Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity

Abstract

We numerically investigate the fractal structure of two-dimensional quantum gravity coupled to matter central charge c for −2 ≤ c ≤ 1. We reformulate Q-state Potts model into the model which can be identified as a weighted percolation cluster model and can make continuous change of Q, which relates c, on the dynamically triangulated lattice. The c-dependence of the critical coupling is measured from the percolation probability and susceptibility. The c-dependence of the string susceptibility of the quantum surface is evaluated and has very good agreement with the theoretical predictions. The c-dependence of the fractal dimension based on the finite size scaling hypothesis is measured and has excellent agreement with one of the theoretical predictions previously proposed except for the region near c ≈ 1.

Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity

“…2 " 1 2 p3 `?17q « 3.56. Since then all numerical estimates reported for these and other models are consistent with Watabiki's prediction [17,19,20,21,22]. The most accurate estimates are summarized in Table 1.…”

“…If (19) is satisfied then k n nÑ8 " c pn{n 0 q´1 {dγ (20) for some constant c « 1. To get a first idea of the rate of convergence to the asymptotics (20), we plot in Figure 9 the logarithmic ratios logpn 0 {nq logpk n {k n 0 q…”

“…with statistical error bars for the four models using both the graph distance (purple) and its dual (green). The advantage of considering the different distance measurements becomes clear upon inspecting these plots: the deviations from the scaling relation (20) appear with different sign, allowing one in principle to estimate the exponent d γ by eye at the point where the two curves converge. It is also immediately clear that the data is incompatible with d W γ in the case of bipolar-oriented and Schnyder-wood-decorated triangulations, and is much closer to d DG γ .…”

“…It is also immediately clear that the data is incompatible with d W γ in the case of bipolar-oriented and Schnyder-wood-decorated triangulations, and is much closer to d DG γ . To accurately estimate d γ , we make an ansatz for the leading-order correction to (20) of the form…”

The Hausdorff Dimension of Two-Dimensional Quantum Gravity

“…1−c , was then proposed [15] and claimed to fit well results from numerical simulations, in particular for c ∈ [−2, 0] [16][17][18]. However, it has been consistently observed that, at least for c ≥ 0, a value D H ≈ 4 is also plausible [19].…”