Path Integral over Reparametrizations: Levy Flights versus Random Walks

Details Path Integral over Reparametrizations: Levy Flights versus Random Walks Path Integral over Reparametrizations: Levy Flights versus Random Walks

2009-11-05

arxiv.org/abs/0911.1083v3

Nucl.Phys.B834:453-470,2010

Physics

High Energy Physics

High Energy Physics - Theory

22pp., RevTex, 9 figures; v2: sect.IV modified; v3: minor changes, to appear in Nucl. Phys. B

Scientific paper

10.1016/j.nuclphysb.2010.03.003

We investigate the properties of the path integral over reparametrizations (= the boundary value of the Liouville field in open string theory). Discretizing the path integral, we apply the Metropolis-Hastings algorithm to numerical simulations of a proper (subordinator) stochastic process and find that typical trajectories are not Brownian but rather have discontinuities of the type of Levy's flights. We study a fractal structure of these trajectories and show that their Hausdorff dimension is zero. We confirm thereby the discretization and heuristic consideration of QCD scattering amplitudes by analytical and numerical calculations. We also perform Monte Carlo simulations of the path integral over reparametrization in the effective-string ansatz for a circular Wilson loop and discuss their subtleties associated with the discretization of Douglas' functional.

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