Random Walks and Quantum Gravity in Two Dimensions

Details Random Walks and Quantum Gravity in Two Dimensions Random Walks and Quantum Gravity in Two Dimensions

Dec 1998

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998phrvl..81.5489d&link_type=abstract

Physical Review Letters, Volume 81, Issue 25, December 21, 1998, pp.5489-5492

Mathematics

Probability

22

Scientific paper

We consider L planar random walks (or Brownian motions) of large length t, starting at neighboring points, and the probability PL\(t\)~t-ζL that their paths do not intersect. By a 2D quantum gravity method, i.e., a nonlinear map to an exact solution on a random surface, I establish our former conjecture that ζL = 124\(4L2-1\). This also applies to the half plane where ζ~L = L3\(1+2L\), as well as to nonintersection exponents of unions of paths. Mandelbrot's conjecture for the Hausdorff dimension DH = 4/3 of the frontier of a Brownian path follows from ζ3/2.

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