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Commutation relations for Schramm-Loewner evolutions. (English) Zbl 1137.82009
Schramm-Loewner evolutions (SLE) describe a one-parameter family of growth processes in the plane which have special conformal invariance properties. Here one considers the case when one has several SLE in the same simply connected domain, which are invariant in distribution under global time reparametrization. After reviewing some commutation relations in SLE, one considers more especially commutation at the level of infinitesimal operators. The necessary infinitesimal conditions (on the solutions) are converted into integrability conditions which themselves lead to PDE’s. Restriction formulae are obtained in the chordal case, and multiply connected domains are considered.

##### MSC:
 82B43 Percolation
Full Text:
##### References:
 [1] Aizenman, Duke Math J 99 pp 419– (1999) [2] ; On chordal and bilateral SLE in multiply connected domains. Preprint. arXiv:math.PR/0503178, 2005. [3] The dimension of SLE curves. Preprint. arXiv:math.PR/0211322, 2002. [4] ; The full scaling limit of two-dimensional critical percolation. Preprint. arXiv:math.PR/0504036, 2005. [5] Cardy, J Phys A 36 pp l379– (2003) [6] Cardy, J Phys A 36 pp 12343– (2003) [7] Dubédat, Comm Math Phys 245 pp 627– (2004) [8] Dubédat, Probab Theory Related Fields 134 pp 453– (2006) [9] Dubédat, J Stat Phys 123 pp 1183– (2006) [10] Dubédat, Ann Probab 33 pp 223– (2005) [11] Fomin, Trans Amer Math Soc 353 pp 3563– (2001) [12] Kozdron, Electron J Probab 10 pp 1442– (2005) · Zbl 1110.60046 [13] Lawler, J Amer Math Soc 16 pp 917– (2003) [14] Lawler, Illinois J Math 50 pp 701– (2006) [15] Conformally invariant processes in the plane. Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, R.I., 2005. [16] Lawler, Ann Probab 32 pp 939– (2004) [17] Lawler, Probab Theory Related Fields 128 pp 565– (2004) [18] Rohde, Ann of Math (2) 161 pp 883– (2005) [19] Schramm, Israel J Math 118 pp 221– (2000) [20] Schramm, Electron Comm Probab 6 pp 115– (2001) · Zbl 1008.60100 [21] Smirnov, C R Acad Sci Paris Sér I Math 333 pp 239– (2001) · Zbl 0985.60090 [22] Werner, Ann Fac Sci Toulouse Math (6) 13 pp 121– (2004) · Zbl 1059.60099 [23] Random planar curves and Schramm-Loewner evolutions. Lectures on probability theory and statistics, 107–195. Lecture Notes in Math, 1840. Springer, Berlin, 2004. [24] Werner, Probab Surv 2 pp 145– (2005) [25] Generating random spanning trees more quickly than the cover time. Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), 296–303. ACM, New York, 1996. · Zbl 0946.60070 [26] Random Loewner chains in Riemann surfaces. Thesis. California Institute of Technology, Pasadena, Calif., 2004. [27] Zhan, Probab Theory Related Fields 129 pp 340– (2004)
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