Related papers: Tightness for Thick Points in two dimensions

Let $C^*_{\epsilon,S^2}$ denote the cover time of the two dimensional sphere by a Wiener sausage of radius $\epsilon$. We prove that $$\sqrt{C^{*}_{\epsilon,S^2} } -\sqrt{\frac{2A_{S^2}}{\pi}}(\log \epsilon^{-1}-\frac14\log\log…

Let $T_1^{(\mu)}$ be the first hitting time of the point 1 by the Bessel process with index $\mu\in \R$ starting from $x>1$. Using an integral formula for the density $q_x^{(\mu)}(t)$ of $T_1^{(\mu)}$, obtained in Byczkowski, Ryznar (Studia…

In this paper, we study the edge behavior of Dyson Brownian motion with general $\beta$. Specifically, we consider the scenario where the averaged initial density near the edge, on the scale $\eta_*$, is lower bounded by a square root…

Let $U\subseteq\mathbf{C}$ be a bounded domain with smooth boundary and let $F$ be an instance of the continuum Gaussian free field on $U$ with respect to the Dirichlet inner product $\int_U\nabla f(x)\cdot \nabla g(x)\,dx$. The set…

Packings of equal disks in the plane are known to have density at most $\pi/\sqrt{12}$, although this density is never achieved in the square torus, which is what we call the plane modulo the square lattice. We find packings of disks in a…

The distribution of the first-passage time (FPT)$T_a$ for a Brownian particle with drift $\mu$ subject to hitting an absorber at a level $a>0$ is well-known and given by its density $\gamma(t) = \frac{a}{\sqrt{2 \pi t^3} } e^{-\frac{(a-\mu…

We consider probability measures, $d\mu=w(\theta) \f{d\theta}{2\pi} +d\mu_\s$, on the unit circle, $\partial\bbD$, with Verblunsky coefficients, $\{\alpha_j\}_{j=0}^\infty$. We prove for $\theta_1\neq\theta_2$ in $[0,2\pi)$ and…

We compute the joint distribution of the site and the time at which a $d$-dimensional standard Brownian motion $B_t$ hits the surface of the ball $ U(a) =\{|{\bf x}|<a\}$ for the first time. The asymptotic form of its density is obtained…

We consider Brownian motion in a circular disk $\Omega$, whose boundary $\p\Omega$ is reflecting, except for a small arc, $\p\Omega_a$, which is absorbing. As $\epsilon=|\partial \Omega_a|/|\partial \Omega|$ decreases to zero the mean time…

The present work is devoted to the study of the large time behaviour of a critical Brownian diffusion in two dimensions, whose drift is divergence-free, ergodic and given by the curl of the 2-dimensional Gaussian Free Field. We prove the…

We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and…

We obtain bounds for the expected loss of torsional rigidity of a cylinder $\Omega_L=(-L/2,L/2) \times \Omega\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $\Omega_L,$ and runs until the first time…

Recently, a theoretical framework known as {\it ballistic macroscopic fluctuation theory} has been developed to study large-scale fluctuations and correlations in many-body systems exhibiting ballistic transport. In this paper, we review…

Combining analytic calculations, computer simulations, and classical density functional theory we determine the interfacial tension of orientable two-dimensional hard rectangles near a curved hard wall. Both a circular cavity holding the…

Let $d$ be any positive and non square integer. We prove an upper bound for the first two moments of the length $T(d)$ of the period of the continued fraction expansion for $\sqrt{d}$. This allows to improve the existing results for the…

For a spectrally positive strictly stable process with index in (1,2), the paper obtains i) the density of the time when the process makes first exit from an interval by hitting the interval's lower end point before jumping over its upper…

We study the problem of finding an $\epsilon$-fixed point of a contraction map $f:[0,1]^k\mapsto[0,1]^k$ under both the $\ell_\infty$-norm and the $\ell_1$-norm. For both norms, we give an algorithm with running time $O(\log^{\lceil…

The pressure and the viscosity in two-dimensional sheared granular assemblies are investigated numerically. The behavior of both pressure and viscosity is smoothly changing qualitatively when starting from a mono-disperse hard-disk system…

We provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard-spheres as the number of particles $N$ goes to infinity and their diameter $\varepsilon$ simultaneously goes to $0$, in the fast…

We study the survival probability and the corresponding first passage time density of fractional Brownian motion confined to a two-dimensional open wedge domain with absorbing boundaries. By analytical arguments and numerical simulation we…