A Test of a Conjecture of Cardy

Abstract

In reference to Werner's measure on self-avoiding loops on Riemann surfaces, Cardy conjectured a formula for the measure of all homotopically nontrivial loops in a finite type annular region with modular parameter $\rho$. Ang, Remy and Sun have announced a proof of this conjecture using random conformal geometry. Cardy's formula implies that the measure of the set of homotopically nontrivial loops in the punctured plane which intersect $S^1$ equals $\frac{2\pi}{\sqrt{3}}$. 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 sum which, in a limit, calculates the measure of the set of loops which avoid a ray. Each term in the sum involves finding the transfinite diameter of a slit domain. This is numerically accessible using the remarkable Schwarz-Christoffel package developed by Driscoll and Trefethen. Our calculations suggest this sum is around $\pi$, consistent with Cardy's formula.