Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Test of a Conjecture of Cardy

Van Higgs, Doug Pickrell

TL;DR

The paper investigates Cardy's conjecture for Werner's measure of homotopically nontrivial loops by recasting the problem as an inclusion/exclusion computation over slit-domain conformal maps and evaluating the resulting transfinite diameters with Schwarz–Christoffel mappings. It presents a numerical program that sums alternating terms up to $n=22$, with preliminary evidence suggesting the limit approaches $\pi$, in line with Cardy’s formula, while also exploring the integrability of a height function on universal Teichmüller space and the consistency of Werner measure normalizations. The work situates Cardy’s formula within conformal restriction theory, specifically via the continuum $O(n)$ loop model, and connects numerical conformal mapping techniques to deep geometric questions about loop measures. Overall, it provides both numerical corroboration and theoretical structure (via height-function integrability and normalization equivalence) for Cardy’s conjecture and related Werner-measure questions.

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 $ρ$. 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 $\frac{2π}{\sqrt{3}}$. 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 $π$, consistent with Cardy's formula.

A Test of a Conjecture of Cardy

TL;DR

The paper investigates Cardy's conjecture for Werner's measure of homotopically nontrivial loops by recasting the problem as an inclusion/exclusion computation over slit-domain conformal maps and evaluating the resulting transfinite diameters with Schwarz–Christoffel mappings. It presents a numerical program that sums alternating terms up to , with preliminary evidence suggesting the limit approaches , in line with Cardy’s formula, while also exploring the integrability of a height function on universal Teichmüller space and the consistency of Werner measure normalizations. The work situates Cardy’s formula within conformal restriction theory, specifically via the continuum loop model, and connects numerical conformal mapping techniques to deep geometric questions about loop measures. Overall, it provides both numerical corroboration and theoretical structure (via height-function integrability and normalization equivalence) for Cardy’s conjecture and related Werner-measure questions.

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 . 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 equals . 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 , consistent with Cardy's formula.
Paper Structure (8 sections, 8 theorems, 58 equations, 1 figure)

This paper contains 8 sections, 8 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Plan of the paper
  3. An origin story for Cardy's conjecture
  4. Applying inclusion/exclusion
  5. Some examples
  6. Numerically testing Cardy's formula
  7. Integrability of a height function
  8. Normalizations of Werner's family of measures

Key Result

Theorem 1.1

There exists a nontrivial family of locally finite measures $\{\mu_S\}$ on self-avoiding loops on Riemann surfaces which satisfies conformal restriction. This family is unique up to multiplication by an overall positive constant.

Figures (1)

  • Figure 1: The last 5 outputted values are 3.0140, 3.0236, 3.0318, 3.0390, and 3.0451.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Theorem 5.1
  • Theorem 6.1