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Conformal fractal geometry and boundary quantum gravity. (English) Zbl 1068.60019

Lapidus, Michel L. (ed.) et al., Fractal geometry and applications: A jubilee of Benoît Mandelbrot. Multifractals, probability and statistical mechanics, applications. In part the proceedings of a special session held during the annual meeting of the American Mathematical Society, San Diego, CA, USA, January 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3638-2/v.2; 0-8218-3292-1/set). Proceedings of Symposia in Pure Mathematics 72, Pt. 2, 365-482 (2004).
The paper contains 12 sections. Section 1 (Introduction) is devoted to the description of history and some results about the problems considered in the paper. In Section 2 the values of the intersection exponents of random walks or Brownian paths from quantum gravity are established. In Section 3 the critical properties of an arbitrary set of mixing simple random walks or Brownian paths and self-avoiding walks are studied. The multifractal spectrum of the harmonic measure near Brownian paths or self-avoiding ones is studied in Section 4, including the case of double-sided potential. Section 5 yields the related multifractal spectrum for percolation clusters. In Section 6 the general solution for the multifractal potential distrubution near any conformal fractal in 2D is presented, which allows determination of the Hausdorff dimension of the frontier.
In Section 7 a higher multifractality is considered. Section 8 describes the more subtle mixed multifractal spectrum associated with the local rotations and singularities along a conformally-invariant curve, as seen by harmonic measure. In Section 9 Hausdorff dimension of \(O(N)\) lines, Potts cluster boundaries, and the stochastic Loewner evolutions (SLE) traces are given. Conformally invariant paths have quite different critical properties and obey different quantum gravity rules, depending on whether they are simple paths or not. The next sections (10–12) are devoted to the elucidation of this difference, and its treatment within a unified framwork.
For the entire collection see [Zbl 1055.37003].

MSC:

60D05 Geometric probability and stochastic geometry
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
05C80 Random graphs (graph-theoretic aspects)
60J65 Brownian motion
60J45 Probabilistic potential theory
30C85 Capacity and harmonic measure in the complex plane
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B43 Percolation
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