arXiv subjects

Constanza Rojas-Molina

Publications and source records attributed to Constanza Rojas-Molina.

14 recordsLinked to original sources

arXiv2026

Dynamical delocalization in disordered 2D Chern insulators

We show the existence of energies exhibiting dynamical delocalization in discrete 2D Chern insulators perturbed by a random potential in a general setting. Our proof exploits two main features of the model: jumps in the integer value of the Chern character and continuity of averaged spectral projections in both energy and disorder parameters. This allows us to show robustness of the topological index in the presence of disorder, which, combined with existing methods to prove dynamical localization, allows us to provide detailed information on the phase diagram of the model. The novelty of our approach is that we are able to show dynamical delocalization in the disorder parameter, and not only in the energy parameter, which allows to prove Anderson metal-insulator transition even when spectral gaps close due to the strength of disorder.

math-ph↗

arXiv2025

Phase transition in the Integrated Density of States of the Anderson model arising from a supersymmetric sigma model

We study the Integrated Density of States (IDS) of the random Schrödinger operator appearing in the study of certain reinforced random processes in connection with a supersymmetric sigma-model. We rely on previous results on the supersymmetric sigma-model to obtain lower and upper bounds on the asymptotic behavior of the IDS near the bottom of the spectrum in all dimension. We show a phase transition for the IDS between weak and strong disorder regime in dimension larger or equal to three, that follows from a phase transition in the corresponding random process and supersymmetric sigma-model. In particular, we show that the IDS does not exhibit Lifshitz tails in the strong disorder regime, confirming a recent conjecture. This is in stark contrast with other disordered systems, like the Anderson model. A Wegner type estimate is also derived, giving an upper bound on the IDS and showing the regularity of the function.

math-ph↗

arXiv2024

Decay of the Green's function of the fractional Anderson model and connection to long-range SAW

We prove a connection between the Green's function of the fractional Anderson model and the two point function of a self-avoiding random walk with long range jumps, adapting a strategy proposed by Schenker in 2015. This connection allows us to exploit results from the theory of self-avoiding random walks to improve previous bounds known for the fractional Anderson model at strong disorder. In particular, we enlarge the range of the disorder parameter where spectral localization occurs. Moreover we prove that the decay of Green's function at strong disorder for any $0<α<1$ is arbitrarily close to the decay of the massive resolvent of the corresponding fractional Laplacian, in agreement with the case of the standard Anderson model $α=1$. We also derive upper and lower bounds for the resolvent of the discrete fractional Laplacian with arbitrary mass $m\geq 0,$ that are of independent interest.

math-ph↗

arXiv2021

Lifshitz tails for random diagonal perturbations of Laurent matrices

We study the Integrated Density of States of one-dimensional random operators acting on $\ell^2(\mathbb Z)$ of the form $T + V_ω$ where $T$ is a Laurent (also called bi-infinite Toeplitz) matrix and $V_ω$ is an Anderson potential generated by i.i.d. random variables. We assume that the operator $T$ is associated to a bounded, Hölder-continuous symbol $f$, that attains its minimum at a finite number of points. We allow for $f$ to attain its minima algebraically. The resulting operator $T$ is long-range with weak (algebraic) off-diagonal decay. We prove that this operator exhibits Lifshitz tails at the lower edge of the spectrum with an exponent given by the Integrated Density of States of $T$ at the lower spectral edge. The proof relies on generalizations of Dirichlet-Neumann bracketing to the long-range setting and a generalization of Temple's inequality to degenerate ground state energies.

math-ph↗

arXiv2021

Localisation for Delone operators via Bernoulli randomisation

Delone operators are Schrödinger operators in multi-dimensional Euclidean space with a potential given by the sum of all translates of a given "single-site potential" centred at the points of a Delone set. In this paper, we use randomisation to study dynamical localisation for families of Delone operators. We do this by suitably adding more points to a Delone set and by introducing i.i.d. Bernoulli random variables as coupling constants at the additional points. The resulting non-ergodic continuum Anderson model with Bernoulli disorder is accessible to the latest version of the multiscale analysis. The novel ingredient here is the initial length-scale estimate whose proof is hampered due to the non-periodic background potential. It is obtained by the use of a quantitative unique continuation principle. As applications we obtain both probabilistic and topological statements about dynamical localisation. Among others, we show that Delone sets for which the associated Delone operators exhibit dynamical localisation at the bottom of the spectrum are dense in the space of Delone sets.

math-ph↗

arXiv2020

Lifshitz tails for the fractional Anderson model

We consider the $d$-dimensional fractional Anderson model $(-Δ)^α+ V_ω$ on $\ell^2(\mathbb Z^d)$ where $0<α\leq 1$. Here $-Δ$ is the negative discrete Laplacian and $V_ω$ is the random Anderson potential consisting of iid random variables. We prove that the model exhibits Lifshitz tails at the lower edge of the spectrum with exponent $ d/ (2α)$. To do so, we show among other things that the non-diagonal matrix elements of the negative discrete fractional Laplacian are negative and satisfy the two-sided bound $$ \frac{c_{α,d}}{|n-m|^{d+2α}} \leq -(-Δ)^α(n,m)\leq \frac{C_{α,d}}{|n-m|^{d+2α}} $$ for positive constants $c_{α,d}$, $C_{α,d}$ and all $n\neq m\in\mathbb Z^d$.

