The Potts-q random matrix model : loop equations, critical exponents, and rational case

TL;DR

This paper extends the analysis of Potts-q random matrix models to arbitrary q, deriving loop equations, algebraic resolvent equations for specific q values, and calculating critical exponents and string susceptibility.

Contribution

It introduces a general set of loop equations for Potts-q models, generalizes algebraic resolvent equations for specific q, and computes critical exponents and string susceptibility for 0 to 4 q.

Findings

Loop equations valid for any q.

Algebraic equations for resolvent at specific q values.

Critical exponents and string susceptibility for 0 ≤ q ≤ 4.

Abstract

In this article, we study the q-state Potts random matrix models extended to branched polymers, by the equations of motion method. We obtain a set of loop equations valid for any arbitrary value of q. We show that, for q=2-2 \cos {l \over r} \pi (l, r mutually prime integers with l < r), the resolvent satisfies an algebraic equation of degree 2 r -1 if l+r is odd and r-1 if l+r is even. This generalizes the presently-known cases of q=1, 2, 3. We then derive for any 0 \leq q \leq 4 the Potts-q critical exponents and string susceptibility.