The limiting behavior of some infinitely divisible exponential dispersion models

Consider an exponential dispersion model (EDM) generated by a probability $\mu$ on $[0,\infty )$ which is infinitely divisible with an unbounded L\'{e}vy measure $\nu$. The Jorgensen set (i.e., the dispersion parameter space) is then $\mathbb{R}^{+}$, in which case the EDM is characterized by two parameters: $\theta _{0}$ the natural parameter of the associated natural exponential family and the Jorgensen (or dispersion) parameter $t$. Denote by $EDM(\theta _{0},t)$ the corresponding distribution and let $Y_{t}$ is a r.v. with distribution $EDM(\theta_0,t)$. Then if $\nu ((x,\infty ))\sim -\ell \log x$ around zero we prove that the limiting law $F_0$ of $Y_{t}^{-t}$ as $t\rightarrow 0$ is of a Pareto type (not depending on $\theta_0$) with the form $F_0(u)=0$ for $u<1$ and $1-u^{-\ell }$ for $u\geq 1$. Such a result enables an approximation of the distribution of $Y_{t}$ for relatively small values of the dispersion parameter of the corresponding EDM. Illustrative examples are provided.Comment: 8 page

Monte Carlo Methods for Insurance Risk Computation

In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function. This specific modelling is supported by data of the aggregated claim distribution of an insurance company. Large tail probabilities are important as they reflect the risk of large losses, however, analytic or numerical expressions are not available. We propose several simulation algorithms which are based on an asymptotic analysis of the distribution of the counting variable and on the reproducibility property of the claim distribution. The aggregated sum is simulated efficiently by importancesampling using an exponential cahnge of measure. We conclude by numerical experiments of these algorithms.Comment: 26 pages, 4 figure

Anomalous thermalization in ergodic systems

It is commonly believed that quantum isolated systems satisfying the eigenstate thermalization hypothesis (ETH) are diffusive. We show that this assumption is too restrictive, since there are systems that are asymptotically in a thermal state, yet exhibit anomalous, subdiffusive thermalization. We show that such systems satisfy a modified version of the ETH ansatz and derive a general connection between the scaling of the variance of the offdiagonal matrix elements of local operators, written in the eigenbasis of the Hamiltonian, and the dynamical exponent. We find that for subdiffusively thermalizing systems the variance scales more slowly with system size than expected for diffusive systems. We corroborate our findings by numerically studying the distribution of the coefficients of the eigenfunctions and the offdiagonal matrix elements of local operators of the random field Heisenberg chain, which has anomalous transport in its thermal phase. Surprisingly, this system also has non-Gaussian distributions of the eigenfunctions, thus directly violating Berry's conjecture.Comment: 5 pages, 3 figures; generalized derivations and introduced analogy with Thouless tim

Slow Dynamics in a Two-Dimensional Anderson-Hubbard Model

We study the real-time dynamics of a two-dimensional Anderson--Hubbard model using nonequilibrium self-consistent perturbation theory within the second-Born approximation. When compared with exact diagonalization performed on small clusters, we demonstrate that for strong disorder this technique approaches the exact result on all available timescales, while for intermediate disorder, in the vicinity of the many-body localization transition, it produces quantitatively accurate results up to nontrivial times. Our method allows for the treatment of system sizes inaccessible by any numerically exact method and for the complete elimination of finite size effects for the times considered. We show that for a sufficiently strong disorder the system becomes nonergodic, while for intermediate disorder strengths and for all accessible time scales transport in the system is strictly subdiffusive. We argue that these results are incompatible with a simple percolation picture, but are consistent with the heuristic random resistor network model where subdiffusion may be observed for long times until a crossover to diffusion occurs. The prediction of slow finite-time dynamics in a two-dimensional interacting and disordered system can be directly verified in future cold atoms experimentsComment: Title change and minor changes in the tex

Absence of dynamical localization in interacting driven systems

Using a numerically exact method we study the stability of dynamical localization to the addition of interactions in a periodically driven isolated quantum system which conserves only the total number of particles. We find that while even infinitesimally small interactions destroy dynamical localization, for weak interactions density transport is significantly suppressed and is asymptotically diffusive, with a diffusion coefficient proportional to the interaction strength. For systems tuned away from the dynamical localization point, even slightly, transport is dramatically enhanced and within the largest accessible systems sizes a diffusive regime is only pronounced for sufficiently small detunings.Comment: Scipost resubmission. 14 pages, 4 figures. Changes to the figures. Corrects a few typo

Multifractality and its role in anomalous transport in the disordered XXZ spin-chain

The disordered XXZ model is a prototype model of the many-body localization transition (MBL). Despite numerous studies of this model, the available numerical evidence of multifractality of its eigenstates is not very conclusive due severe finite size effects. Moreover it is not clear if similarly to the case of single-particle physics, multifractal properties of the many-body eigenstates are related to anomalous transport, which is observed in this model. In this work, using a state-of-the-art, massively parallel, numerically exact method, we study systems of up to 24 spins and show that a large fraction of the delocalized phase flows towards ergodicity in the thermodynamic limit, while a region immediately preceding the MBL transition appears to be multifractal in this limit. We discuss the implication of our finding on the mechanism of subdiffusive transport.Comment: 13 pages, 8 figure