### 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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References

1. Airault H., Malliavin P., Thalmaier A., Brownian measures on Jordan-Virasoro curves associated to the Weil-Petersson metric, J. Funct. Anal. 259 (2010), 3037-3079.
2. Astala K., Jones P., Kupiainen A., Saksman E., Random curves by conformal welding, C. R. Math. Acad. Sci. Paris 348 (2010), 257-262, arXiv:0912.3423.
3. Bauer R.O., A simple construction of Werner measure from chordal ${\rm SLE}_{8/3}$, Illinois J. Math. 54 (2010), 1429-1449, arXiv:0902.1626.
4. Benoist S., Dubédat J., An ${\rm SLE}_2$ loop measure, arXiv:1405.7880.
5. Bishop C.J., Conformal welding and Koebe's theorem, Ann. of Math. 166 (2007), 613-656.
6. Cardy J., The ${\rm O}(n)$ model on the annulus, J. Stat. Phys. 125 (2006), 1-21, math-ph/0604043.
7. Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997.
8. Duren P.L., Univalent functions, Grundlehren der Mathematischen Wissenschaften, Vol. 259, Springer-Verlag, New York, 1983.
9. Duren P.L., Schiffer M., The theory of the second variation in extremum problems for univalent functions, J. Analyse Math. 10 (1962/1963), 193-252.
10. Hille E., Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass. - New York - Toronto, Ont., 1962.
11. Kac V.G., Raina A.K., Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, Vol. 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987.
12. Kirillov A.A., Yuriev D.V., Representations of the Virasoro algebra by the orbit method, J. Geom. Phys. 5 (1988), 351-363.
13. Kontsevich M., Suhov Y., On Malliavin measures, SLE, and CFT, Proc. Steklov Inst. Math. 258 (2007), 100-146, math-ph/0609056.
14. Michael E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182.
15. Segal G., The definition of conformal field theory, in Topology, Geometry and Quantum Field Theory, Proceedings of the Symposium in Honour of the 60th Birthday of Graeme Segal (Oxford, June 24-29, 2002), London Mathematical Society Lecture Note Series, Vol. 308, Editor U. Tillmann, Cambridge University Press, Cambridge, 2004, 421-577.
16. Simon B., OPUC on one foot, Bull. Amer. Math. Soc. (N.S.) 42 (2005), 431-460, math.SP/0502485.
17. Werner W., The conformally invariant measure on self-avoiding loops, J. Amer. Math. Soc. 21 (2008), 137-169, math.PR/0511605.