Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity

TL;DR

This study numerically explores how the fractal dimension of two-dimensional quantum gravity surfaces varies with matter central charge c, confirming theoretical predictions across a broad c range.

Contribution

It introduces a reformulation of the Q-state Potts model as a weighted percolation cluster model, enabling continuous variation of c on a dynamically triangulated lattice.

Findings

c-dependence of critical coupling matches theoretical predictions

String susceptibility results agree well with theory

Fractal dimension measurements align with predictions except near c≈1

Abstract

We numerically investigate the fractal structure of two-dimensional quantum gravity coupled to matter central charge c for $-2 \leq c \leq 1$. We reformulate Q-state Potts model into the model which can be identified as a weighted percolation cluster model and can make continuous change of Q, which relates c, on the dynamically triangulated lattice. The c-dependence of the critical coupling is measured from the percolation probability and susceptibility. The c-dependence of the string susceptibility of the quantum surface is evaluated and has very good agreement with the theoretical predictions. The c-dependence of the fractal dimension based on the finite size scaling hypothesis is measured and has excellent agreement with one of the theoretical predictions previously proposed except for the region near $c\approx 1$.