Random Walks and Quantum Gravity in Two Dimensions

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 ζL0ex0ex=0ex0ex1/24(4L²-1). This also applies to the half plane where ̃ \~ζL0ex0ex=0ex0exL/3(1+2L), as well as to nonintersection exponents of unions of paths. Mandelbrot's conjecture for the Hausdorff dimension DH0ex0ex=0ex0ex4/3 of the frontier of a Brownian path follows from ζ3/2.