Geometric and spectral properties of causal maps
We study the random planar map obtained from a critical, finite variance, Galton–Watson plane tree by adding the horizontal connections between successive vertices at each level. This random graph is closely related to the well-known causal dynamical triangulation that was introduced by Ambjørn and Loll and has been studied extensively by physicists. We prove that the horizontal distances in the graph are smaller than the vertical distances, but only by a subpolynomial factor: The diameter of the set of vertices at level n is both o(n) and n^(1−o(1)). This enables us to prove that the spectral dimension of the infinite version of the graph is almost surely equal to 2, and consequently the random walk is diffusive almost surely. We also initiate an investigation of the case in which the offspring distribution is critical and belongs to the domain of attraction of an α-stable law for α ∈ (1,2), for which our understanding is much less complete.
Additional Information© 2020 EMS Publishing House. NC acknowledges support from the Institut Universitaire de France, ANR Graal (ANR-14-CE25-0014), ANR Liouville (ANR-15-CE40-0013) and ERC GeoBrown. TH and AN were supported by ISF grant 1207/15 and ERC grant 676970 RandGeom. TH was also supported by a Microsoft Research PhD Fellowship and he thanks Tel Aviv University and Université Paris-Sud Orsay for their hospitality during visits in which this work was carried out. TH also thanks Jian Ding for bringing the problem of resistance growth in the CDT to his attention. Lastly, we warmly thank the anonymous referees for many valuable comments on the manuscript.
Accepted Version - 1710.03137.pdf