Abstract
Schramm-Loewner Evolutions (SLEs) have proved an efficient way to describe a single continuous random conformally invariant interface in a simply-connected planar domain; the admissible probability distributions are parameterized by a single positive parameter κ. As shown in, Ref. 8 the coexistence of n interfaces in such a domain implies algebraic (“commutation”) conditions. In the most interesting situations, the admissible laws on systems of n interfaces are parameterized by κ and the solution of a particular (finite rank) holonomic system.
The study of solutions of differential systems, in particular their global behaviour, often involves the use of integral representations. In the present article, we provide Euler integral representations for solutions of holonomic systems arising from SLE commutation. Applications to critical percolation (general crossing formulae), Loop-Erased Random Walks (direct derivation of Fomin’s formulae in the scaling limit), and Uniform Spanning Trees are discussed. The connection with conformal restriction and Poissonized non-intersection for chordal SLEs is also studied.
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References
P. Appell and J. Kampé de Fériet. Fonctions hypergéometriques et hypersphériques. Polynômes d’Hermite. (Gauthier-Villars, Paris, 1926).
F. Camia and C. M. Newman. The full scaling limit of two-dimensional critical percolation. preprint, arXiv:math.PR/0504036.
J. L. Cardy. Critical percolation in finite geometries. J. Phys. A 25(4):L201–L206 (1992).
J. L. Cardy. Conformal invariance and percolation. preprint, arXiv:math-ph/0103018 (2001).
J. L. Cardy. Crossing formulae for critical percolation in an annulus. J. Phys. A 35(41):L565–L572 (2002).
P. Di Francesco, P. Mathieu and D. Sénéchal. Conformal field theory. Graduate Texts in Contemporary Physics. (Springer-Verlag, New York, 1997).
V. S. Dotsenko and V. A. Fateev. Conformal algebra and multipoint correlation functions in 2D statistical models. Nuclear Phys. B 240(3):312–348 (1984).
J. Dubédat. Commutation relations for SLE. preprint, arXiv:math.PR/0411299 (2004).
J. Dubédat. Excursion decompositions for SLE and Watts’ crossing formula. Probab. Theory Related Fields, to appear, http://dx.doi.org/10.1007/s00440-005-0446-3, (2004).
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi. Higher transcendental functions. Vol. I. Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman.
H. M. Farkas and I. Kra. Riemann surfaces, volume 71 of Graduate Texts in Mathematics. 2nd edn. (Springer-Verlag, New York, 1992).
S. Fomin. Loop-erased walks and total positivity. Trans. Amer. Math. Soc. 353(9):3563–3583 (electronic) (2001).
R. Friedrich and J. Kalkkinen. On conformal field theory and stochastic Loewner evolution. Nuclear Phys. B 687(3):279–302 (2004).
G. Grimmett. Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften. 2nd edn. (Springer-Verlag, Berlin, 1999).
M. J. Kozdron and G. F. Lawler. Estimates of random walk exit probabilities and application to loop-erased random walk. Electron. J. Probab. 10:1442–1467 (electronic) (2005).
G. Lawler, O. Schramm and W. Werner. Conformal restriction: the chordal case. J. Amer. Math. Soc. 16(4):917–955 (electronic) (2003).
G. F. Lawler. The Laplacian-b random walk and the Schramm-Loewner evolution. Illinois J. Math., to appear.
G. F. Lawler. Conformally invariant processes in the plane, volume 114 of Mathematical Surveys and Monographs. (American Mathematical Society, Providence, RI, 2005).
G. F. Lawler, O. Schramm and W. Werner. On the scaling limit of planar self-avoiding walk. In Fractal geometry and application. A jubilee of Benoit Mandelbrot, AMS Proc. Symp. Pure Math. 2002.
G. F. Lawler, O. Schramm and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B):939–995 (2004).
G. F. Lawler and W. Werner. The Brownian loop soup. Probab. Theory Related Fields 128(4):565–588 (2004).
H. T. Pinson. Critical percolation on the torus. J. Statist. Phys. 75(5–6):1167–1177 (1994).
S. Rohde and O. Schramm. Basic properties of SLE. Ann. of Math. (2) 161(2):883–924 (2005).
M. Saito, B. Sturmfels, and N. Takayama. Gröbner deformations of hypergeometric differential equations, volume 6 of Algorithms and Computation in Mathematics. (Springer-Verlag, Berlin, 2000).
O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118:221–288 (2000).
S. Smirnov. Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3):239–244 (2001).
G. M. T. Watts. A crossing probability for critical percolation in two dimensions. J. Phys. A 29(14):L363–L368 (1996).
W. Werner. Random planar curves and Schramm-Loewner evolutions. In Lectures on probability theory and statistics, volume 1840 of Lecture Notes in Math., pp. 107–195. Springer, Berlin, 2004.
W. Werner. Conformal restriction and related questions. Probab. Surv. 2:145–190 (electronic) (2005).
D. B. Wilson. Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pp. 296–303, New York, 1996. ACM.
M. Yoshida. Fuchsian differential equations. Aspects of Mathematics, E11. Friedr. Vieweg & Sohn, Braunschweig, 1987. With special emphasis on the Gauss-Schwarz theory.
D. Zhan. Random Loewner chains in Riemann surfaces. PhD thesis (2004).
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Dubédat, J. Euler Integrals for Commuting SLEs. J Stat Phys 123, 1183–1218 (2006). https://doi.org/10.1007/s10955-006-9132-9
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DOI: https://doi.org/10.1007/s10955-006-9132-9