Duality Transformations for Two-Dimensional Directed Percolation and Resistance Problems

1981 ◽  
Vol 47 (18) ◽  
pp. 1238-1241 ◽  
Author(s):  
Deepak Dhar ◽  
Mustansir Barma ◽  
Mohan K. Phani
Keyword(s):  
Directed Percolation ◽  
Two Dimensional ◽  
Duality Transformations

On two-dimensional directed percolation

1988 ◽  
Vol 21 (19) ◽  
pp. 3815-3832 ◽  
Author(s):  
J W Essam ◽  
A J Guttmann ◽  
K De'Bell
Keyword(s):  
Directed Percolation ◽  
Two Dimensional

Directed percolation in the two-dimensional continuum

1985 ◽  
Vol 32 (1) ◽  
pp. 527-529 ◽  
Author(s):  
I. Balberg ◽  
N. Binenbaum
Keyword(s):  
Directed Percolation ◽  
Two Dimensional

scholarly journals Critical behavior of density-driven and shear-driven reversible–irreversible transitions in cyclically sheared vortices

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
S. Maegochi ◽  
K. Ienaga ◽  
S. Okuma
Keyword(s):  
Power Law ◽  
Critical Behavior ◽  
Particle Density ◽  
Universality Class ◽  
Time Dependent ◽  
Directed Percolation ◽  
Two Dimensional ◽  
Significant Drop ◽  
Cyclic Shear ◽  
Vortex Density

AbstractRandom assemblies of particles subjected to cyclic shear undergo a reversible–irreversible transition (RIT) with increasing a shear amplitude d or particle density n, while the latter type of RIT has not been verified experimentally. Here, we measure the time-dependent velocity of cyclically sheared vortices and observe the critical behavior of RIT driven by vortex density B as well as d. At the critical point of each RIT, $$B_{\mathrm {c}}$$ B c and $$d_{\mathrm {c}}$$ d c , the relaxation time $$\tau $$ τ to reach the steady state shows a power-law divergence. The critical exponent for B-driven RIT is in agreement with that for d-driven RIT and both types of RIT fall into the same universality class as the absorbing transition in the two-dimensional directed-percolation universality class. As d is decreased to the average intervortex spacing in the reversible regime, $$\tau (d)$$ τ ( d ) shows a significant drop, indicating a transition or crossover from a loop-reversible state with vortex-vortex collisions to a collisionless point-reversible state. In either regime, $$\tau (d)$$ τ ( d ) exhibits a power-law divergence at the same $$d_{\mathrm {c}}$$ d c with nearly the same exponent.

Duality of the Two-dimensional Ising Model

2020 ◽  
pp. 161-188
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Phase Transition ◽  
High Temperature ◽  
Ising Model ◽  
Statistical Models ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Duality Transformations ◽  
Peierls Argument ◽  
Chapter Deal ◽  
Sum Formula

Chapter 4 begins by discussing the Peierls argument, which allows us to prove the existence of a phase transition in the two-dimensional Ising model. The remaining sections of the chapter deal with duality transformations (duality in square, hexagonal and triangular lattices) that link the low- and high-temperature phases of several statistical models. Particularly important is the proof of the so-called star-triangle identity. This identity will be crucial in the later discussion of the transfer matrix of the Ising model. Finally, it covers the aspect of duality in two dimensions. An appendix provides information about the Poisson sum formula.

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