Fate of 2D Kinetic Ferromagnets and Critical Percolation Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

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Estimates of critical percolation probabilities for a set of two-dimensional lattices

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100050325 ◽

1972 ◽

Vol 71 (1) ◽

pp. 97-106 ◽

Cited By ~ 16

Author(s):

D. G. Neal

Keyword(s):

Monte Carlo ◽

Two Dimensions ◽

Two Dimensional ◽

Critical Percolation ◽

Site Percolation

AbstractThis paper describes new detailed Monte Carlo investigations into bond and site percolation problems on the set of eleven regular and semi-regular (Archimedean) lattices in two dimensions.

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Approximations of boundary crossing probabilities for a Brownian motion

Journal of Applied Probability ◽

10.1239/jap/1032374752 ◽

1999 ◽

Vol 36 (4) ◽

pp. 1019-1030 ◽

Cited By ~ 47

Author(s):

Alex Novikov ◽

Volf Frishling ◽

Nino Kordzakhia

Keyword(s):

Brownian Motion ◽

Power Series ◽

Numerical Integration ◽

Piecewise Linear ◽

Boundary Crossing ◽

Square Root ◽

Girsanov Transformation ◽

Piecewise Linear Approximations ◽

Infinite Power ◽

Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

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A PERCOLATION MODEL OF DIAGENESIS

International Journal of Modern Physics C ◽

10.1142/s0129183102003176 ◽

2002 ◽

Vol 13 (03) ◽

pp. 319-331 ◽

Cited By ~ 7

Author(s):

S. S. MANNA ◽

T. DATTA ◽

R. KARMAKAR ◽

S. TARAFDAR

Keyword(s):

Numerical Simulation ◽

Critical Behavior ◽

Sedimentary Rocks ◽

Two Dimensions ◽

Percolation Model ◽

Type Model ◽

Stable Configuration ◽

Critical Percolation ◽

Restructuring Process ◽

Simulation Results

The restructuring process of diagenesis in the sedimentary rocks is studied using a percolation type model. The cementation and dissolution processes are modeled by the culling of occupied sites in rarefied and growth of vacant sites in dense environments. Starting from sub-critical states of ordinary percolation the system evolves under the diagenetic rules to critical percolation configurations. Our numerical simulation results in two dimensions indicate that the stable configuration has the same critical behavior as the ordinary percolation.

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