Analyticity of intersection exponents for planar Brownian motion

We show that the intersection exponents for planar Brownian motions are analytic. More precisely, let B and B′ be independent planar Brownian motions started from distinct points, and define the exponent ξ(1, λ) by E [ P [ B[0, t] ∩B[0, t] = ∅ ∣∣ B[0, t] ]λ ] ≈ t, t → ∞. Then the mapping λ 7→ ξ(1, λ) is real analytic in (0,∞). The same result is proved for the exponents ξ(k, λ) where k is a positive integer. In combination with the determination of ξ(k, λ) for integer k ≥ 1 and real λ ≥ 1 in our previous papers, this gives the value of ξ(k, λ) also for λ ∈ (0, 1) and the disconnection exponents limλց0 ξ(k, λ). In particular, it shows that limλց0 ξ(2, λ) = 2/3 and concludes the proof of the following result that had been conjectured by Mandelbrot: the Hausdorff dimension of the outer boundary of B[0, 1] is 4/3 almost surely.