scholarly journals An Exactly Solvable Anisotropic Directed Percolation Model in Three Dimensions

1998 ◽  
Vol 81 (8) ◽  
pp. 1646-1649 ◽  
Author(s):  
R. Rajesh ◽  
Deepak Dhar
Keyword(s):  
Percolation Model ◽  
Three Dimensions ◽  
Directed Percolation ◽  
Exactly Solvable

scholarly journals A class of exactly solvable rationally extended non-central potentials in two and three dimensions

2018 ◽  
Vol 59 (6) ◽  
pp. 062103 ◽  
Author(s):  
Nisha Kumari ◽  
Rajesh Kumar Yadav ◽  
Avinash Khare ◽  
Bhabani Prasad Mandal
Keyword(s):  
Three Dimensions ◽  
Exactly Solvable

scholarly journals Sharpness of the Phase Transition for the Orthant Model

2021 ◽  
Vol 24 (4) ◽  
Author(s):  
Thomas Beekenkamp
Keyword(s):  
Phase Transition ◽  
Random Walk ◽  
Critical Value ◽  
Percolation Model ◽  
Critical Threshold ◽  
Directed Percolation ◽  
Sharp Threshold ◽  
Shape Theorem ◽  
Exponentially Small

AbstractThe orthant model is a directed percolation model on $\mathbb {Z}^{d}$ ℤ d , in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.

THE QUANTIZATION RULE AND MASLOV INDEX

2010 ◽  
Vol 24 (22) ◽  
pp. 2303-2316
Author(s):  
XIAO-YAN GU
Keyword(s):  
Harmonic Oscillator ◽  
Maslov Index ◽  
Three Dimensions ◽  
Energy Spectra ◽  
Quantization Rule ◽  
Solvable Potentials ◽  
Rule Approach ◽  
Exactly Solvable Potentials ◽  
Exactly Solvable ◽  
Arbitrary Dimensions

Within extensions of the new quantization rule approach in arbitrary dimensions, the Maslov indices and energy spectra of some exactly solvable potentials are presented. We find that the Maslov index for the harmonic oscillator in three dimensions agrees well with those obtained by other methods.

Spatial dispersion of interstellar civilizations: a probabilistic site percolation model in three dimensions

2012 ◽  
Vol 12 (1) ◽  
pp. 45-52 ◽  
Author(s):  
Thomas W. Hair ◽  
Andrew D. Hedman
Keyword(s):  
Cellular Automaton ◽  
Uniform Distribution ◽  
Spatial Dispersion ◽  
Three Dimensional ◽  
Percolation Model ◽  
Three Dimensions ◽  
Site Percolation ◽  
Original Seed ◽  
Probabilistic Cellular Automaton

AbstractA model of the spatial emergence of an interstellar civilization into a uniform distribution of habitable systems is presented. The process of emigration is modelled as a three-dimensional probabilistic cellular automaton. An algorithm is presented which defines both the daughter colonies of the original seed vertex and all subsequent connected vertices, and the probability of a connection between any two vertices. The automaton is analysed over a wide set of parameters for iterations that represent up to 250 000 years within the model's assumptions. Emigration patterns are characterized and used to evaluate two hypotheses that aim to explain the Fermi Paradox. The first hypothesis states that interstellar emigration takes too long for any civilization to have yet come within a detectable distance, and the second states that large volumes of habitable space may be left uninhabited by an interstellar civilization and Earth is located in one of these voids.

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