Transfer matrix formalism for two-dimensional quantum gravity and fractal structures of space-time

Kawai, Hiroyuki; Kawamoto, Noboru; Mogami, Tsuguo; Watabiki, Yoshiyuki

doi:10.1016/0370-2693(93)91131-6

Cited by 189 publications

(287 citation statements)

References 17 publications

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“…[6]. In the meantime Kawai, Kawamoto, Mogami and Watabiki tried to understand the fractal nature of quantum gravity from the Matrix model point of view and succeeded to derive the transfer matrix of the quantum surface of two-dimensional pure gravity (c = 0) [7]. The formulation made it possible to derive the fractal dimension of pure gravity to be exactly d F = 4 which is consistent with the value of the first and third formulae.…”

Section: Introductionmentioning

confidence: 79%

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

Kawamoto

Yotsuji

2002

Nuclear Physics B

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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.

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Section: Introductionmentioning

confidence: 79%

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

Kawamoto

Yotsuji

2002

Nuclear Physics B

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“…Taking a generic approach δg = −g c αǫ 6 , δz = −z c βǫ 6 , we arrive at δV = −V c (α − β) 1 3 ǫ 2 . Accordingly, the characteristic equation (6.5) reads…”

Section: Tricritical Point and Fractal Dimensionmentioning

confidence: 99%

Geodesic distance in planar graphs

2003

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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. 03/031

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“…The Laplace transform of the proper-time evolution kernel is obtained as [6] N(ζ, ζ ′ ; D; t) = 1 ζ ′ + A(ζ; D; t) .…”

Section: Two-point Functions With Fixed Geodesic Distancesmentioning

confidence: 99%

“…In quantum gravity we can use the geodesic distance, which is general coordinate invariant. Recently a formalism has been developed, which enables us to introduce the geodesic distance [6,7]. Using this formalism we calculate correlation functions with fixed geodesic distances between the operators.…”

Section: Introductionmentioning

confidence: 99%

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Operator product expansion in two-dimensional quantum gravity

et al. 1996

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We consider correlation functions of operators and the operator product expansion in two-dimensional quantum gravity. First we introduce correlation functions with geodesic distances between operators kept fixed. Next by making two of the operators closer, we examine if there exists an analog of the operator product expansion in ordinary field theories. Our results suggest that the operator product expansion holds in quantum gravity as well, though special care should be taken regarding the physical meaning of fixing geodesic distances on a fluctuating geometry.

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Transfer matrix formalism for two-dimensional quantum gravity and fractal structures of space-time

Cited by 189 publications

References 17 publications

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

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

Geodesic distance in planar graphs

Operator product expansion in two-dimensional quantum gravity

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