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 $P_L\left(t\right)\sim t^{-{\zeta }_L}$ 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 ${\zeta }_L=\frac{1}{24}({4L}^2-1\mathrm{})$. This also applies to the half plane where ${\tilde{\zeta }}_L=\frac{L}{3}(1+2L)$, as well as to nonintersection exponents of unions of paths. Mandelbrot's conjecture for the Hausdorff dimension $D_H=4/3\mathrm{}$ of the frontier of a Brownian path follows from ${\zeta }_{3/2}$.