A Test of a Conjecture of Cardy

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Authors: Higgs, Van, Pickrell, Doug
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/213181
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: A Test of a Conjecture of Cardy. Van Higgs and Doug Pickrell. SIGMA 21 (2025), 034, 12 pages

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 . 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 that intersect ¹ equals 2π/√3. This set is the disjoint union of the set of loops that avoid a ray from the unit circle to infinity and its complement. There is an inclusion/exclusion sum which, in the limit, calculates the measure of the set of loops that 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 π, consistent with Cardy's formula.