Fractal Structure of 4D Euclidean Simplicial Manifold

TL;DR

This paper investigates the fractal properties of 4D Euclidean simplicial manifolds generated by dynamical triangulation, using a novel box-counting method based on geodesic distance, revealing insights into their fractal dimension and phase structure.

Contribution

It introduces a new approach to measure the fractal dimension of 4D manifolds via geodesic distances and box-counting, providing consistent results with random walk models and exploring effects of matter fields.

Findings

Fractal dimension slightly exceeds 4 near the critical point.

The box-counting method effectively characterizes the fractal structure.

Correlation functions reveal phase-dependent behavior of the manifold.

Abstract

The fractal properties of four-dimensional Euclidean simplicial manifold generated by the dynamical triangulation are analyzed on the geodesic distance D between two vertices instead of the usual scale between two simplices. In order to make more unambiguous measurement of the fractal dimension, we employ a different approach from usual, by measuring the box-counting dimension which is computed by counting the number of spheres with the radius D within the manifold. The numerical result is consistent to the result of the random walk model in the branched polymer region. We also measure the box-counting dimension of the manifold with additional matter fields. Numerical results suggest that the fractal dimension takes value of slightly more than 4 near the critical point. Furthermore, we analyze the correlation functions as functions of the geodesic distance. Numerically, it is suggested that the fractal structure of four-dimensional simplicial manifold can be properly analyzed in terms of the distance between two vertices. Moreover, we show that the behavior of the correlation length regards the phase structure of 4D simplicial manifold.