Great progress in the understanding of conformally invariant scaling limits of stochastic models, has been given by the Stochastic Löwner Evolutions (SLE). This approach has been pioneered by Schramm [46] and by Lawler, Schramm and Werner [31]. It describes a one-parameter family of conformally invariant measures of curves in the plane or a two-dimensional domain. This family is commonly re-ferred to as SLEκ, where κ parametrizes the family. It has been shown to be the scaling limit of many well-known and less well-known statistical lattice models. These models are typically members of the families of critical and tricritical [40] q-state Potts models [61] and of O(n) models [17], or believed to be in the corresponding universality class.
SLE describes the scaling limit of various open, non-crossing, stochastic paths on the lattice, which are, at least on one side, attached to the boundary. Therefore its application to polygons is restricted in various ways. In the first place it describes only the scaling limit. In many studies of lattice polygons, of course, the scaling limit is considered the most interesting aspect. The restriction to open paths attached to the boundary is more severe. This restriction has been lifted to some extent by recursively considering domains bounded by closed paths resulting from a previous SLE process. This approach applies only to paths that have a tendency to touch themselves (without, of course crossing), and this generalization is not the subject of this chapter. In most cases the paths under consideration by their nature occur in extensive numbers. However, one may concentrate on one of them, and treat the interaction with the others only as an ingredient that defines the stochastic measure of the path under consideration. This, in fact is precisely what SLE does.
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References
L. V. Ahlfors. Conformal invariants: topics in geometric function theory. McGraw-Hill, New York, 1973.
M. Bauer and D. Bernard. SLEκ growth processes and conformal field theories. Phys. Lett. B 543:135–138, 2002. arXiv: math-ph/0206028.
M. Bauer and D. Bernard. Conformal Field Theories of Stochastic Loewner Evolutions. Comm. Math. Phys. 239:493–521, 2003. arXiv: hep-th/0210015.
M. Bauer and D. Bernard. SLE martingales and the Virasoro algebra. Phys. Lett. B 557:309–316, 2003. arXiv: hep-th/0301064.
M. Bauer and D. Bernard. Conformal transformations and the SLE partition function martin-gale. Ann. Henri Poincaré 5:289–326. arXiv: math-ph/0305061.
M. Bauer, D. Bernard, J. Houdayer. Dipolar SLE's. J. Stat. Mech. 0503:P001, 2005. arXiv:math-ph/0411038.
R. J. Baxter. Exactly solved models in statistical mechanics. Academic Press, London, 1982.
R. J. Baxter. q colourings of the triangular lattice. J. Phys. A 19:2821–2839, 1986.
R. J. Baxter, S. B. Kelland, and F. Y. Wu. Equivalence of the Potts model or Whitney polyno-mial with an ice-type model. J. Phys. A 9:397–406, 1976.
V. Beffara. The dimension of the SLE curves. Ann. Prob. 36:1421–1452, 2002. arXiv: math.PR/0211322.
V. Beffara. Hausdorff dimensions for SLE6. Ann. Prob. 32:2606–2629, 2002. arXiv: math.PR/0204208.
F. Camia and C. M. Newman. Continuum nonsimple loops and 2D critical percolation. J. Stat. Phys. 37:157–173, 2004. arXiv: math.PR/0308122.
F. Camia, C. M. Newman. SLE(6) and CLE(6) from Critical Percolation. arXiv:math/0611116.
J. Cardy. Conformal invariance. In C. Domb and J. L. Lebowitz, editors, Phase transitions and critical phenomena, volume 11, pages 55–126. Academic Press, London, 1987.
J. Cardy. Stochastic Loewner Evolution and Dyson's Circular Ensembles. J. Phys. A 36:L379–L408, 2003. arXiv: math-ph/0301039.
J. Cardy. SLE for theoretical physicists. Ann. Phys. 318:81–118, 2005.
E. Domany, D. Mukamel, B. Nienhuis and A. Schwimmer. Duality relations and equivalences for models with O(n) and cubic symmetry. Nucl. Phys. B190:279, 1981.
B. Duplantier. Harmonic measure exponents for two-dimensional percolation. Phys. Rev. Lett. 82:3940–3943, 1999.
B. Duplantier. Conformally invariant fractals and potential theory. Phys. Rev. Lett. 84:1363–1367, 2000.
B. Duplantier and K-H. Kwon. Conformal invariance and intersections of random walks. Phys. Rev. Lett. 61:2514–2517, 1988.
R. Friedrich and W. Werner. Conformal fields, restriction properties, degenerate representa-tions and SLE. C. R. Acad. Sci. Paris Ser. I 335:947–952, 2002. arXiv: math.PR/0209382.
R. Friedrich and W. Werner. Conformal restriction, highest-weight representations and SLE. Comm. Math. Phys. 243:105–122, 2003. arXiv: math-ph/0301018.
C. M. Fortuin and P. W. Kasteleyn. On the random cluster model 1: Introduction and relation to other models. Physica 57:536–564, 1972.
W. Kager, B. Nienhuis, L.P. Kadanoff. Exact solutions for Loewner evolutions. J. Stat. Phys. 115:805–822, 2004. arXiv: math-ph/0309006.
W. Kager, B. Nienhuis. A guide to stochastic Löwner evolution and its application. J. Stat. Phys. 115:1149–1229, 2004. arXiv: math-ph/0312056.
