Super-Brownian Motion and Critical Spatial Stochastic Systems

2004 ◽  
Vol 47 (2) ◽  
pp. 280-297 ◽  
Author(s):  
Ed Perkins
Keyword(s):  
Brownian Motion ◽  
Limit Theorems ◽  
Stochastic Systems ◽  
Contact Process ◽  
Voter Model ◽  
Oriented Percolation ◽  
Short Introduction ◽  
Super Brownian Motion

AbstractThis article is a short introduction to super-Brownian motion. Some of its properties are discussed but our main objective is to describe a number of limit theorems which show super-Brownian motion is a universal limit for rescaled spatial stochastic systems at criticality above a critical dimenson. These systems include the voter model, the contact process and critical oriented percolation.

Lower bounds for the critical probability in percolation models with oriented bonds

1980 ◽  
Vol 17 (04) ◽  
pp. 979-986 ◽  
Author(s):  
Lawrence Gray ◽  
John C. Wierman ◽  
R. T. Smythe
Keyword(s):  
Lower Bounds ◽  
Oriented Percolation ◽  
Critical Probability ◽  
Critical Percolation ◽  
Simple Method

In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.

Approximate upper bounds for the critical probability of oriented percolation in two dimensions based on rapidly mixing Markov chains

1997 ◽  
Vol 34 (04) ◽  
pp. 859-867
Author(s):  
Béla Bollabás ◽  
Alan Stacey
Keyword(s):  
Monte Carlo ◽  
Markov Chains ◽  
Monte Carlo Simulations ◽  
Statistical Tests ◽  
Upper Bounds ◽  
Two Dimensions ◽  
Oriented Percolation ◽  
Critical Probability ◽  
Confidence Levels

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc < 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc ≈ 0.6445, this bound is fairly tight.

scholarly journals Random Walk on the Incipient Infinite Cluster for Oriented Percolation in High Dimensions

2008 ◽  
Vol 278 (2) ◽  
pp. 385-431 ◽  
Author(s):  
Martin T. Barlow ◽  
Antal A. Járai ◽  
Takashi Kumagai ◽  
Gordon Slade
Keyword(s):  
Random Walk ◽  
Oriented Percolation ◽  
High Dimensions ◽  
Infinite Cluster ◽  
Incipient Infinite Cluster

Stochastic growth models: bounds on critical values

1992 ◽  
Vol 29 (1) ◽  
pp. 11-20 ◽  
Author(s):  
R. Durrett
Keyword(s):  
Interacting Particle Systems ◽  
Growth Models ◽  
Upper Bounds ◽  
Particle Systems ◽  
Critical Values ◽  
Oriented Percolation ◽  
Finite Sets ◽  
Stochastic Growth ◽  
Interacting Particle ◽  
Stochastic Growth Models

We give upper bounds on the critical values for oriented percolation and some interacting particle systems by computing their behavior on small finite sets.

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