相关论文: Nontrivial upper bound for chemical distance on th…
The classical definitions of the Incipient Infinite Cluster (IIC) of percolation consist in conditioning the origin on being connected to radius $n$ and letting $n$ go to infinity. We provide a short proof of that convergence in the planar…
Percolation models with multiple percolating clusters have attracted much attention in recent years. Here we use Monte Carlo simulations to study bond percolation on $L_{1}\times L_{2}$ planar random lattices, duals of random lattices, and…
The ellipses model is a continuum percolation process in which ellipses with random orientation and eccentricity are placed in the plane according to a Poisson point process. A parameter $\alpha$ controls the tail distribution of the major…
In this communication with computer simulation we evaluate simple cubic random-site percolation thresholds for neighbourhoods including the nearest neighbours (NN), the next-nearest neighbours (2NN) and the next-next-nearest neighbours…
The problem of percolation along sites of square lattice is studied. The number of contours being external boundaries for finite clusters has been estimated using geometric considerations. This estimation makes it possible to determine more…
Using an inverse of the standard linear congruential random number generator, large randomly occupied lattices can be visited by a random walker without having to determine the occupation status of every lattice site in advance. In seven…
We study non-uniform percolation in a two-dimensional cluster growth model with multiple seeds. With increasing concentration of seeds, the percolation threshold is found to increase monotonically, while the exponents for correlation…
The probability distributions of the masses of the clusters spanning from top to bottom of a percolating lattice at the percolation threshold are obtained in all dimensions from two to five. The first two cumulants and the exponents for the…
We consider boundary roughness for the ``droplet'' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least $l^2$, for large $l$. The…
Let $C$ be a binary code of length $n$ with distances $0<d_1<\cdots<d_s\le n$. In this note we prove a general upper bound on the size of $C$ without any restriction on the distances $d_i$. The bound is asymptotically optimal.
We consider upper exponential bounds for the probability of the event that an absolute deviation of sample mean from mathematical expectation p is bigger comparing with some ordered level epsilon. These bounds include 2 coefficients {alpha,…
In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in $d$-dimensional Euclidean space $\mathbb{R}^d$ were obtained,…
In this paper we consider statistical estimates of threshold and strength of percolation clusters on square lattices. The percolation threshold $p_c$ and the strength of percolation clusters $P_\infty$ for a square lattice with $(1,…
We consider a family of percolation models in which geometry and connectivity are defined by two independent random processes. Such models merge characteristics of discrete and continuous percolation. We develop an algorithm allowing…
We present structural properties of two-dimensional polymers as far as they can be described by percolation theory. The percolation threshold, critical exponents and fractal dimensions of clusters are determined by computer simulation and…
In long-range percolation on $\mathbb{Z}^d$, points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta \geq 0$ is a parameter. As $d$ and $\alpha$ vary, the model…
For semiclassical problems we establish upper bounds on the number of resonances in boxes of size $h$ along the real axis, in terms of the dimension of the set of trapped trajectories. The proof uses second microlocalization.
How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of $N=uL^d$ steps on a $d$-dimensional hypercubic lattice of size $L^d$ (with…
We study infinite ``$+$'' or ``$-$'' clusters for an Ising model on an connected, transitive, non-amenable, planar, one-ended graph $G$ with finite vertex degree. If the critical percolation probability $p_c^{site}$ for the i.i.d.~Bernoulli…
We derive refined entropy upper bounds for $q$-ary $B_2$ codes by exploiting the Fourier structure of the i.i.d. difference distribution $D=X-Y$. Since the pmf of $D$ is an autocorrelation, its Fourier series is a nonnegative trigonometric…