Zero-temperature dynamics for the ferromagnetic Ising model on random graphs

2002 ◽  
Vol 310 (3-4) ◽  
pp. 275-284 ◽  
Author(s):  
Olle Häggström
Keyword(s):  
Ising Model ◽  
Random Graphs ◽  
Zero Temperature ◽  
Temperature Dynamics ◽  
Ferromagnetic Ising Model

Zero-temperature ising spin dynamics on the homogeneous tree of degree three

2000 ◽  
Vol 37 (03) ◽  
pp. 736-747 ◽  
Author(s):  
C. Douglas Howard
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Spin Dynamics ◽  
Spin Chains ◽  
Homogeneous Tree ◽  
Zero Temperature ◽  
Spin Configuration ◽  
Ising Spin ◽  
Temperature Dynamics ◽  
Ferromagnetic Ising Model

We investigate zero-temperature dynamics for a homogeneous ferromagnetic Ising model on the homogeneous tree of degree three (𝕋) with random (i.i.d. Bernoulli) spin configuration at time 0. Letting θ denote the probability that any particular vertex has a +1 initial spin, for infinite spin clusters do not exist at time 0 but we show that infinite ‘spin chains’ (doubly infinite paths of vertices with a common spin) exist in abundance at any time ϵ > 0. We study the structure of the subgraph of 𝕋 generated by the vertices in time-ϵ spin chains. We show the existence of a phase transition in the sense that, for some critical θ c with spin chains almost surely never form for θ < θc but almost surely do form in finite time for θ > θc . We relate these results to certain quantities of physical interest, such as the t → ∞ asymptotics of the probability that any particular vertex changes spin after time t.

Zero-temperature ising spin dynamics on the homogeneous tree of degree three

2000 ◽  
Vol 37 (3) ◽  
pp. 736-747 ◽  
Author(s):  
C. Douglas Howard
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Spin Dynamics ◽  
Spin Chains ◽  
Homogeneous Tree ◽  
Zero Temperature ◽  
Spin Configuration ◽  
Ising Spin ◽  
Temperature Dynamics ◽  
Ferromagnetic Ising Model

We investigate zero-temperature dynamics for a homogeneous ferromagnetic Ising model on the homogeneous tree of degree three (𝕋) with random (i.i.d. Bernoulli) spin configuration at time 0. Letting θ denote the probability that any particular vertex has a +1 initial spin, for infinite spin clusters do not exist at time 0 but we show that infinite ‘spin chains’ (doubly infinite paths of vertices with a common spin) exist in abundance at any time ϵ > 0. We study the structure of the subgraph of 𝕋 generated by the vertices in time-ϵ spin chains. We show the existence of a phase transition in the sense that, for some critical θc with spin chains almost surely never form for θ < θc but almost surely do form in finite time for θ > θc. We relate these results to certain quantities of physical interest, such as the t → ∞ asymptotics of the probability that any particular vertex changes spin after time t.

scholarly journals PERSISTENCE IN THE ZERO-TEMPERATURE DYNAMICS OF THE Q-STATES POTTS MODEL ON UNDIRECTED-DIRECTED BARABÁSI–ALBERT NETWORKS AND ERDÖS–RÉNYI RANDOM GRAPHS

2008 ◽  
Vol 19 (12) ◽  
pp. 1777-1785 ◽  
Author(s):  
F. P. FERNANDES ◽  
F. W. S. LIMA
Keyword(s):  
Random Graphs ◽  
Potts Model ◽  
Finite Size ◽  
Glauber Dynamics ◽  
The Other ◽  
Finite Size Effect ◽  
Zero Temperature ◽  
Temperature Dynamics ◽  
Short Times ◽  
Q Values

The zero-temperature Glauber dynamics is used to investigate the persistence probability P(t) in the Potts model with Q = 3, 4, 5, 7, 9, 12, 24, 64, 128, 256, 512, 1024, 4096, 16 384, …, 230 states on directed and undirected Barabási–Albert networks and Erdös–Rényi (ER) random graphs. In this model, it is found that P(t) decays exponentially to zero in short times for directed and undirected ER random graphs. For directed and undirected BA networks, in contrast it decays exponentially to a constant value for long times, i.e., P(∞) is different from zero for all Q values (here studied) from Q = 3, 4, 5, …, 230; this shows "blocking" for all these Q values. Except that for Q = 230 in the undirected case P(t) tends exponentially to zero; this could be just a finite-size effect since in the other "blocking" cases you may have only a few unchanged spins.

scholarly journals GROUND STATES OF ONE-DIMENSIONAL LONG-RANGE FERROMAGNETIC ISING MODEL WITH EXTERNAL FIELD

2012 ◽  
Vol 26 (03) ◽  
pp. 1150014 ◽  
Author(s):  
AZER KERIMOV
Keyword(s):  
Phase Diagram ◽  
Ising Model ◽  
External Field ◽  
Lattice Points ◽  
Temperature Phase ◽  
Zero Temperature ◽  
One Dimensional ◽  
Ferromagnetic Ising Model ◽  
The One ◽  
Temperature Phase Diagram

A zero-temperature phase-diagram of the one-dimensional ferromagnetic Ising model is investigated. It is shown that at zero temperature spins of any compact collection of lattice points with identically oriented external field are identically oriented.

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