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Geodesic distance in planar graphs. (English) Zbl 1022.05022
Summary: We derive the exact generating function for planar maps (genus zero fatgraphs) with vertices of arbitrary even valence and with two marked points at a fixed geodesic distance. This is done in a purely combinatorial way based on a bijection with decorated trees, leading to a recursion relation on the geodesic distance. The latter is solved exactly in terms of discrete soliton-like expressions, suggesting an underlying integrable structure. We extract from this solution the fractal dimensions at the various (multi)-critical points, as well as the precise scaling forms of the continuum two-point functions and the probability distributions for the geodesic distance in (multi)-critical random surfaces. The two-point functions are shown to obey differential equations involving the residues of the KdV hierarchy.

MSC:
05C12 Distance in graphs
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[1] Tutte, W.; Tutte, W.; Tutte, W.; Tutte, W., A census of planar maps, Canad. J. math., Canad. J. math., Canad. J. math., Canad. J. math., 15, 249-271, (1963) · Zbl 0115.17305
[2] Brézin, E.; Itzykson, C.; Parisi, G.; Zuber, J.-B., Planar diagrams, Commun. math. phys., 59, 35-51, (1978) · Zbl 0997.81548
[3] Di Francesco, P.; Ginsparg, P.; Zinn-Justin, J., 2D gravity and random matrices, Phys. rep., 254, 1-131, (1995)
[4] Eynard, B., Random matrices, Saclay Lecture Notes (2000) · Zbl 0925.82121
[5] Knizhnik, V.G.; Polyakov, A.M.; Zamolodchikov, A.B.; David, F.; Distler, J.; Kawai, H., Conformal field theory and 2D quantum gravity, Mod. phys. lett. A, Mod. phys. lett. A, Nucl. phys. B, 321, 509-527, (1989)
[6] Kawai, H.; Kawamoto, N.; Mogami, T.; Watabiki, Y., Transfer matrix formalism for two-dimensional quantum gravity and fractal structures of spacetime, Phys. lett. B, 306, 19-26, (1993)
[7] Ambjørn, J.; Watabiki, Y., Scaling in quantum gravity, Nucl. phys. B, 445, 129-144, (1995) · Zbl 1006.83015
[8] Ambjørn, J.; Jurkiewicz, J.; Watabiki, Y., On the fractal structure of two-dimensional quantum gravity, Nucl. phys. B, 454, 313-342, (1995) · Zbl 0925.83006
[9] Chassaing, P.; Schaeffer, G., Random planar lattices and integrated super-Brownian excursion, Probab. Theory Related Fields (2002), in press · Zbl 1025.60004
[10] See also, G. Schaeffer, Conjugaison d’arbres et cartes combinatoires aléatoires, PhD Thesis, Université Bordeaux I (1998) · Zbl 0885.05076
[11] Bouttier, J.; Di Francesco, P.; Guitter, E., Census of planar maps: from the one-matrix model solution to a combinatorial proof, Nucl. phys. B, 645, 477-499, (2002) · Zbl 0999.05052
[12] Bousquet-Mélou, M.; Schaeffer, G., Enumeration of planar constellations, Adv. appl. math., 24, 337-368, (2000) · Zbl 0955.05004
[13] Jimbo, M.; Miwa, T., Solitons and infinite-dimensional Lie algebras, Publ. res. inst. math. sci. Kyoto univ., 19, 3, 943-1001, (1983), Eq. (2.12) · Zbl 0557.35091
[14] Staudacher, M., The yang – lee edge singularity on a dynamical planar random surface, Nucl. phys. B, 336, 349-362, (1990)
[15] Gelfand, I.; Dikii, L., Fractional powers of operators and Hamiltonian systems, Funct. anal. appl., 10, 4, 13, (1976) · Zbl 0346.35085
[16] Kawamoto, N.; Yotsuji, K., Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity, Nucl. phys. B, 644, 533-567, (2002) · Zbl 0999.83017
[17] Bouttier, J.; Di Francesco, P.; Guitter, E., Counting colored random triangulations, Nucl. phys. B, 641, 519-532, (2002) · Zbl 0998.05019
[18] Bousquet-Mélou, M.; Schaeffer, G., The degree distribution in bipartite planar maps: application to the Ising model
[19] Bouttier, J.; Di Francesco, P.; Guitter, E., Combinatorics of hard particles on planar maps, Nucl. phys. B, 655, 313-341, (2002) · Zbl 1009.82005
[20] J. Bouttier, P. Di Francesco, E. Guitter, in preparation
[21] Delmas, J.-F., Computation of moments for the length of the one dimensional ISE support, (2002)
[22] Di Francesco, P.; Guitter, E., Critical and multicritical semi-random (1+d)-dimensional lattices and hard objects in d dimensions, J. phys. A: math. gen., 35, 897-927, (2002) · Zbl 0993.82026
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