  We derive the exact value of intersection exponents between planar Brownian
motions or random walks, confirming predictions from theoretical physics by
Duplantier and Kwon. Let B and B' be independent Brownian motions (or simple
random walks) in the plane, started from distinct points. We prove that the
probability that the paths B[0,t] and B'[0,t] do not intersect decays like
t^{-5/8}. More precisely, there is a constant c>0 such that if |B(0) - B'(0)|
=1, for all t \ge 1, c^{-1} t^{-5/8} \le \P[ B[0,t] \cap B'[0,t] = \emptyset ]
\le c t^{-5/8}. One consequence is that the set of cut-points of B[0,1] has
Hausdorff dimension 3/4 almost surely. The values of other exponents are also
derived. Using an analyticity result, which is to be established in a
forthcoming paper, it follows that the Hausdorff dimension of the outer
boundary of B[0,1] is 4/3, as conjectured by Mandelbrot. The proofs are based
on a study of SLE_6 (stochastic Loewner evolution with parameter 6), a recently
discovered process which conjecturally is the scaling limit of critical
percolation cluster boundaries. The exponents of SLE_6 are calculated, and they
agree with the physicists' predictions for the exponents for critical
percolation and self-avoiding walks. From the SLE_6 exponents the Brownian
intersection exponents are then derived.
