Related papers: Brownian Bridge Asymptotics for the Subcritical Be…
Percolation on a five-dimensional simple hypercubic (sc(5)) lattice with extended neighborhoods is investigated by means of extensive Monte Carlo simulations, using an effective single-cluster growth algorithm. The critical exponents,…
We study mixed site-bond directed percolation on 2D and 3D lattices by using time-dependent simulations. Our results are compared with rigorous bounds recently obtained by Liggett and by Katori and Tsukahara. The critical fractions…
In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the $d$-dimensional body-centered cubic (BCC) lattice $\mathbb{L}^d$ and the set of non-negative integers…
We study the isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ for supercritical bond percolation on $\mathbb{Z}^d$ with $d\geq 3$. Specifically, we consider the subgraphs of $\textbf{C}_\infty \cap [-n,n]^d$ which have…
A method to treat a N-component percolation model as effective one component model is presented by introducing a scaled control variable $p_{+}$. In Monte Carlo simulations on $16^{3}$, $32^{3}$, $64^{3}$ and $128^{3}$ simple cubic lattices…
We consider first-passage percolation on $\mathbb{Z}^2$ with i.i.d. weights, whose distribution function satisfies $F(0) = p_c = 1/2$. This is sometimes known as the "critical case" because large clusters of zero-weight edges force passage…
Let L_n denote the lowest crossing of the 2n \times 2n square box B(n) centered at the origin for critical site percolation on Z^2 or critical site percolation on the triangular lattice imbedded in Z^2, and denote by Q_n the set of pivotal…
We prove central limit theorems (CLTs) for topological functionals of Bernoulli bond percolation on infinite graphs beyond the Euclidean lattice $\mathbb{Z}^{d}$. For quasi-transitive graphs of subexponential growth, we show that the number…
We consider first-passage percolation on the edges of $\mathbb{Z}^2 \times \{1, \cdots, k\},$ namely the slab $\mathbb{S}_k$ of width $k$. Each edge is assigned independently a passage time of either 0 (with probability $p_c(\mathbb{S}_k)$)…
We investigate Laplacians on supercritical bond-percolation graphs with different boundary conditions at cluster borders. The integrated density of states of the Dirichlet Laplacian is found to exhibit a Lifshits tail at the lower spectral…
The stochastic addition of either vertices or connections in a network leads to the observation of the percolation transition, a structural change with the appearance of a connected component encompassing a finite fraction of the system.…
In this paper, we study independent (Bernoulli) bond percolation in dimensions $d \ge 2$, focusing on the maximum diameter of finite clusters in the non-critical regime ($p\neq p_c$). We prove that the maximum diameter $R_n$ satisfies $R_n…
In this thesis, I am going to consider Bernoulli percolation on graphs admitting vertex-transitive actions of groups of isometries of d-dimensional hyperbolic spaces H^d. In the first chapter, I give an overview concerning percolation and…
We present a general method for predicting bond percolation thresholds and critical surfaces for a broad class of two-dimensional periodic lattices, reproducing many known exact results and providing excellent approximations for several…
We study a model of nonintersecting Brownian bridges on an interval with either absorbing or reflecting walls at the boundaries, focusing on the point in space-time at which the particles meet the wall. These processes are determinantal,…
In this paper, we consider Bernoulli percolation on a locally finite, transitive and infinite graph (e.g. the hypercubic lattice $\mathbb{Z}^d$). We prove the following estimate, where $\theta_n(p)$ is the probability that there is a path…
We prove the asymptotic independence of the empirical process $\alpha_n = \sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $\beta_n = n (F_n(\tau+\frac{\cdot}{n})-F_n(\tau))$, where $F$ is an arbitrary cdf,…
For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For…
We consider first passage percolation with i.i.d. weights on edges of the d-dimensional cubic lattice. Under the assumptions that a weight is equal to zero with probability smaller than the critical probability of bond percolation in the…
Consider N Brownian bridges B_i:[-N,N] -> R, B_i(-N) = B_i(N) = 0, 1 <= i <= N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a limit as N -> infinity of these curves scaled around (0,2^{1/2} N)…