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Dimension result and KPZ formula for two-dimensional multiplicative cascade processes. (English) Zbl 1298.60046

The famous KPZ formula was first derived by V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov [“Fractal structure of 2D-quantum gravity”, Modern Phys. Lett. A 3, 819–826 (1988), MR0947880] via Liouville quantum gravity in a light cone gauge, building upon an earlier work of A. M. Polyakov [“Quantum gravity in two dimensions”, Modern Phys. Lett. A 2, 893–898 (1987), MR0913671]. Shortly after, F. David [“Conformal field theories coupled to 2-D gravity in the conformal gauge”, Modern Phys. Lett. A 3, No. 17 1651–1656 (1988), MR0981529] provided an alternative heuristic derivation of the KPZ formula by using Liouville field theory in the so-called conformal gauge. The KPZ formula has had great influence on string theory and conformal field theory, and it plays a core role in studying the connections of two-dimensional quantum gravity to random planar maps, two-dimensional lattice models, random matrix theory and Schramm-Loewner evolution. In a recent inspiring paper [Invent. Math. 185, No. 2, 333–393 (2011; Zbl 1226.81241), arXiv:0808.1560], B. Duplantier and S. Sheffield have provided (in a mathematically rigorous way) a geometrical KPZ formula under a similar framework as used in [MR0981529]. In the present paper, the author proves a Hausdorff dimension result for the image of two-dimensional multiplicative cascade processes, and obtains from this result a KPZ-type formula which normally has one point of phase transition.

MSC:

60G18 Self-similar stochastic processes
60G57 Random measures
28A78 Hausdorff and packing measures
28A80 Fractals
60K40 Other physical applications of random processes
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics

Citations:

Zbl 1226.81241
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References:

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