Covering by random intervals and one-dimensional continuum percolation
- Univ. of Sussex, Brighton (England)
A brief historical introduction is given to the problem of covering a line by random overlapping intervals. The problem for equal intervals was first solved by Whitworth in the 1890s. A brief resume is given of his solution. The advantages of the present author's approach, which uses a Poisson process, are outlined, and a solution is derived by Laplace transforms. The asymptotic behavior as the line becomes long is calculated and is related to the one-dimensional continuum percolation problem. It is shown that as long as the mean interval size is finite, the probability of complete coverage decays exponentially, so that the critical percolation probability p{sub c} = 1. However, as soon as the mean interval size becomes infinite, the critical percolation probability p{sub c} switches to 0. This is in accord with previous results for a lattice model by Chinese workers, but differs from those of Schulman. A possible reason for the discrepancy is a difference in boundary conditions.
- OSTI ID:
- 5504835
- Journal Information:
- Journal of Statistical Physics; (USA), Vol. 55:1-2; ISSN 0022-4715
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
SUPERCONDUCTIVITY AND SUPERFLUIDITY
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CRYSTAL MODELS
RANDOMNESS
BOUNDARY CONDITIONS
CRYSTAL LATTICES
HISTORICAL ASPECTS
ISING MODEL
LAPLACE TRANSFORMATION
ONE-DIMENSIONAL CALCULATIONS
POISSON EQUATION
PROBABILITY
STATISTICAL MECHANICS
STOCHASTIC PROCESSES
TOPOLOGICAL MAPPING
CRYSTAL STRUCTURE
DIFFERENTIAL EQUATIONS
EQUATIONS
INTEGRAL TRANSFORMATIONS
MAPPING
MATHEMATICAL MODELS
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
TRANSFORMATIONS
656002* - Condensed Matter Physics- General Techniques in Condensed Matter- (1987-)
657002 - Theoretical & Mathematical Physics- Classical & Quantum Mechanics