Improved Lower Bounds for the Critical Probability of Oriented Bond Percolation in Two Dimensions

We present a coupled decreasing sequence of random walks on Z that dominate the edge process of oriented bond percolation in two dimensions. Using the concept of random walk in a strip, we describe an algorithm that generates an increasing sequence of lower bounds that converges to the critical probability of oriented percolation p_c. From the 7th term on, these lower bounds improve upon 0.6298, the best rigorous lower bound at present, establishing 0.63328 as a rigorous lower bound for p_c. Finally, a Monte Carlo simulation technique is presented; the use thereof establishes 0.64450 as a non-rigorous five-digit-precision (lower) estimate for p_c. Mathematics Subject Classification (1991): 60K35 Supported by CNPq (grant N.301637/91-1). Supported by a grant from CNPq.     

Nucleation near the spinodal in long-range Ising models

Properties of metastable long-range Ising models (LRIMs) are studied for deep quenches near the mean-field spinodal with Monte Carlo simulations using Glauber dynamics. The theory of spinodal-assisted nucleation is found to agree well with the data. Nucleating droplets are shown to have the same structure as large clusters in random long-range bond percolation.     

Ising model in a quasiperiodic transverse field,percolation, and contact processes in quasiperiodic environments

Quantum Ising models in a transverse field are related to continuous-time percolation processes whose oriented percolation versions are contact processes. We study such models in the presence of quasiperiodic disorder and prove localization in the ground state, no percolation, and extinction, respectively, for sufficiently large disorder.     

No directed fractal percolation in zero area

We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {p _k}; that is, for allk ⩾ 1, at thek th stage every retained square of side lengthN ¹⁻ k is partitioned intoN ² congruent subsquares, and each of these is retained with probabilityp _k. independent of all others. We show that if Π_k p _k =0 (i.e., if the area of the limiting set vanishes a.s.), then a.s. the limiting set contains no directed crossings of the unit square (a directed crossing is a path that crosses the unit square from left to right, and moves only up, down, and to the right).     

Scale-free random graphs and Potts model

We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertexi has a prescribed weight P_i ∝ i^-μ (0 < μ< 1) and an edge can connect verticesi andj with rateP _i P _j . Corresponding equilibrium ensemble is identified and the problem is solved by theq → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, whereλ = 1 +μ ^-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finiteN shows double peaks.     

ASYMPTOTIC BEHAVIOR OF BOOTSTRAP SPECTRAL WINDOW ESTIMATION

ThisprojectissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionLetX~{X(t);t=0,FI,12,'}beazerthmeantunitvariancestationaryGaussianprocess,theautocovariancefunction7(u)=EX(n u)X(n)satisfySupposethatthespectraldensityfunctionisf(A)andspectraldistributionfunctionisF(A)oftheprocessX,wherewerestrictAwithin11(if=[--x,7]).Fromtheassumptionsabove,wecaneasilyseethatthef(A)isjustaprobabilitydensityfunction,andF(A)isaprobabilitydistributionfunction.LetX(1),'tX(n)betheobserv…     

Local bootstrap

The local bootstrap method draws bootstrap samples from a neighborhood of each data point. Unlike the common bootstrap, it is especially useful for heteroscedastic data. This paper considers local bootstrap of y for a data set {(x _i , y _i )} with the neighborhood defined by the window in a kernel regression model. It shows that for each x, the bootstrap distribution and moments converge almost surely to the true distribution and moments of y respectively, and this convergence is also uniform in x within the data set.     

Numerical study of local and global persistence in directed percolation

The local persistence probability P _l (t) that a site never becomes active up to time t, and the global persistence probability P _g (t) that the deviation of the global density from its mean value does not change its sign up to time t are studied in a (1+1)-dimensional directed percolation process by Monte-Carlo simulations. At criticality, starting from random initial conditions, P _l (t) decays algebraically with the exponent . The value is found to be independent of the initial density and the microscopic details of the dynamics, suggesting is an universal exponent. The global persistence exponent is found to be equal or larger than . This contrasts with previously known cases where . It is shown that in the special case of directed-bond percolation, P _l (t) can be related to a certain return probability of a directed percolation process with an active source (wet wall). Received: 15 December 1997 / Revised: 6 April 1998 / Accepted: 29 May 1998