Fragmented perspective of self-organized criticality and disorder in log gravity

Abstract

We use a statistical model to discuss nonequilibrium fragmentation phenomena taking place in the stochastic dynamics of the log sector in log gravity. From the canonical Gibbs model, a combinatorial analysis reveals an important aspect of the $n$-particle evolution previously shown to generate a collection of random partitions according to the Ewens distribution realized in a disconnected double Hurwitz number in genus zero. By treating each possible partition as a member of an ensemble of fragmentations, and ensemble averaging over all partitions with the Hurwitz number as a special case of the Gibbs distribution, a resulting distribution of cluster sizes appears to fall as a power of the size of the cluster. Dynamical systems that exhibit a distribution of sizes giving rise to a scale-invariant power-law behavior at a critical point possess an important property called self-organized criticality. As a corollary, the log sector of log gravity is a self-organized critical system at the critical point $μl =1$. A similarity between self-organized critical systems, spin glass models and the dynamics of the log sector which exhibits aging behavior reminiscent of glassy systems is pointed out by means of the Pòlya distribution, also known to classify various models of (randomly fragmented) disordered systems, and by presenting the cluster distribution in the log sector of log gravity as a distinguished member of this probability distribution. We bring arguments from a probabilistic perspective to discuss the disorder in log gravity, largely anticipated through the conjectured AdS$_3$/LCFT$_2$ correspondence.