Critical scaling in the matrix model on the Bethe tree

TL;DR

This paper analyzes a matrix model on a Bethe tree, revealing critical scaling behaviors and deriving explicit solutions using a Riemann-Hilbert approach, with implications for induced QCD models.

Contribution

It extends the Riemann-Hilbert method to analyze critical scaling in the matrix model on the Bethe tree, providing explicit solutions at specific parameters.

Findings

Critical scaling exponent $\gamma_{str}$ derived for different D values.

Explicit solutions constructed at D=1/2 and D=∞.

Identifies edge singularity behavior in the density of the matrix model.

Abstract

The matrix model with a Bethe-tree embedding space (coinciding at large $N$ with the Kazakov-Migdal ``induced QCD'' model \cite{KM}) is investigated. We further elaborate the Riemann-Hilbert approach of \rf{Mig1} assuming certain holomorphic properties of the solution. The critical scaling (an edge singularity of the density) is found to be $\gamma_{str} = -\frac{1}{\pi} \arcos D$, for $|D|<1$, and $\gamma_{str} = -\frac{1}{\pi} \arcos \frac{D}{2D-1}$, for $D>1$. Explicit solutions are constructed at $D=\frac{1}{2}$ and $D=\infty$.