Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 034, 12 pages      arXiv:2401.03600      https://doi.org/10.3842/SIGMA.2025.034

A Test of a Conjecture of Cardy

Van Higgs and Doug Pickrell
Mathematics Department, University of Arizona, Tucson AZ 85721, USA

Received January 25, 2025, in final form May 04, 2025; Published online May 09, 2025

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.

Key words: Werner measure; Cardy conjecture; transfinite diameter; Schwarz-Christoffel.

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References

1. Ang M., Remy G., Sun X., The moduli of annuli in random conformal geometry, Ann. Sci. Éc. Norm. Supér, to appear, arXiv:2203.12398.
2. Cardy J., Scaling and renormalization in statistical physics, Cambridge Lecture Notes Phys., Vol. 5, Cambridge University Press, Cambridge, 1996.
3. Cardy J., The ${\rm O}(n)$ model on the annulus, J. Stat. Phys. 125 (2006), 1-21, arXiv:math-ph/0604043.
4. Chavez A., Pickrell D., Werner's measure on self-avoiding loops and welding, SIGMA 10 (2014), 081, 42 pages, arXiv:1401.2675.
5. Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Grad. Texts Contemp. Phys., Springer, New York, 1997.
6. Driscoll T.A., Trefethen L.N., Schwarz-Christoffel mapping, Cambridge Monogr. Appl. Comput. Math., Vol. 8, Cambridge University Press, Cambridge, 2002.
7. Hille E., Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, Boston, Mass., 1962.
8. Michael E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182.
9. Saleur H., Bauer M., On some relations between local height probabilities and conformal invariance, Nuclear Phys. B 320 (1989), 591-624.
10. Werner W., The conformally invariant measure on self-avoiding loops, J. Amer. Math. Soc. 21 (2008), 137-169, arXiv:math.PR/0511605.