Ergodic Concepts for a Self-Organizing Trivalent Spin Network: A Path to $(2+1)$-dimensional Black Hole Entropy

Abstract

We consider, from a dynamical systems point of view, a trivalent spin network model in Loop Quantum Gravity presenting self-organized criticality (SOC), arising from a spin propagation dynamics. We obtain a partition function for the domains of stability connecting gauge non-invariant avalanches, leading to an entropy formula for the asymptotic SOC state. The microscopic origin of this SOC entropy is therefore given by the excitation-relaxation spin dynamics in the avalanche cycle. The puncturing of TSN edges participating in the avalanche are counted in terms of an ensemble perimeter over the implicit avalanches. By identifying this perimeter with that of an isolated $(2+1)$-dim. black hole horizon, we conjecture that the SOC entropy reduces to the Bekenstein-Hawking perimeter-entropy law for the Ba\~nados, Teitelboim, and Zanelli (BTZ) black hole, by an appropriate adjustment of a potential function based on the thermodynamical formalism of Sinai, Ruelle and Bowen.