2003/09/11 by H. S. Egawa, H.S. Egawa, S. Horata +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Box counting #Combinatorics #Euclidean geometry #Fractal #Fractal analysis #Fractal dimension #Fractal dimension on networks #Geodesic #Geometry #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Physics #Pure mathematics #Riemannian manifold #Simplicial complex #Simplicial manifold #Theoretical and Computational Physics #Topological and Geometric Data Analysis #hep-lat
paper · pdf · doi:10.1016/s0920-5632(03)02714-2
published as Nucl.Phys.Proc.Suppl.129:791-793,2004 · 3 figures(eps), Lattice2003(Gravity)
arxiv created 2003/09/11 · openalex publication_date 2004/03/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.