On the spectral dimension of causal triangulations

TL;DR

This paper studies the spectral and Hausdorff dimensions of infinite causal triangulations, showing they almost surely have Hausdorff dimension 2 and spectral dimension less than or equal to 2, with some cases exactly 2, relevant to 2D quantum gravity.

Contribution

It introduces the uniform infinite causal triangulation ensemble, proves its equivalence to infinite trees, and establishes spectral and Hausdorff dimension results relevant to quantum gravity models.

Findings

Hausdorff dimension almost surely equals 2

Spectral dimension is almost surely ≤ 2

Reduced models have spectral dimension exactly 2

Abstract

We introduce an ensemble of infinite causal triangulations, called the uniform infinite causal triangulation, and show that it is equivalent to an ensemble of infinite trees, the uniform infinite planar tree. It is proved that in both cases the Hausdorff dimension almost surely equals 2. The infinite causal triangulations are shown to be almost surely recurrent or, equivalently, their spectral dimension is almost surely less than or equal to 2. We also establish that for certain reduced versions of the infinite causal triangulations the spectral dimension equals 2 both for the ensemble average and almost surely. The triangulation ensemble we consider is equivalent to the causal dynamical triangulation model of two-dimensional quantum gravity and therefore our results apply to that model.