Related papers: Heavy tails and one-dimensional localization
We extend the Lyapunov function technique, a fundamental tool for investigating asymptotic stability and existence of attractors for ordinary differential equations, by introducing the notion of a {\it strong Lyapunov function} for an…
Let $L$ be a one-to-one operator of type $\omega$ in $L^2(\mathbb{R}^n)$, with $\omega\in[0,\,\pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$…
Given $\rho\in[0,1]$, we consider for $\varepsilon\in(0,1]$ the nonautonomous viscoelastic equation with a singularly oscillating external force $$ \partial_{tt} u-\kappa(0)\Delta u - \int_0^\infty \kappa'(s)\Delta u(t-s) d s…
We introduce and prove local Wegner estimates for continuous generalized Anderson Hamiltonians, where the single-site random variables are independent but not necessarily identically distributed. In particular, we get Wegner estimates with…
Consider a random Schr\"odinger-type operator of the form $H:=-H_X+V+\xi$ acting on a general graph $\mathscr G=(\mathscr V,\mathscr E)$, where $H_X$ is the generator of a Markov process $X$ on $\mathscr G$, $V$ is a deterministic potential…
The probability distribution of finite-time Lyapunov exponents provides an important characterization of dynamical attractors. We study such distributions for strange nonchaotic attractors (SNAs) created through several different mechanisms…
This paper introduces novel frameworks for large deviations and metastability analysis in heavy-tailed stochastic dynamical systems. We develop and apply these frameworks within the context of stochastic difference equation $X^\eta_{j+1}(x)…
We consider the product of spectral projections $$ \Pi_\epsilon(\lambda) = 1_{(-\infty,\lambda-\epsilon)}(H_0) 1_{(\lambda+\epsilon,\infty)}(H) 1_{(-\infty,\lambda-\epsilon)}(H_0) $$ where $H_0$ and $H$ are the free and the perturbed…
We prove a localization theorem for continuous ergodic Schr\"odinger operators $ H_\omega := H_0 + V_\omega $, where the random potential $ V_\omega $ is a nonnegative Anderson-type perturbation of the periodic operator $ H_0$. We consider…
We construct random Schr\"odinger operators, called Anderson Hamiltonians, with Dirichlet and Neumann boundary conditions for a fairly general class of singular random potentials on bounded domains. Furthermore, we construct the integrated…
In this paper, we prove a power-law version dynamical localization for a random operator $\mathrm{H}_{\omega}$ on $\mathbb{Z}^d$ with long-range hopping. In breif, for the linear Schr\"odinger equation…
This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for…
Heavy-tailed fluctuations and power law statistics pervade physics, finance, and economics, yet their origin is often ascribed to systems poised near criticality. Here we show that such behavior can emerge far from instability through a…
We provide a detailed study of the spectral properties of the linear operator $H(\varepsilon)=-(\varepsilon^{2}\chi_{\Omega_{\varepsilon}}+\chi_{\Omega^{c}_{\varepsilon}})\Delta$ modeling, through the wave equation…
Let P be the operator $-\Delta+V$ on R^d, where $V$ is a real potential with several inverse square singularities. The usual non-trapping type high-frequency inequality on the truncated resolvent of $P$ is shown, using semi-classical…
As put forth by Kerov in the early 1990s and elucidated in subsequent works, numerous properties of Wigner random matrices are shared by certain linear maps playing an important r\^ole in the representation theory of the symmetric group. We…
We study numerically the transport and localization properties of waves in ordered and disordered ladder-shaped lattices with local $\mathcal{PT}$ symmetry. Using a transfer matrix method, we calculate the transmittance and the reflectance…
The threshold behaviour of negative eigenvalues for Schr\"{o}dinger operators of the type $$ H_\lambda=-\frac{d^2}{dx^2}+U(x)+\lambda\alpha_\lambda V(\alpha_\lambda x) $$ is considered. The potentials $U$ and $V$ are real-valued bounded…
We consider the Schroedinger equation with a supersymmetric random potential, where the superpotential is a Levy noise. We focus on the problem of computing the so-called complex Lyapunov exponent, whose real and imaginary parts are,…
We propose a novel complex-analytic method for sums of i.i.d. random variables that are heavy-tailed and integer-valued. The method combines singularity analysis, Lindel\"of integrals, and bivariate saddle points. As an application, we…