Continuum Percolation with Steps in an Annulus
نویسندگان
چکیده
R 2 according to a Poisson process with intensity 1 and let GA be the random graph with vertex set {xi} and edges xixj whenever xi − xj ∈ A. We show that if the area of A is large, then GA almost surely has an infinite component. Moreover, if we fix ε, increase r and let nc = nc(ε) be the area of A when this infinite component appears, then nc → 1 as ε → 0. This is in contrast to the case of a “square” annulus where we show that nc is bounded away from 1.
منابع مشابه
Conformal two-boundary loop model on the annulus
We study the two-boundary extension of a loop model—corresponding to the dense phase of the O(n) model, or to the Q = n state Potts model—in the critical regime −2 < n ≤ 2. This model is defined on an annulus of aspect ratio τ . Loops touching the left, right, or both rims of the annulus are distinguished by arbitrary (real) weights which moreover depend on whether they wrap the periodic direct...
متن کاملFractal Structure of Equipotential Curves on a Continuum Percolation Model
We numerically investigate the electric potential distribution over a twodimensional continuum percolation model between the electrodes. The model consists of overlapped conductive particles on the background with an infinitesimal conductivity. Using the finite difference method, we solve the generalized Laplace equation and show that in the potential distribution, there appear the quasi-equipo...
متن کاملContinuum percolation with steps in the square or the disc
In 1961 Gilbert defined a model of continuum percolation in which points are placed in the plane according to a Poisson process of density one, and two are joined if one lies within a disc of area A about the other. We prove some good bounds on the critical area Ac for percolation in this model. The proof is in two parts: first we give a rigorous reduction of the problem to a finite problem, an...
متن کاملFinite-Size Scaling in Two-dimensional Continuum Percolation Models
We test the universal finite-size scaling of the cluster mass order parameter in two-dimensional (2D) isotropic and directed continuum percolation models below the percolation threshold by computer simulations. We found that the simulation data in the 2D continuum models obey the same scaling expression of massM to sample size L as generally accepted for isotropic lattice problems, but with a p...
متن کامل2 00 6 The O ( n ) model on the annulus
We use Coulomb gas methods to propose an explicit form for the scaling limit of the partition function of the critical O(n) model on an annulus, with free boundary conditions, as a function of its modulus. This correctly takes into account the magnetic charge asymmetry and the decoupling of the null states. It agrees with all known results for special values of n and gives new formulae for perc...
متن کامل