Critical behavior in $c=1$ matrix model with branching interactions

TL;DR

This paper explores the phase structure of a $c=1$ matrix model with branching interactions, revealing three distinct phases and multi-critical points, which enhance understanding of critical phenomena in matrix field theories.

Contribution

It introduces a $c=1$ matrix model with branching interactions and identifies multiple phases, including a new intermediate phase and multi-critical points, advancing the study of critical phenomena in matrix models.

Findings

Identified three distinct phases: gravity, branched polymer, and intermediate.

Discovered multi-critical points in the generalized interaction model.

Analyzed phase transitions and critical behavior in the model.

Abstract

Motivated by understanding the phase structure of $d >1$ strings we investigate the $c=1$ matrix model with $g' (\tr M(t)^{2})^{2}$ interaction which is the simplest approximation of the model expected to describe the critical phenomena of the large-$N$ reduced model of odd-dimensional matrix field theory. We find three distinct phases: (i) an ordinary $c=1$ gravity phase, (ii) a branched polymer phase and (iii) an intermediate phase. Further we can also analyse the one with slightly generalized $ g^{(2)} (\frac{1}{N}\tr M(t)^{2})^{2} +g^{(3)} (\frac{1}{N}\tr M(t)^{2})^{3} + \cdots + g^{(n)} (\frac{1}{N}\tr M(t)^{2})^{n} $ interaction. As a result the multi-critical versions of the phase (ii) are found.