
SIGMA 10 (2014), 081, 42 pages arXiv:1401.2675
https://doi.org/10.3842/SIGMA.2014.081
Werner's Measure on SelfAvoiding 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 selfavoiding loops on Riemann surfaces is determined by a single measure $\mu_0$ on selfavoiding 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 selfavoiding 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.
pdf (537 kb)
tex (38 kb)
References

Airault H., Malliavin P., Thalmaier A., Brownian measures on JordanVirasoro curves associated to the WeilPetersson metric, J. Funct. Anal. 259 (2010), 30373079.

Astala K., Jones P., Kupiainen A., Saksman E., Random curves by conformal welding, C. R. Math. Acad. Sci. Paris 348 (2010), 257262, arXiv:0912.3423.

Bauer R.O., A simple construction of Werner measure from chordal ${\rm SLE}_{8/3}$, Illinois J. Math. 54 (2010), 14291449, arXiv:0902.1626.

Benoist S., Dubédat J., An ${\rm SLE}_2$ loop measure, arXiv:1405.7880.

Bishop C.J., Conformal welding and Koebe's theorem, Ann. of Math. 166 (2007), 613656.

Cardy J., The ${\rm O}(n)$ model on the annulus, J. Stat. Phys. 125 (2006), 121, mathph/0604043.

Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Graduate Texts in Contemporary Physics, SpringerVerlag, New York, 1997.

Duren P.L., Univalent functions, Grundlehren der Mathematischen Wissenschaften, Vol. 259, SpringerVerlag, New York, 1983.

Duren P.L., Schiffer M., The theory of the second variation in extremum problems for univalent functions, J. Analyse Math. 10 (1962/1963), 193252.

Hille E., Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.  New York  Toronto, Ont., 1962.

Kac V.G., Raina A.K., Bombay lectures on highest weight representations of infinitedimensional Lie algebras, Advanced Series in Mathematical Physics, Vol. 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987.

Kirillov A.A., Yuriev D.V., Representations of the Virasoro algebra by the orbit method, J. Geom. Phys. 5 (1988), 351363.

Kontsevich M., Suhov Y., On Malliavin measures, SLE, and CFT, Proc. Steklov Inst. Math. 258 (2007), 100146, mathph/0609056.

Michael E., Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152182.

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 2429, 2002), London Mathematical Society Lecture Note Series, Vol. 308, Editor U. Tillmann, Cambridge University Press, Cambridge, 2004, 421577.

Simon B., OPUC on one foot, Bull. Amer. Math. Soc. (N.S.) 42 (2005), 431460, math.SP/0502485.

Werner W., The conformally invariant measure on selfavoiding loops, J. Amer. Math. Soc. 21 (2008), 137169, math.PR/0511605.

