Decay of Correlation Rate in the Mean Field Limit of Point Vortices Ensembles

Finite volume corrections and decay of correlations in the canonical ensemble

We consider a classical system of $N$ particles confined in a box $Λ\subset\mathbb{R}^d$ interacting via a finite range pair potential. Given the validity of the cluster expansion in the canonical ensemble we compute the error between the finite and the infinite volume free energy and estimate it to be bounded by the area of the surface of the box's boundary over its volume. We also compute the truncated two-point correlation function and find that the contribution from the ideal gas case is of the order $1/N$ while the contribution of the interactions is of the order $1/|Λ|$ plus an exponenti…

点涡动力学的吉布斯平衡涨落

Gibbs Equilibrium Fluctuations of Point Vortex Dynamics

研究N个点涡在有界域内零总环流系统的统计特性，证明当N趋于无穷时，空间时间涨落场满足一种保留高斯能量-涡量 Ensemble 的二维欧拉动力学。

Level statistics for one-dimensional Schrödinger operators and Gaussian beta ensemble

We study the level statistics for two classes of 1-dimensional random Schrödinger operators : (1) for operators whose coupling constants decay as the system size becomes large, and (2) for operators with critically decaying random potential. As a byproduct of (2) with our previous result \cite{KN} imply the coincidence of the limits of circular and Gaussian beta ensembles.

平面上的塞伯格-维顿漩涡方程解的存在性

On the Existence of Solutions to the Seiberg-Witten Vortex Equations with Exponential Decay on the Plane

本文研究了平面上的希奇金型降维塞伯格-维顿方程模空间，证明其包含指数衰减解和多项式增长解。

Exponential Decay of Matrix $Φ$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles

In the study of Markovian processes, one of the principal achievements is the equivalence between the $Φ$-Sobolev inequalities and an exponential decrease of the $Φ$-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix $Φ$-entropies. This result also specializes to spectral gap inequalities and modified logarithmic Sobolev inequalities in the random matrix setting. To establish the main result, we define a non-commutative generalization of the carré du champ operator, and prove a …

Bielectron vortices in two-dimensional Dirac semimetals

Searching for new states of matter and unusual quasiparticles in emerging materials and especially low-dimensional systems is one of the major trends in contemporary condensed matter physics. Dirac materials, which host quasiparticles which are described by ultrarelativistic Dirac-like equations, are of a significant current interest from both a fundamental and applied physics perspective. Here we show that a pair of two-dimensional massless Dirac-Weyl fermions can form a bound state independently of the sign of the inter-particle interaction potential, as long as this potential decays at larg…

Aharonov-Bohm effect with many vortices

The Aharonov-Bohm effect is the prime example of a zero-field-strength configuration where a non-trivial vector potential acquires physical significance, a typical quantum mechanical effect. We consider an extension of the traditional A-B problem, by studying a two-dimensional medium filled with many point-like vortices. Systems like this might be present within a Type II superconducting layer in the presence of a strong magnetic field perpendicular to the layer, and have been studied in different limits. We construct an explicit solution for the wave function of a scalar particle moving withi…