Abstract
The Brownian loop measure is a conformally invariant measure on loops in the plane that arises when studying the Schramm–Loewner evolution (SLE). When an SLE curve in a domain evolves from an interior point, it is natural to consider the loops that hit the curve and leave the domain, but their measure is infinite. We show that there is a related normalized quantity that is finite and invariant under Möbius transformations of the plane. We estimate this quantity when the curve is small and the domain simply connected. We then use this estimate to prove a formula for the Radon–Nikodym derivative of reversed radial SLE with respect to whole-plane SLE.
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Dubédat, J.: SLE and the free field: partition functions and couplings. J. Am. Math. Soc. 22(4), 995–1054 (2009)
Lawler, G.F.: Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, vol. 114. Am. Math. Soc., Providence (2005)
Lawler, G.F.: Partition functions, loop measure, and versions of SLE. J. Stat. Phys. 134(5–6), 813–837 (2009)
Lawler, G.F.: Schramm-Loewner evolution (SLE). In: Sheffield, S., Spencer, T. (eds.) Statistical Mechanics. IAS/Park City Math. Ser., vol. 16, pp. 231–295. Am. Math. Soc., Providence (2009)
Lawler, G.F.: Continuity of radial and two-sided radial SLE κ at the terminal point. arXiv:1104.1620v1 (2011)
Lawler, G.F.: Defining SLE in multiply connected domains with the Brownian loop measure. arXiv:1108.4364v1 (2011)
Lawler, G.F., Werner, W.: The Brownian loop soup. Probab. Theory Relat. Fields 128(4), 565–588 (2004)
Lawler, G.F., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917–955 (2003)
Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004)
Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161(2), 883–924 (2005)
Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000)
Werner, W.: The conformally invariant measure on self-avoiding loops. J. Am. Math. Soc. 21(1), 137–169 (2008)
Zhan, D.: Reversibility of chordal SLE. Ann. Probab. 36(4), 1472–1494 (2008)
Zhan, D.: Reversibility of whole-plane SLE. arXiv:1004.1865v2 (2010)
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G.F. Lawler research supported by National Science Foundation grant DMS-0907143.
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Field, L.S., Lawler, G.F. Reversed Radial SLE and the Brownian Loop Measure. J Stat Phys 150, 1030–1062 (2013). https://doi.org/10.1007/s10955-013-0729-5
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DOI: https://doi.org/10.1007/s10955-013-0729-5