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

SIGMA 10 (2014), 081, 42 pages      arXiv:1401.2675      https://doi.org/10.3842/SIGMA.2014.081

Werner's Measure on Self-Avoiding Loops and Welding

Angel Chavez and Doug Pickrell
Mathematics Department, University of Arizona, Tucson, AZ 85721, USA

Received February 18, 2014, in final form July 31, 2014; Published online August 04, 2014

Abstract
Werner's conformally invariant family of measures on self-avoiding loops on Riemann surfaces is determined by a single measure $\mu_0$ on self-avoiding loops in ${\mathbb C} \setminus\{0\}$ which surround $0$. Our first major objective is to show that the measure $\mu_0$ is infinitesimally invariant with respect to conformal vector fields (essentially the Virasoro algebra of conformal field theory). This makes essential use of classical variational formulas of Duren and Schiffer, which we recast in representation theoretic terms for efficient computation. We secondly show how these formulas can be used to calculate (in principle, and sometimes explicitly) quantities (such as moments for coefficients of univalent functions) associated to the conformal welding for a self-avoiding loop. This gives an alternate proof of the uniqueness of Werner's measure. We also attempt to use these variational formulas to derive a differential equation for the (Laplace transform of) the ''diagonal distribution'' for the conformal welding associated to a loop; this generalizes in a suggestive way to a deformation of Werner's measure conjectured to exist by Kontsevich and Suhov (a basic inspiration for this paper).

Key words: loop measures; conformal welding; conformal invariance; moments; Virasoro algebra.

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