The stochastic geometry of invasion percolation

J. T. Chayes, L. Chayes, C. M. Newman

Research output: Contribution to journalArticle

Abstract

Invasion percolation, a recently introduced stochastic growth model, is analyzed and compared to the critical behavior of standard d-dimensional Bernoulli percolation. Various functions which measure the distribution of values accepted into the dynamically growing invaded region are studied. The empirical distribution of values accepted is shown to be asymptotically unity above the half-space threshold and linear below the point at which the expected cluster size diverges for the associated Bernoulli problem. An acceptance profile is defined and shown to have corresponding behavior. Quantities related to the geometry of the invaded region are studied, including the surface to volume ratio and the volume fraction. The former is shown to have upper and lower bounds in terms of the above defined critical points, and the latter is bounded above by the probability of connection to infinity at the half-space threshold. Provided that the critical regimes of Bernoulli percolation possess their anticipated properties, as is known to be the case in two dimensions, these results verify numerical predictions on the acceptance profile, establish the existence of a sharp surface to volume ratio and show that the invaded region has zero volume fraction. Large-time asymptotics are analyzed in terms of the probability that the invaded region accepts a value greater than x at time n. This quantity is shown to be bounded below by h(x)exp[-c(x)n(d-1)/d] for x above threshold, and to have an upper bound of the same form for x larger than a particular value (which coincides with the threshold in d=2). For two dimensions, it is also established that the infinite-time invaded region is essentially independent of initial conditions.

Original languageEnglish (US)
Pages (from-to)383-407
Number of pages25
JournalCommunications in Mathematical Physics
Volume101
Issue number3
DOIs
StatePublished - Sep 1985

Fingerprint

Invasion Percolation
Stochastic Geometry
Bernoulli
thresholds
geometry
half spaces
Volume Fraction
acceptability
Half-space
Two Dimensions
Large Time Asymptotics
Region Growing
Empirical Distribution
profiles
Growth Model
Critical Behavior
Diverge
infinity
Stochastic Model
unity

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

The stochastic geometry of invasion percolation. / Chayes, J. T.; Chayes, L.; Newman, C. M.

In: Communications in Mathematical Physics, Vol. 101, No. 3, 09.1985, p. 383-407.

Research output: Contribution to journalArticle

Chayes, J. T. ; Chayes, L. ; Newman, C. M. / The stochastic geometry of invasion percolation. In: Communications in Mathematical Physics. 1985 ; Vol. 101, No. 3. pp. 383-407.
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