On the spectral dimension of causal triangulations

Details On the spectral dimension of causal triangulations On the spectral dimension of causal triangulations

2009-08-25

arxiv.org/abs/0908.3643v2

Physics

Mathematical Physics

22 pages, 6 figures; typos fixed, one extra figure, references updated

Scientific paper

10.1007/s10955-010-9968-x

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.

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