Critical Dynamics of Random Surfaces: Time Evolution of Area and Genus

TL;DR

This paper investigates the critical dynamics of random surfaces in conformal field theories, focusing on how their area and genus evolve over time, revealing phase transitions and statistical properties of these geometric features.

Contribution

It extends the study of random surfaces from equilibrium to dynamic behavior, proposing models for social network evolution and analyzing phase transitions in surface genus.

Findings

Area follows Cox Ingersol Ross process

Planar surfaces shrink while higher genus surfaces grow

Two phases: planar-dominated and foamy, diverging genus

Abstract

Conformal field theories with central charge $c\le1$ on random surfaces have been extensively studied in the past. Here, this discussion is extended from their equilibrium distribution to their critical dynamics. This is motivated by the conjecture that these models describe the time evolution of certain social networks that are self-driven to a critical point. This paper focuses on the dynamics of the overall area and the genus of the surface. The time evolution of the area is shown to follow a Cox Ingersol Ross process. Planar surfaces shrink, while higher genus surfaces grow to a size of order of the inverse cosmological constant. The time evolution of the genus is argued to lead to two different phases, dominated by (i) planar surfaces, and (ii) ``foamy'' surfaces, whose genus diverges. In phase (i), which exhibits critical phenomena, time variations of the order parameter are approximately t-distributed with 4 or more degrees of freedom.