Exponential Decay for the Near-Critical Scaling Limit of the Planar Ising Model

We consider the Ising model at its critical temperature with external magnetic field ha^15/8 on the square lattice with lattice spacing a . We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a ↓ 0 . As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a = 1 , the mass (inverse correlation length) is of order h^8/15 as h ↓ 0 ; for the Euclidean field, it equals exactly Ch^8/15 for some C . Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Scaling Limit for the Ant in High-Dimensional Labyrinths

We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e., the behavior of the “ant in the labyrinth.” It is natural to conjecture that the scaling limit for random walks on large critical random graphs exists in high dimensions and is universal. This scaling limit is simply the natural Brownian motion on the integrated super-Brownian excursion. We give here a set of four natural, sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, and spread-out percolation in high enough dimension. In a companion paper, we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above in dimensions larger than 14. © 2019 Wiley Periodicals, Inc.     

Quasi-neutral Limit of a Nonlinear Drift Diffusion Model for Semiconductors

The limit of the vanishing Debye length (the charge neutral limit) in a nonlinear bipolar drift-diffusion model for semiconductors without a pn-junction (i.e., with a unipolar background charge) is studied. The quasi-neutral limit (zero-Debye-length limit) is determined rigorously by using the so-called entropy functional which yields appropriate uniform estimates.     

半导体中非线性漂流扩散模型的拟中性极限:快扩散情形

本文研究无Pn-联结的非线性双极半导体漂流扩散模型的消失Debye长度极限(即粒子中性极限)问题．使用熵方法和弱紧性方法从数学上严格证明了快扩散情形的拟中性极限．     

Poisson-Type Limit Theorems for Eigenvalues of Finite-Volume Anderson Hamiltonians

We consider the spectral problem for the random Schrödinger operator on the multidimensional lattice torus increasing to the whole of lattice, with an i.i.d. potential (Anderson Hamiltonian). We prove complete Poisson-type limit theorems for the (normalized) eigenvalues and their locations, provided that the upper tails of the distribution of potential decay at infinity slower than the double exponential tails. For the fractional-exponential tails, the strong influence of the parameters of the model on a specification of the normalizing constants is described.     

Diffusion Limit of a Non-Linear Kinetic Model without the Detailed Balance Principle

 The aim of this paper is the rigorous derivation of a non-linear drift-diffusion model from the Boltzmann equation in a semiconductor device. Collisions are taken into account through the non-linear Pauli operator, without assuming any relation concerning the cross-section (such as the so-called detailed balance principle). The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The same program is applied to the derivation of a non-linear SHE (Spherical Harmonics Expansion) model, when elastic collisions are considered, and is rigorously carried through in the case of zero electric field. (Received 5 October 2000; in revised form 3 September 2001)     

The NLS Limit for Bosons in a Quantum Waveguide

We consider a system of N bosons confined to a thin waveguide, i.e. to a region of space within an \({\epsilon}\)-tube around a curve in \({\mathbb{R}^3}\). We show that when taking simultaneously the NLS limit \({N \to \infty}\) and the limit of strong confinement \({\epsilon \to 0}\), the time-evolution of such a system starting in a state close to a Bose–Einstein condensate is approximately captured by a non-linear Schrödinger equation in one dimension. The strength of the non-linearity in this Gross–Pitaevskii type equation depends on the shape of the cross-section of the waveguide, while the “bending” and the “twisting” of the waveguide contribute potential terms. Our analysis is based on an approach to mean-field limits developed by Pickl (On the time-dependent Gross–Pitaevskii-and Hartree equation. arXiv:0808.1178, 2008).     

Rigorous Derivation of a Nonlinear Diffusion Equation as Fast-Reaction Limit of a Continuous Coagulation-Fragmentation Model with Diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann–Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [5 Carrillo , J. A. , Desvillettes , L. , Fellner , K. ( 2008 ). Fast-reaction limit for the inhomogeneous aizenman-bak model . Kinetic and Related Models 1 : 127 – 137 . [Google Scholar]], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters.     

