  In reference to Werner's measure on random self-avoiding loops on Riemann
surfaces, Cardy has conjectured a formula for the measure of all homotopically
nontrivial loops in a finite type annular region. Cardy's conjecture implies
that the measure of the set of homotopically nontrivial loops in the punctured
plane which intersect the unit circle equals $2\pi/\sqrt{3}$. This set is the
disjoint union of the set of loops which avoid a ray from the unit circle to
infinity and its complement. There is an inclusion/exclusion (sign alternating)
sum which, in a limit, calculates the measure of the set of loops which avoid a
ray. To calculate this exactly (or numerically in a reliable way) it would
suffice to find a formula (or reasonably efficient algorithm) for the
transfinite diameter of a compact set consisting of a finite number of rays
from the origin to the unit circle. This is currently out of my reach. A first
approximation suggests this measure grows logarithmically, inconsistent with
Cardy's conjecture. A better approximation suggests this measure is bounded by
$2\pi/\sqrt{3}$. At this point I have no idea what to believe.
