A modified c=1 matrix model with new critical behavior

TL;DR

This paper introduces a modified $c=1$ matrix model with an additional interaction term, revealing a new critical behavior in the sum over random surfaces, characterized by a different divergence pattern at a special coupling value.

Contribution

The authors propose a modified $c=1$ matrix model with a new interaction term, uncovering a novel critical behavior in the sum over random surfaces not seen in the conventional model.

Findings

New critical behavior with $ ext{sum} o ext{const} imes \Delta^2 imes ext{log}\Delta$

Multiple spherical bubbles touching at points in the planar limit

Different divergence pattern compared to the conventional $c=1$ model

Abstract

By introducing a $\int dt \, g\left(\Tr \Phi^2(t)\right)^2$ term into the action of the $c=1$ matrix model of two-dimensional quantum gravity, we find a new critical behavior for random surfaces. The planar limit of the path integral generates multiple spherical ``bubbles'' which touch one another at single points. At a special value of $g$, the sum over connected surfaces behaves as $\Delta^2 \log\Delta$, where $\Delta$ is the cosmological constant (the sum over surfaces of area $A$ goes as $A^{-3}$). For comparison, in the conventional $c=1$ model the sum over planar surfaces behaves as $\Delta^2/ \log\Delta$.