Eigenfunction Statistics in the Localized Anderson Model

We consider the localized region of the Anderson model and study the distribution of eigenfunctions simultaneously in space and energy. In a natural scaling limit, we prove convergence to a Poisson process. This provides a counterpoint to recent work, [9], which proves repulsion of the localization centres in a subtly different regime. Submitted: December 12, 2005; Revised: April 22, 2006; Accepted: May 3, 2006     

Lifshits Tails in the Hierarchical Anderson Model

We prove that the homogeneous hierarchical Anderson model exhibits a Lifshits tail at the upper edge of its spectrum. The Lifshits exponent is given in terms of the spectral dimension of the homogeneous hierarchical structure. Our approach is based on Dirichlet–Neumann bracketing for the hierarchical Laplacian and a large-deviation argument.     

A Scharfetter–Gummel Type Discretization of the Quantum Drift Diffusion Model

We derive a generalized Scharfetter–Gummel discretization of the Quantum Drift Diffusion Model in one space dimension. The scheme relies on the introduction of a generalized potential which identifies the quantum drift term. Further, the new scheme is stable in the semiclassical limit recovering the SG scheme for the classical drift diffusion equations. Numerical results for a ballistic diode are presented.     

Diffusive Limit for a Quantum Linear Boltzmann Dynamics

In this article, I study the diffusive behavior for a quantum test particle interacting with a dilute background gas. The model that I begin with is a reduced picture for the test particle dynamics given by a quantum linear Boltzmann equation in which the gas particle scattering is assumed to occur through a hard-sphere interaction. The state of the particle is represented by a density matrix that evolves according to a translation-covariant Lindblad equation. The main result is a proof that the particle’s position distribution converges to a Gaussian under diffusive rescaling.     

Limit Theorem for the Spread of Branching Diffusion with Stabilizing Drift

In the present paper we derive a formula describing the limiting behavior of R _t , the position of the rightmost particle over a time interval [0, t] in the one-dimensional branching diffusion with a stabilizing drift, and generalize the result to a multidimensional case.     

Localization for the Anderson Model on Trees with Finite Dimensions

We introduce a family of trees that interpolate between the Bethe lattice and . We prove complete localization for the Anderson model on any member of that family. Submitted: August 8, 2006. Accepted: February 13, 2007.     

Poisson Statistics of Eigenvalues in the Hierarchical Anderson Model

We study the eigenvalue statistics for the hieracharchial Anderson model of Molchanov [21–23,27,28]. We prove Poisson fluctuations at arbitrary disorder, when the the model has a spectral dimension d < 1. The proof is based on Minami’s technique [25] and we give an elementary exposition of the probabilistic arguments. Submitted: October 8, 2007. Accepted: December 17, 2007.     

Absolutely Continuous Spectrum for the Anderson Model on Some Tree-like Graphs

We prove persistence of absolutely continuous spectrum for the Anderson model on a general class of tree-like graphs.     

The Semiclassical Limit in the Quantum Drift-Diffusion Equations with Isentropic Pressure

The semiclassical limit in the transient quantum drift-diffusion equations with isentropic pressure in one space dimension is rigorously proved. The equations are supplemented with homogeneous Neumann boundary conditions. It is shown that the semiclassical limit of this solution solves the classical drift-diffusion model. In the meanwhile, the global existence of weak solutions is proved.     

Smoothness of Correlations in the Anderson Model at Strong Disorder

We study the higher-order correlation functions of covariant families of observables associated with random Schr?dinger operators on the lattice in the strong disorder regime. We prove that if the distribution of the random variables has a density analytic in a strip about the real axis, then these correlation functions are analytic functions of the energy outside of the planes corresponding to coincident energies. In particular, this implies the analyticity of the density of states, and of the current-current correlation function outside of the diagonal. Consequently, this proves that the current-current correlation function has an analytic density outside of the diagonal at strong disorder. Submitted: October 8, 2005; Accepted: February 15, 2006     

A Random Walk with Collapsing Bonds and Its Scaling Limit

Abstract

We introduce a new self-interacting random walk on the integers in a dynamic random environment and show that it converges to a pure diffusion in the scaling limit. We also find a lower bound on the diffusion coefficient in some special cases. With minor changes the same argument can be used to prove the scaling limit of the corresponding walk in ? ^d .