The spectrum of asymptotic Cayley trees
Journal of Physics A: Mathematical and Theoretical IOP Publishing 57 (2024) 215202
Abstract:
We characterize the spectrum of the transition matrix for simple random walk on graphs consisting of a finite graph with a finite number of infinite Cayley trees attached. We show that there is a continuous spectrum identical to that for a Cayley tree and, in general, a non-empty pure point spectrum. We apply our results to studying continuous time quantum walk on these graphs. If the pure point spectrum is nonempty the walk is in general confined with a nonzero probability.From Trees to Gravity
Chapter in Handbook of Quantum Gravity, Springer Nature (2024) 3385-3435
The Spectrum of Asymptotic Cayley Trees
(2023)
From trees to gravity
Chapter in Handbook of Quantum Gravity, Spinger (2023)
Abstract:
In this article, we study two related models of quantum geometry: generic random trees and two-dimensional causal triangulations. The Hausdorff and spectral dimensions that arise in these models are calculated, and their relationship with the structure of the underlying random geometry is explored. Modifications due to interactions with matter fields are also briefly discussed. The approach to the subject is that of classical statistical mechanics, and most of the tools come from probability and graph theory.From Trees to Gravity
(2022)