G. F. Lawler. Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab. 1:1–20, 1996.
G. F. Lawler. The dimension of the frontier of planar Brownian motion. Elect. Comm. Probab. 1:29–47, 1996.
G. F. Lawler. Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions. In Random Walks (Budapest, 1998), Bolyai Society Mathematical Studies, volume 9, pages 219–258, 1999.
G. F. Lawler. An introduction to the Stochastic Loewner Evolution. Available online at URL http://www.math.duke.edu/%7Ejose/esi.html, 2001.
G. F. Lawler and W. Werner. Intersection exponents for planar Brownian motion. Ann. Prob. 27(4):1601–1642, 1999.
G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187(2):237–273, 2001. arXiv: math.PR/9911084.
G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents II: Plane exponents. Acta Math. 187(2):275–308, 2001. arXiv: math.PR/0003156.
G. F. Lawler, O. Schramm, and W. Werner. Values of Brownian intersection exponents III: Two-sided exponents. Ann. Inst. H. Poincaré Statist. 38(1):109–123, 2002. arXiv: math.PR/0005294.
G. F. Lawler, O. Schramm, and W. Werner. Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189:179–201, 2002. arXiv: math.PR/0005295.
G. F. Lawler, O. Schramm, and W. Werner. One-arm exponent for critical 2D percolation. Electron. J. Probab. 7(2):13 pages, 2001. arXiv: math.PR/0108211.
G. F. Lawler, O. Schramm, and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Prob. 32:939–995, 2001. arXiv: math.PR/0112234.
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 Benoît Mandelbrot, Amer. Math. Soc. Proc. Symp. Pure Math. 72, Amer. Math. Soc., Providence RI, 2004. arXiv: math.PR/0204277.
G. F. Lawler, O. Schramm, and W. Werner. Conformal restriction: the chordal case. J. Amer. Math. Soc. 16(4):917–955, 2003. arXiv: math.PR/0209343.
K. Löwner. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89:103–121, 1923.
B. Nienhuis, A.N. Berker, E.K. Riedel and M. Schick. First-and second-order phase transi-tions in Potts models; a renormalization-group solution. Phys. Rev. Lett. 43:737 1979.
B. Nienhuis. Exact critical point and exponents of the O(n) model in two dimensions. Phys. Rev. Lett. 49:1062–1065, 1982.
B. Nienhuis. Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb Gas. Journal of Statistical Physics 34:731–761, 1984. Coulomb Gas formulation of two-dimensional phase transitions. In C. Domb and J. L. Lebowitz, editors, Phase transitions and critical phenomena, volume 11, pages 1–53. Academic Press, London, 1987.
B. Nienhuis. Locus of the tricritical transition in a two-dimensional q-state Potts model. Phys-ica A 177:109–113, 1991.
S. Rohde and O. Schramm. Basic properties of SLE. Ann. Math. 161:879–920, 2005. arXiv: math.PR/0106036.
H. Saleur and B. Duplantier. Exact determination of the percolation hull exponent in two dimentions. Phys. Rev. Lett. 58:2325–2328, 1987.
O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118:221–288, 2000. arXiv: math.PR/9904022.
O. Schramm. A percolation formula. Elect. Comm. Probab. 6:115–120, 2001. arXiv: math.PR/0107096.
O. Schramm and S. Sheffield. The harmonic explorer and its convergence to SLE4. Ann. Prob. 33:2127–2148, 2003. arXiv: math.PR/0310210.
O. Schramm, S. Sheffield, D. B. Wilson. Conformal radii for conformal loop ensembles. arXiv:math/0611687.
S. Sheffield. Exploration trees and conformal loop ensembles. arXiv:math/0609167.
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. A longer version is available at URL http://www.math.kth.se/˜stas/papers/.
S. Smirnov and W. Werner. Critical exponents for two-dimensional percolation. Math. Res. Lett. 8:729–744, 2001. arXiv: math.PR/0109120.
S. Smirnov. Conformal Invariance in random cluster models. I Holomorphic fermions in the Ising model, 2007. arXiv:0708.0039.
W. Werner. Random planar curves and Schramm-Löwner Evolutions. Lecture notes from the 2002 Saint-Flour summer school Springer, 2003. arXiv: math.PR/0303354.
W. Werner. Conformal restriction and related questions. 2003. arXiv: math.PR/0307353.
W. Werner, Girsanov's transformation for SLEκρ processes, intersection exponents and hiding exponents. 2003. arXiv:math/0302115.
W. Werner. The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc. 21:137–169, 2005. arXiv:math/0511605.
W. Werner. Some recent aspects of random conformally invariant systems. arXiv:math/0511268.
W. Werner. SLEs as boundaries of clusters of Brownian loops. C. R. Acad. Sci. Paris to appear. arXiv:math/0308164
D. B. Wilson. Generating random spanning trees more quickly than the cover time. In Proceed-ings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pages 296–303, New York, 1996. ACM.
F.Y. Wu. The Potts model. Rev. Mod. Phys. 54:235, 1982.
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Nienhuis, B., Kager, W. (2009). Stochastic Lowner Evolution and the Scaling Limit of Critical Models. In: Guttman, A.J. (eds) Polygons, Polyominoes and Polycubes. Lecture Notes in Physics, vol 775. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9927-4_15
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