Critical behavior of target space mean field theory

TL;DR

This paper analyzes the critical phenomena of the target space mean field (TSMF) matrix model, revealing two critical curves with distinct exponents and a bicritical point, by summing the free energy to all orders.

Contribution

It provides an exact all-orders expression for the free energy of the TSMF model and characterizes its critical behavior and phase structure.

Findings

Identified two critical curves with exponents alpha=1/2 and gamma_str=-1/2.

Discovered a bicritical point with gamma_str=+1/3.

Derived a transcendental algebraic equation for the free energy.

Abstract

Recently, the free energy of the target space mean field (TSMF) matrix model has been calculated in the low temperature phase, order-by-order in a low temperature expansion. The TSMF model is a matrix model whose discrete target space has an infinite coordination number, and whose free energy assumes a universal form, corresponding to baby universes joined into a tree. Here the free energy is summed to all orders, and expressed through a transcendental algebraic equation, using which we analyze the critical phenomena that occur in the TSMF model. There are two critical curves, at which the matter and the geometry become critical, and for which the critical exponents are alpha = 1/2 and gamma_str = -1/2, respectively. There is a bicritical point where the curves meet, at which gamma_str = +1/3.