Self-Organized Critical Behavior: The Evolution of Frozen Spin Networks Model in Quantum Gravity

Abstract

In quantum gravity, we study the evolution of a two-dimensional planar open frozen spin network, in which the color (i.e. the twice spin of an edge) labeling edge changes but the underlying graph remains fixed. The mainly considered evolution rule, the random edge model, is depending on choosing an edge randomly and changing the color of it by an even integer. Since the change of color generally violate the gauge invariance conditions imposed on the system, detailed propagation rule is needed and it can be defined in many ways. Here, we provided one new propagation rule, in which the involved even integer is not a constant one as in previous works, but changeable with certain probability. In random edge model, we do find the evolution of the system under the propagation rule exhibits power-law behavior, which is suggestive of the self-organized criticality (SOC), and it is the first time to verify the SOC behavior in such evolution model for the frozen spin network. Furthermore, the increase of the average color of the spin network in time can show the nature of inflation for the universe.

Ternary Social Networks: Dynamic Balance and Self-Organized Criticality

“…where N is the size of the network, and M = i σ i (t) /N is the average value of all spins. So the spin flipping probability depends both on the triad relation and the surroundings [18,19].…”

“…Since the values of each spin are more than two, the spin flipping mechanism changes into the spin transition mechanism. For the spin transition mechanism, the spin takes one of the three values, +1, 0 and -1, depending both on the triad relation and the surroundings [18,19]. The Hamiltonian, the spin transition probabilities and the geometrical temperature are illustrated by Eqs.…”

“…The second modification is the judgement of a triad relation. In order to embody the majority principle, we take the judgement as the gauge invariance or the gauge variance [19][20][21][22], which is explained as follows. Under this judgement, the triad relation evolves from one state to the other by changing node's value.…”

“…While the hostile faces are in the majority, the triad relation is in the other state, or the gauge variance state. The explicit expressions of the gauge invariance and the gauge variance states [19][20][21][22] are as follows.…”

“…Since each node has three values, the way of spin changing is the spin transition [18]. Besides, we propose the spin transition depends both on itself and the surroundings [18,19]. Considering the SOC is determined by the structure of the system, we set the geometric temperature as β = 1 T > 0, where…”