

Proofs of localization for random Schrödinger operators with sufficiently regular distribution of the potential can take advantage of the fractional moment method introduced by Aizenman-Molchanov, or use the classical Wegner estimate as part of another method, e.g. the multi-scale analysis introduced by Fröhlich-Spencer and significantly developed by Klein and his collaborators. When the potential distribution is singular, most proofs rely crucially on exponential estimates of events corresponding to finite truncations of the operator in question; these estimates in some sense substitute for the classical Wegner estimate. We introduce a method to "lift" such estimates, which have been obtained for many stationary models, to certain closely related non-stationary models. As an application, we use this method to derive Anderson localization on the 1-D lattice for certain non-stationary potentials along the lines of the non-perturbative approach developed by Jitomirskaya-Zhu in 2019.

math-ph↗

arXiv2024

Localization and unique continuation for non-stationary Schr\"odinger operators on the 2D lattice

We extend methods of Ding and Smart from their breakthrough paper in 2020 which showed Anderson localization for certain random Schr\"odinger operators on $\ell^2(\mathbb{Z}^2)$ via a quantitative unique continuation principle and Wegner estimate. We replace the requirement of identical distribution with the requirement of a uniform bound on the essential range of potential and a uniform positive lower bound on the variance of the variables giving the potential. Under those assumptions, we recover the unique continuation and Wegner lemma results, using Bernoulli decompositions and modifications of the arguments therein. This leads to a localization result at the bottom of the spectrum.

math-ph↗