On Finite Size Effects in $d=2$ Quantum Gravity

TL;DR

This paper systematically studies finite size effects in two-dimensional quantum gravity and coupled Ising models, combining numerical and analytical methods to understand how finite size corrections depend on the string susceptibility.

Contribution

It introduces a combined numerical and analytical approach to analyze finite size effects in 2D quantum gravity and coupled Ising models, including cases with unknown matrix model free energy.

Findings

Finite size corrections depend on string susceptibility.

Analytic methods for singularity analysis are developed for unknown free energy forms.

Fisher-scaling laws are confirmed for the Ising model.

Abstract

A systematic investigation is given of finite size effects in $d=2$ quantum gravity or equivalently the theory of dynamically triangulated random surfaces. For Ising models coupled to random surfaces, finite size effects are studied on the one hand by numerical generation of the partition function to arbitrary accuracy by a deterministic calculus, and on the other hand by an analytic theory based on the singularity analysis of the explicit parametric form of the free energy of the corresponding matrix model. Both these reveal that the form of the finite size corrections, not surprisingly, depend on the string susceptibility. For the general case where the parametric form of the matrix model free energy is not explicitly known, it is shown how to perform the singularity analysis. All these considerations also apply to other observables like susceptibility etc. In the case of the Ising model it is shown that the standard Fisher-scaling laws are reproduced.