×

zbMATH — the first resource for mathematics

Conformal invariance of planar loop-erased random walks and uniform spanning trees. (English) Zbl 1126.82011
Summary: This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain \(D\subsetneqq\mathbb C\) is equal to the radial \(\text{SLE}_2\) path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that \(\partial D\) is a \(C^1\)-simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc \(A\subset\partial D\), is the chordal \(\text{SLE}_8\) path in \(\overline D\) joining the endpoints of \(A\). A by-product of this result is that \(\text{SLE}_8\) is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60G50 Sums of independent random variables; random walks
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ahlfors, L. V. (1973). Conformal Invariants : Topics in Geometric Function Theory . McGraw-Hill, New York. · Zbl 0272.30012
[2] Aizenman, M. and Burchard, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 419–453. · Zbl 0944.60022
[3] Aizenman, M., Burchard, A., Newman, C. M. and Wilson, D. B. (1997). Scaling limits for minimal and random spanning trees in two dimensions. Random Structures Algorithms 15 319–367. · Zbl 0939.60031
[4] Benjamini, I. and Schramm, O. (1996). Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 1219–1238. · Zbl 0862.60053
[5] Cardy, J. L. (1992). Critical percolation in finite geometries. J. Phys. A 25 L201–L206. · Zbl 0965.82501
[6] Collatz, L. (1960). The Numerical Treatment of Differential Equations , 3rd ed. Springer, Berlin. · Zbl 0086.32601
[7] Dehn, M. (1903). Über die Zerlegung von Rechtecken in Rechtecke. Math. Ann. 57 314–332. · JFM 34.0547.02
[8] Dubins, L. E. (1968). On a theorem of Skorohod. Ann. Math. Statist. 39 2094–2097. · Zbl 0185.45103
[9] Dudley, R. M. (1989). Real Analysis and Probability . Wadsworth and Brooks/Cole, Pacific Grove, CA. · Zbl 0686.60001
[10] Duplantier, B. (1987). Critical exponents of Manhattan Hamiltonian walks in two dimensions, from Potts and \(o(n)\) models. J. Statist. Phys. 49 411–431.
[11] Duplantier, B. (1992). Loop-erased self-avoiding walks in two dimensions: Exact critical exponents and winding numbers. Phys. A 191 516–522.
[12] Fomin, S. (2001). Loop-erased walks and total positivity. Trans. Amer. Math. Soc. 353 3563–3583. · Zbl 0973.15014
[13] Fukai, Y. and Uchiyama, K. (1996). Potential kernel for two-dimensional random walk. Ann. Probab. 24 1979–1992. · Zbl 0879.60068
[14] Guttmann, A. and Bursill, R. (1990). Critical exponent for the loop-erased self-avoiding walk by Monte-Carlo methods. J. Statist. Phys. 59 1–9.
[15] Häggström, O. (1995). Random-cluster measures and uniform spanning trees. Stochastic Process. Appl. 59 267–275. · Zbl 0840.60089
[16] He, Z.-X. and Schramm, O. (1998). The \(C^ \infty\)-convergence of hexagonal disk packings to the Riemann map. Acta Math. 180 219–245. · Zbl 0913.30004
[17] Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus . Springer, New York. · Zbl 0638.60065
[18] Kasteleyn, P. W. (1963). A soluble self-avoiding walk problem. Physica 29 1329–1337.
[19] Kenyon, R. (1998). Tilings and discrete Dirichlet problems. Israel J. Math. 105 61–84. · Zbl 0938.60070
[20] Kenyon, R. (2000). The asymptotic determinant of the discrete Laplacian. Acta Math. 185 239–286. · Zbl 0982.05013
[21] Kenyon, R. (2000). Long-range properties of spanning trees. J. Math. Phys. 41 1338–1363. · Zbl 0977.82011
[22] Kenyon, R. (2000). Conformal invariance of domino tilings. Ann. Probab. 28 759–795. · Zbl 1043.52014
[23] Kozma, G. (2002). Scaling limit of loop-erased random walks: A naive approach. Unpublished manuscript.
[24] Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47 655–693. · Zbl 0445.60058
[25] Lawler, G. F. (1991). Intersections of Random Walks . Birkhäuser, Boston. · Zbl 1228.60004
[26] Lawler, G. F. (1999). Loop-erased random walk. In Perplexing Problems in Probability 197–217. Birkhäuser, Boston. · Zbl 0947.60055
[27] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents I. Half-plane exponents. Acta Math. 187 237–273. · Zbl 1005.60097
[28] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents II. Plane exponents. Acta Math. 187 275–308. · Zbl 0993.60083
[29] Lawler, G. F., Schramm, O. and Werner, W. (2002). Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189 179–201. · Zbl 1024.60033
[30] Lawler, G. F., Schramm, O. and Werner, W. (2002). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7 No. 2. · Zbl 1015.60091
[31] Lawler, G. F., Schramm, O. and Werner, W. (2003). Conformal restriction properties: The chordal case. J. Amer. Math. Soc. 16 917–955. · Zbl 1030.60096
[32] Lyons, R. (1998). A bird’s-eye view of uniform spanning trees and forests. In Microsurveys in Discrete Probability 135–162. Amer. Math. Soc., Providence, RI. · Zbl 0909.60016
[33] Majumdar, S. N. (1992). Exact fractal dimension of the loop-erased self-avoiding walk in two dimensions. Phys. Rev. Lett. 68 2329–2331.
[34] Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559–1574. JSTOR: · Zbl 0758.60010
[35] Pommerenke, Ch. (1992). Boundary Behaviour of Conformal Maps . Springer, Berlin. · Zbl 0762.30001
[36] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion . Springer, Berlin. · Zbl 0731.60002
[37] Rohde, S. and Schramm, O. (2001). Basic properties of SLE. Unpublished manuscript. · Zbl 1081.60069
[38] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288. · Zbl 0968.60093
[39] Schramm, O. (2001). A percolation formula. Electron. Comm. Probab. 6 115–120. · Zbl 1008.60100
[40] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239–244. · Zbl 0985.60090
[41] Smirnov, S. (2001). Critical percolation in the plane. I. Conformal invariance and Cardy’s formula. II. Continuum scaling limit. · Zbl 0985.60090
[42] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729–744. · Zbl 1009.60087
[43] Strassen, V. (1967). Almost sure behavior of sums of independent random variables and martingales. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 315–343. Univ. California Press. · Zbl 0201.49903
[44] Wilson, D. B. (1996). Generating random spanning trees more quickly than the cover time. In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing 296–303. ACM, New York. · Zbl 0946.60070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.