Related papers: Sharp asymptotics for arm probabilities in critica…
We give accurate estimates for the bond percolation critical probabilities on seven Archimedean lattices, for which the critical probabilities are unknown, using an algorithm of Newman and Ziff.
For uniform random 4-colorings of graph edges with colors a,b,c,d, every two colors form a 1/2-percolation, and every two overlapping pairs of colors form independent 1/2-percolations. We show joint positive dependence for pairs of colors…
We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the…
The purpose of this article is twofold. On the one hand, we prove asymptotic formulas for the quantitative distribution of rational points on any smooth non-split projective quadratic surface. We obtain the optimal error term for the real…
In this paper, we compute the next-nearest-neighboring site percolation (Connections exist not only between nearest-neighboring sites, but also between next-nearest-neighboring sites.) probabilities Pc on the two-dimensional Sierpinski…
We deal with a planar random flight $\{(X(t),Y(t)),0<t\leq T\}$ observed at $n+1$ equidistant times $t_i=i\Delta_n,i=0,1,...,n$. The aim of this paper is to estimate the unknown value of the parameter $\lambda$, the underlying rate of the…
We prove that the supercritical phase of Voronoi percolation on $\mathbb{R}^d$, $d\geq 3$, is well behaved in the sense that for every $p>p_c(d)$ local uniqueness of macroscopic clusters happens with high probability. As a consequence,…
Prompted by an example arising in critical percolation, we study some reflected Brownian motions in symmetric planar domains and show that they are intertwined with one-dimensional diffusions. In the case of a wedge, the reflected Brownian…
We obtain asymptotic approximations for the probability density function of the product of two correlated normal random variables with non-zero means and arbitrary variances. As a consequence, we deduce asymptotic approximations for the…
In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our…
In this paper, the local asymptotic estimation for the supremum of a random walk and its applications are presented. The summands of the random walk have common long-tailed and generalized strong subexponential distribution. This…
We consider full scaling limits of planar nearcritical percolation in the Quad-Crossing-Topology introduced by Schramm and Smirnov. We show that two nearcritical scaling limits with different parameters are singular with respect to each…
We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we will consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to…
New calculations to over ten million time steps have revealed a more complex diffusive behavior than previously reported, of a point particle on a square and triangular lattice randomly occupied by mirror or rotator scatterers. For the…
This material is dedicated to the estimation of the chromatic number and chromatic class of the conjugated triangulation (first conversion) and also of the second conversion of the planar triangulation. Also this paper introduces some new…
Using the Coulomb Fluid method, this paper derives central limit theorems (CLTs) for linear spectral statistics of three "spiked" Hermitian random matrix ensembles. These include Johnstone's spiked model (i.e., central Wishart with spiked…
Motivated by critical planar percolation, we investigate a ``backbone'' event of planar Brownian motion, i.e.~the existence of two disjoint subpaths on the Brownian trajectory connecting the $\varepsilon$-neighborhood of the starting point…
The breakup of glass and alumina plates due to planar impacts on one of their lateral sides is studied. Particular attention is given to investigating the spatial location of the cracks within the plates. Analysis based on a…
We derive a universal, exact asymptotic form of the splitting probability for symmetric continuous jump processes, which quantifies the probability $ \pi_{0,\underline{x}}(x_0)$ that the process crosses $x$ before 0 starting from a given…
The charge asymmetry due to higher-order QED corrections in elastic lepton-proton scattering is estimated without employing the ultrarelativistic approximation. Our calculation is performed by generalizing the soft-photon approximation…