Random walks, polymers, percolation, 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 non linear map onto a random Riemann surface, the former conjecture that ζL = 124 (4L2 − 1) is established. This also applies to the half-plane where ζL = L3 (1 + 2L), as well as to non-intersection exponents of unions of paths. Mandelbrot's conjecture for the Hausdorff dimension DH = 43 of the frontier of a Brownian path follows from DH = 2 − ξ32. We then consider in two dimensions the most general star-shaped copolymer, mixing random (RW) or self-avoiding walks (SAW) with specific mutual avoidance interactions thereof. Its exact conformal scaling dimensions in the plane are derived. The harmonic measures (or electrostatic potential, or diffusion field) near a RW or a SAW, or near a critical percolation cluster are also considered. Their moments exhibit a multifractal spectrum. The generalized dimensions D (n) as well as the function ƒ (α) are derived, and are shown to be all identical. These are examples of exact conformal multifractality.