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In an infinite sequence of independent identically distributed continuous random variables we study the number of strings of two subsequent records interrupted by a given number of non-records. By embedding in a marked Poisson process we prove that these counts are independent and Poisson distributed. Also the distribution of the number of uninterrupted strings of records is considered.

In the first part of this article, we proved a local version of the circular law up to the finest scale $N^{-1/2+ \e}$ for non-Hermitian random matrices at any point $z \in \C$ with ||z|−1|>c for any c>0 independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case $|z|-1=\oo(1)$. Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge $|z|-1=\oo(1)$ up to scale $N^{-1/4+ \e}$.

We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\Z/L\Z)^d$. The entries $h_{xy}=\overline h_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$, which satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any fixed $n\in \N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$, with an error $\OO(W^{-nd/2})$, and use it to prove a local law on the Green's function. This $T$-expansion was the main tool to prove the delocalization and quantum diffusion of random band matrices for dimensions $d\ge 8$ in part I \cite{PartI_high} of this series.

Abstract We investigate a system of N Brownian particles with the Coulomb interaction in any dimension \(d\ge 2\), and we assume that the initial data are independent and identically distributed with a common density \(\rho _0\) satisfying \(\int _{\mathbb {R}^{d}}\rho _0\ln \rho _0\,\hbox {d}x<\infty \) and \(\rho _0\in L^{\frac{2d}{d+2}} (\mathbb {R}^{d}) \cap L^1(\mathbb {R}^{d}, (1+|x|^2)\,\hbox {d}x)\). We prove that there exists a unique global strong solution for this interacting partsicle system and there is no collision among particles almost surely. For \(d=2\), we rigorously prove the propagation of chaos for this particle system globally in time without any cutoff in the following sense. When \(N\rightarrow \infty \), the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is the unique weak solution to the mean-field Poisson–Nernst–Planck equation of single component.