Related papers: Heavy tails and one-dimensional localization

We study sampling from a target distribution ${\nu_* = e^{-f}}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm. For any potential function $f$ whose tails behave like ${\|x\|^\alpha}$ for ${\alpha \in [1,2]}$, and has…

In this article we prove an upper bound for the Lyapunov exponent $\gamma(E)$ and a two-sided bound for the integrated density of states $N(E)$ at an arbitrary energy $E>0$ of random Schr\"odinger operators in one dimension. These…

We consider the fourth order Schr\"odinger operator $H=\Delta^2+V(x)$ in three dimensions with real-valued potential $V$. Let $H_0=\Delta^2$, if $V$ decays sufficiently and there are no eigenvalues or resonances in the absolutely continuous…

The present paper is concerned with the stationary workload of queues with heavy-tailed (regularly varying) characteristics. We adopt a transform perspective to illuminate a close connection between the tail asymptotics and heavy-traffic…

With $\xi$ the number of triangles in the usual (Erd\H{o}s-R\'enyi) random graph $G(m,p)$, $p>1/m$ and $\eta>0$, we show (for some $C_{\eta}>0$) $$\Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}].$$ This is tight up to…

In this paper we study one-dimensional Jacobi operators on the lattice with a potential given by the skew shift. We show that the large deviation theorem takes place for Diophantine frequency and sufficiently large disorder. Combining the…

Consider operators $L^{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega \subset \mathbb{R}^N$. Assume that $V\in C^{1,1}(\Omega)$ and $V$ satisfies $V(x) \leq \overline{a} \mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and a second…

We uncover universal statistical properties of the trajectories of heavy inertial particles in three-dimensional, statistically steady, homogeneous, and isotropic turbulent flows by extensive direct numerical simulations. We show that the…

We establish Anderson localization for quasiperiodic operator families of the form $$ (H(x)\psi)(m)=\psi(m+1)+\psi(m-1)+\lambda v(x+m\alpha)\psi(m) $$ for all $\lambda>0$ and all Diophantine $\alpha$, provided that $v$ is a $1$-periodic…

In this paper we generalize and improve results proven for acoustic operators in \cite{jmp,long}. It deals with the behavior of the integrated density of states of random divergence operators of the form…

We consider non-convex stochastic optimization using first-order algorithms for which the gradient estimates may have heavy tails. We show that a combination of gradient clipping, momentum, and normalized gradient descent yields convergence…

In this paper, we investigate random operators on $\mathbb{Z}^d$ with H\"older continuously distributed potentials and the long-range hopping. The hopping amplitude decays with the inter-particle distance $\|\bm x\|$ as…

We study the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in $\mathbb{R}^n$. We show that under minimal assumptions on $X$, for $N \gtrsim n$ and with high…

For $a \ge - {( \frac{{d}}{2}- 1)^2} $ and $2\sigma= {{d - 2}}-( {{{(d - 2)}^2} + 4a})^{1/2}$, let $$\begin{cases}\mathcal{H}_{a}= - \Delta + \frac{a} {{{{ | x |}^2}}},\\ \mathcal{\widetilde{H}}_{\sigma}= 2\big( { - \Delta + \frac{{{\sigma…

We establish precise asymptotics near zero of the integrated density of states for the random Schr\"{o}dinger operators $(-\Delta)^{\alpha/2} + V^{\omega}$ in $L^2(\mathbb R^d)$ for the full range of $\alpha\in(0,2]$ and a fairly large…

We construct the effective field theory for a single massive higher-spin particle in flat spacetime. Positivity bounds of the S-matrix force the cutoff of the theory to be well below the naive strong-coupling scale, forbid any potential and…

This paper is organized in three parts closely related to closure properties of heavy-tailed distributions and heavy-tailed random vectors. In the first part we consider two random variables X and Y with distributions F and G respectively.…

We study Lifshitz tails for random Schr\"odinger operators where the random potential is alloy type in the sense that the single site potentials are independent, identically distributed, but they may have various function forms. We suppose…

Stable subordinators, and more general subordinators possessing power law probability tails, have been widely used in the context of subdiffusions, where particles get trapped or immobile in a number of time periods, called constant…

In these lecture notes, we review some recent developments on the relation between the macroscopic entropy of four-dimensional BPS black holes and the microscopic counting of states, beyond the thermodynamical, large charge limit. After a…