# Spectral Methods and Running Scales in Causal Dynamical Triangulations

## TL;DR

This paper investigates how the spectrum of the Laplace-Beltrami operator on spatial slices in Causal Dynamical Triangulations reveals the behavior of length scales and critical phenomena near phase transitions, aiding the understanding of continuum limits.

## Contribution

It introduces a spectral analysis approach to study the geometrical properties and scale behavior in CDT, providing new insights into phase transitions and continuum limit prospects.

## Key findings

- Lowest eigenvalues indicate scale running with theory parameters
- Spectral properties reveal critical behavior near phase transitions
- Method offers a new probe for continuum limit in CDT

## Abstract

The spectrum of the Laplace-Beltrami operator, computed on the spatial slices of Causal Dynamical Triangulations, is a powerful probe of the geometrical properties of the configurations sampled in the various phases of the lattice theory. We study the behavior of the lowest eigenvalues of the spectrum and show that this can provide information about the running of length scales as a function of the bare parameters of the theory, hence about the critical behavior around possible second order transition points in the CDT phase diagram, where a continuum limit could be defined.