Geometric and spectral properties of causal maps

TL;DR

This paper analyzes the geometric and spectral properties of a class of random planar maps derived from Galton-Watson trees, establishing their diffusive behavior and spectral dimension, with extensions to stable offspring distributions.

Contribution

It proves subpolynomial bounds on horizontal distances, confirms the spectral dimension is almost surely 2, and explores cases with stable offspring distributions.

Findings

Horizontal distances are smaller than vertical distances by a subpolynomial factor.

Spectral dimension of the infinite graph is almost surely 2.

Random walk on the graph is diffusive almost surely.

Abstract

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 Ambj{\o}rn 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 that 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 $\alpha$-stable law for $\alpha \in (1,2)$, for which our understanding is much less complete.