# Fixed versus random triangulations in 2D simplicial Regge calculus

## TL;DR

This paper compares fixed and random triangulations in 2D quantum gravity using Regge calculus, analyzing their effects on geometric expectations and string susceptibility exponents.

## Contribution

It introduces a systematic comparison between fixed and random triangulations in 2D quantum gravity within the Regge calculus framework, highlighting size-dependent differences.

## Key findings

- Difference in <R^2> depends on lattice size and surface area
- String susceptibility exponent γ'_str matches theoretical predictions
- Estimate for γ_str appears overly negative

## Abstract

We study 2D quantum gravity on spherical topologies using the Regge calculus approach with the $dl/l$ measure. Instead of a fixed non-regular triangulation which has been used before, we study for each system size four different random triangulations, which are obtained according to the standard Voronoi-Delaunay procedure. We compare both approaches quantitatively and show that the difference in the expectation value of $R^2$ between the fixed and the random triangulation depends on the lattice size and the surface area $A$. We also try again to measure the string susceptibility exponents through a finite-size scaling Ansatz in the expectation value of an added $R^2$ interaction term in an approach where $A$ is held fixed. The string susceptibility exponent $\gamma_{str}'$ is shown to agree with theoretical predictions for the sphere, whereas the estimate for $\gamma_{str}$ appears to be too negative.