The sharp amplitude of point non-disconnection for planar Brownian motion

Let B be a planar Brownian motion started at the origin and stopped when it exits the unit disk, and let f(x) be the probability that B disconnects a fixed point at distance √x from the origin from the unit circle. Relative to one hypothesis, the large-internal-weight normalisation of Zamolodchikov's recursion for conformal blocks, we prove that 1−f(x)=C1(x1/8−(1/8)x5/8+(1/16)x9/8−(5/128)x13/8)+O(x17/8) (x↓0), where C1=(24/9 31/3 π3/2 e1/6)/(A2 Γ(1/3)2)=(24/9 31/3 π3/2 e2ζ′(−1))/(Γ(1/3)2), and A is the Glaisher–Kinkelin constant. Since 1−f(R−2) is the probability that a Brownian motion started on the unit circle reaches the circle of radius R without disconnecting the origin from it, C1 is the limit of R1/4 times this probability. We also prove that 1−f is the sum of an absolutely convergent series of conformal-block contributions near 0 and that f is real-analytic on (0,1), and we obtain the non-disconnection profile in every simply connected domain, two representations of the expected area of the filled path, and the amplitudes of the expansion of f at x=1 as products of Gamma values. MSC 2020: 60J65 (primary); 60D05, 81T40, 60G57 (secondary) Files: the paper and its Supplement (PDF), the LaTeX sources of both (ZIP), and the programs of the numerical checks with their reference outputs (ZIP).