Scaling analyses of the spectral dimension in 3-dimensional causal dynamical triangulations

TL;DR

This paper investigates the scale-dependent spectral dimension in 3D causal dynamical triangulations, revealing finite behavior, scale reduction near 2, and implications for quantum geometry and renormalization.

Contribution

It provides the first comprehensive scaling analysis of the spectral dimension in 3D CDT, showing its finite nature and relation to phase transitions and quantum geometry.

Findings

Spectral dimension scales trivially with diffusion constant.

Spectral dimension remains finite in the infinite volume limit.

Spectral dimension approaches 2 near phase transition.

Abstract

The spectral dimension measures the dimensionality of a space as witnessed by a diffusing random walker. Within the causal dynamical triangulations approach to the quantization of gravity, the spectral dimension exhibits novel scale-dependent dynamics: reducing towards a value near 2 on sufficiently small scales, matching closely the topological dimension on intermediate scales, and decaying in the presence of positive curvature on sufficiently large scales. I report the first comprehensive scaling analysis of the small-to-intermediate scale spectral dimension for the test case of the causal dynamical triangulations of 3-dimensional Einstein gravity. I find that the spectral dimension scales trivially with the diffusion constant. I find that the spectral dimension is completely finite in the infinite volume limit, and I argue that its maximal value is exactly consistent with the topological dimension of 3 in this limit. I find that the spectral dimension reduces further towards a value near 2 as this case's bare coupling approaches its phase transition, and I present evidence against the conjecture that the bare coupling simply sets the overall scale of the quantum geometry. On the basis of these findings, I advance a tentative physical explanation for the dynamical reduction of the spectral dimension observed within causal dynamical triangulations: branched polymeric quantum geometry on sufficiently small scales. My analyses should facilitate attempts to employ the spectral dimension as a physical observable with which to delineate renormalization group trajectories in the hope of taking a continuum limit of causal dynamical triangulations at a nontrivial ultraviolet fixed point.