Numerical study for the \(c\)-dependence of fractal dimension in two-dimensional quantum gravity. (English) Zbl 0999.83017
Summary: We numerically investigate the fractal structure of two-dimensional quantum gravity coupled to a matter central charge \(c\) for \(-2\leqslant c\leqslant 1\). We reformulate the \(Q\)-state Potts model into a model which can be identified as a weighted percolation cluster model and can make continuous changes 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 \(c\approx 1\).
MSC:
| 83C27 | Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory |
| 83C45 | Quantization of the gravitational field |
Keywords:
\(Q\)-state Potts model; weighted percolation cluster model; dynamically triangulated latticeReferences:
| [1] | Knizhnik, V.; Polyakov, A. M.; Zamolodchikov, A., Mod. Phys. Lett. A, 3, 819 (1988) |
| [2] | Distler, J.; Hlousek, Z.; Kawai, H., Mod. Phys. Lett. A, 5, 1093 (1990) |
| [3] | Kawai, H.; Ninomiya, M., Nucl. Phys. B, 336, 115 (1990) |
| [4] | Agishtein, M. E.; Migdal, A. A., Nucl. Phys. B, 350, 690 (1991) |
| [5] | Kawamoto, N.; Kazakov, V. A.; Saeki, Y.; Watabiki, Y., Phys. Rev. Lett., 68, 2113 (1992) · Zbl 0969.81627 |
| [6] | Kawamoto, N., Current topics in theoretical physics, (Cho, Y. M., First Asia-Pacific Winter School for Theoretical Physics (1993), World Scientific) |
| [7] | Kawai, H.; Kawamoto, N.; Mogami, T.; Watabiki, Y., Phys. Lett. B, 306, 19 (1993) |
| [8] | Watabiki, Y., Phys. Lett. B, 346, 46 (1995) |
| [9] | Ambjørn, J.; Watabiki, Y., Nucl. Phys. B, 445, 129 (1995) · Zbl 1006.83015 |
| [10] | Ambjørn, J.; Kristjansen, C. F.; Watabiki, Y., Nucl. Phys. B, 504, 555 (1997) · Zbl 0925.83029 |
| [11] | Oda, S.; Tsuda, N.; Yukawa, T., Prog. Theor. Phys., 99, 875 (1998) |
| [12] | Jain, S.; Mathur, S. D., Phys. Lett. B, 286, 239 (1992) |
| [13] | Ambjørn, J.; Jain, S.; Thorleifsson, G., Phys. Lett. B, 307, 34 (1993) |
| [14] | Ambjørn, J.; Jurkiewicz, J.; Watabiki, Y., Nucl. Phys. B, 454, 313 (1995) · Zbl 0925.83006 |
| [15] | Ambjørn, J.; Anagnostopoulos, K. N.; Ichihara, T.; Jensen, L.; Kawamoto, N.; Watabiki, Y.; Yotsuji, K., Phys. Lett. B, 397, 177 (1997) |
| [16] | Ambjørn, J.; Anagnostopoulos, K. N.; Ichihara, T.; Jensen, L.; Kawamoto, N.; Watabiki, Y.; Yotsuji, K., Nucl. Phys. B, 511, 673 (1998) · Zbl 0947.81558 |
| [17] | Ambjørn, J.; Anagnostopoulos, K. N., Nicl. Phys. B, 497, 445 (1997) |
| [18] | Ambjørn, J.; Thorleifsson, G., Phys. Lett. B, 323, 7 (1994) |
| [19] | Anagnostopoulos, K. N.; Bialas, P.; Thorleifsson, G., J. Stat. Phys., 94, 321 (1999) · Zbl 0958.82021 |
| [20] | Potts, R. B., Proc. Cambridge Philos. Soc., 48, 106 (1952) · Zbl 0048.45601 |
| [21] | Fortuin, C. M.; Kasteleyn, P. W., Physica, 57, 536 (1972) |
| [22] | Ambjørn, J.; Thorleifsson, G.; Wexler, M., Nucl. Phys. B, 439, 187 (1995) · Zbl 0990.82521 |
| [23] | Ambjørn, J.; Thorleifsson, G.; Wexler, M., Nucl. Phys. B, 439, 187 (1995) · Zbl 0990.82521 |
| [24] | Distler, J.; Kawai, H., Nucl. Phys. B, 321, 509 (1989) |
| [25] | Christensen, S. M.; Duff, M. J., Phys. Lett. B, 79, 213 (1978) |
| [26] | Ising, E., Phys., 31, 253 (1925) · Zbl 1439.82056 |
| [27] | Baxter, R. J.; Kelland, S. B.; Wu, F. Y., J. Phys. A, 9, 397 (1976) · Zbl 0321.05140 |
| [28] | Tutte, W. T., J. Combin. Theory, 2, 301 (1967) · Zbl 0147.42902 |
| [29] | Wu, F. Y., J. Phys. A, 10, L113 (1977) |
| [30] | Temperley, J. N.V.; Lieb, E. H., Proc. R. Soc. London Ser. A, 322, 251 (1971) · Zbl 0211.56703 |
| [31] | Dotsenko, Vl. S., Nucl. Phys. B, 235, 54 (1984) |
| [32] | Dotsenko, Vl. S.; Fateev, V. A., Nucl. Phys. B, 240, 312 (1984) |
| [33] | Glauber, R. J., J. Math. Phys., 4, 294 (1963) · Zbl 0145.24003 |
| [34] | Sweeny, M., Phys. Rev. B, 27, 4445 (1983) |
| [35] | Boulatov, D. V.; Kazakov, V. A.; Kostov, I. K.; Migdal, A. A., Nucl. Phys. B, 275, 641 (1986) · Zbl 0968.81540 |
| [36] | Kazakov, V. A., Phys. Lett. A, 119, 140 (1986) |
| [37] | Boulatov, D. V.; Kazakov, V. A., Phys. Lett. B, 186, 379 (1987) |
| [38] | Stauffer, D., Introduction to Percolation Theory (1985), Taylor & Francis · Zbl 0990.82530 |
| [39] | Fisher, M. E.; Barber, M. N., Phys. Rev. Lett., 28, 1516 (1972) |
| [40] | Barber, M. N.; Selke, W., J. Phys. A, 15, L617 (1982) |
| [41] | Kihara, T.; Midzuno, Y.; Shizume, T., J. Phys. Soc. Jpn., 9, 681 (1954) |
| [42] | Kim, D.; Joseph, R., J. Phys. C, 7, L167 (1974) |
| [43] | Kawai, H., Nucl. Phys. B (Proc. Suppl.), 26, 93 (1992) · Zbl 0957.81654 |
| [44] | Ambjørn, J.; Durhuus, B.; Jonsson, T., Quantum Geometry (1997), Cambridge Univ. Press · Zbl 0993.82500 |
| [45] | Ambjørn, J.; Durhuus, B.; Jonsson, T., Phys. Lett. B, 244, 403 (1990) |
| [46] | Durhuus, B., Nucl. Phys. B, 426, 203 (1994) · Zbl 1049.82521 |
| [47] | Jurkiewicz, J.; Krzywicki, A., Phys. Lett. B, 392, 291 (1997) |
| [48] | Ambjørn, J.; Boulatov, D. V.; Kawamoto, N.; Watabiki, Y., Phys. Lett. B, 480, 319 (2000) · Zbl 0960.83027 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
