On the width of handles in two-dimensional quantum gravity

TL;DR

This paper investigates the behavior of the shortest non-contractible loop in 2D quantum gravity, showing it diverges at criticality with a critical exponent of 1/2, consistent with branched polymer models.

Contribution

It provides a detailed analysis of the scaling behavior of the shortest loop in 2D quantum gravity and establishes the critical exponent as 1/2, supported by comparisons with branched polymers.

Findings

The average length of the shortest non-contractible loop diverges at the critical point.

The critical exponent for this divergence is 1/2.

The same critical exponent is observed in branched polymer models.

Abstract

We discuss the average length l of the shortest non-contractible loop on surfaces in the two-dimensional pure quantum gravity ensemble. The value of $\gamma_{str}$ and the explicit form of the loop functions indicate that l diverges at the critical point. Scaling arguments suggest that the critical exponent of l is 1/2. We show that this value of the critical exponent is also obtained for branched polymers where the calculation is straightforward.