Fractal structure of 4D euclidean simplicial 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.

Asymptotic safety of higher derivative quantum gravity non-minimally coupled with a matter system

“…Asymptotically safe gravity in viewpoint of the fractal space-time structure has been studied in [110][111][112][113]. Analyses with lattice simulation have been performed in [114][115][116][117][118][119][120][121][122][123][124]. 6 See related studies [138][139][140][141].…”

Non-minimal coupling in Higgs–Yukawa model with asymptotically safe gravity

“…There havebeen attempts to read off the number of effective degrees of freedom (namely the spectral dimension) from the RG flows of the theory, in order to test whether asymptotically safe gravity can be achieved or not: such attempts have been made in [69,[120][121][122][123] using the FRG and in [124,125] using the lattice simulation; see also [126][127][128][129][130] for related studies. 9 An explicit derivation is shown in section 5.…”