math.PR↗

arXiv2020

Random Schrödinger Operators and Anderson localization in aperiodic media

In this note we review some results on localization and related properties for random Schrödinger operators arising in aperiodic media. These include the Anderson model associated to disordered quasycrystals and also the so-called Delone operators, operators associated to deterministic aperiodic structures.

math-ph↗

arXiv2018

Random Schrödinger Operators on discrete structures

The Anderson model serves to study the absence of wave propagation in a medium in the presence of impurities, and is one of the most studied examples in the theory of quantum disordered systems. In these notes we give a review of the spectral and dynamical properties of the Anderson Model on discrete structures, like the $d$-dimensional square lattice and the Bethe lattice, and the methods used to prove localization. These notes are based on a course given at the CIMPA School "Spectral Theory of Graphs and Manifolds" in Kairouan, 2016.

math-ph↗

arXiv2017

Characterization of the metal-insulator transport transition for the two-particle Anderson model

We extend to the two-particle Anderson model the characterization of the metal-insulator transport transition obtained in the one-particle setting by Germinet and Klein. We show that, for any fixed number of particles, the slow spreading of wave packets in time implies the initial estimate of a modified version of the Bootstrap Multiscale Analysis. In this new version, operators are restricted to boxes defined with respect to the pseudo-distance in which we have the slow spreading. At the bottom of the spectrum, within the regime of one-particle dynamical localization, we show that this modified multiscale analysis yields dynamical localization for the two-particle Anderson model, allowing us to obtain a characterization of the metal-insulator transport transition for the two-particle Anderson model at the bottom of the spectrum.

math-ph↗

arXiv2015

Ergodicity and dynamical localization for Delone-Anderson operators

We study the ergodic properties of Delone-Anderson operators, using the framework of randomly coloured Delone sets and Delone dynamical systems. In particular, we show the existence of the integrated density of states and, under some assumptions on the geometric complexity of the underlying Delone sets, we obtain information on the almost-sure spectrum of the family of random operators. We then exploit these results to study the Lifshitz-tail behaviour of the integrated density of states of a Delone-Anderson operator at the bottom of the spectrum. Furthermore, we use Lifshitz-tail estimates as an input for the multi-scale analysis to prove dynamical localization.

math-ph↗

arXiv2013

The Anderson model with missing sites

In the present note we show dynamical localization for an Anderson model with missing sites in a discrete setting at the bottom of the spectrum in arbitrary dimension $d$. In this model, the random potential is defined on a relatively dense subset of $\mathbb Z^d$, not necessarily periodic, i.e., a Delone set in $\mathbb Z^d$. To work in the lower band edge we need no further assumption on the geometric complexity of the Delone set. We use a spatial averaging argument by Bourgain-Kenig to obtain a uniform Wegner estimate and an initial length scale estimate, which yields localization through the Multiscale Analysis for non ergodic models. This argument gives an explicit dependence on the maximal distance parameter of the Delone set for the Wegner estimate. We discuss the case of the upper spectral band edge and the arising need of imposing the (complexity) condition of strict uniform pattern frequency on the Delone set.

math-ph↗

arXiv2012

Scale-free unique continuation estimates and applications to random Schrödinger operators

We prove a unique continuation principle or uncertainty relation valid for Schrödinger operator eigenfunctions, or more generally solutions of a Schrödinger inequality, on cubes of side $L\in 2\NN+1$. It establishes an equi-distribution property of the eigenfunction over the box: the total $L^2$-mass in the box of side $L$ is estimated from above by a constant times the sum of the $L^2$-masses on small balls of a fixed radius $δ>0$ evenly distributed throughout the box. The dependence of the constant on the various parameters entering the problem is given explicitly. Most importantly, there is no $L$-dependence. This result has important consequences for the perturbation theory of eigenvalues of Schrödinger operators, in particular random ones. For so-called Delone-Anderson models we deduce Wegner estimates, a lower bound for the shift of the spectral minimum, and an uncertainty relation for spectral projectors.

math.SP↗

arXiv2011

Characterization of the Anderson metal-insulator transition for non ergodic operators and application

We study the Anderson metal-insulator transition for non ergodic random Schrödinger operators in both annealed and quenched regimes, based on a dynamical approach of localization, improving known results for ergodic operators into this more general setting. In the procedure, we reformulate the Bootstrap Multiscale Analysis of Germinet and Klein to fit the non ergodic setting. We obtain uniform Wegner Estimates needed to perform this adapted Multiscale Analysis in the case of Delone-Anderson type potentials, that is, Anderson potentials modeling aperiodic solids, where the impurities lie on a Delone set rather than a lattice, yielding a break of ergodicity. As an application we study the Landau operator with a Delone-Anderson potential and show the existence of a mobility edge between regions of dynamical localization and dynamical delocalization.

math-ph↗

arXiv2010

Dynamical mobility edge for various random Landau Hamiltonians

We review recent results obtained within the framework of the integer quantum Hall effect in the spirit of the work of Germinet, Klein, Schenker in \cite{GKS}. Landau Hamiltonians perturbed by random electric or magnetic perturbations are shown to exhibit a dynamical mobility edge, that is a transition between a regime of dynamical localization and a regime of non trivial transport at a minimal rate. The focus is put on three situations of interest: 1) unbounded ergodic electric potentials, for which Landau gaps are filled; 2) non ergodic electric potentials; 3) random magnetic potentials.

math-ph↗