Scaling limits of loop-erased random walks and uniform spanning trees. (English) Zbl 0968.60093

The author introduces and analyses the so-called stochastic Löwner evolution (SLE), also named Schramm’s process by other authors, which is the conjectured scaling limit for at least two interesting models from statistical mechanics. The existence of a scaling limit for these two models has not been proven yet, but the existence of scaling limits along suited subsequences is easily verified. Under the assumption that the limit actually exists and is conformal invariant, the author identifies them in terms of the SLE and derives some almost sure properties.
The notion of conformal invariance is around in the physicist’s literature since a few decades, but has not been specified yet for many important models. The present paper gives a mathematically rigorous sense to the conformal invariance for two interesting models, the loop-erased random walk (LERW) and for the uniform spanning tree (UST). (The author announces to describe also the conjectured scaling limit of critical site percolation by similar means in a forthcoming paper.) Together with recent results on non-intersection exponents for Brownian motions, obtained by the author in collaboration with Lawler and Werner, these are the first examples of this kind and represent a breakthrough in the mathematical understanding of critical phenomena of two-dimensional models from statistical mechanics.
The LERW is a discrete-time random process on \(\mathbb Z^d\) (here: \(d=2\)) evolving in time where any loop that the trajectory closes is immediately removed, such that we obtain a self-avoiding random path. The UST is a random cycle-free connected subgraph of a given finite graph \(G\) that contains all the vertices and has the uniform distribution on the set of all such graphs. The notion of a UST may naturally be extended to infinite graphs \(G\), and in the present paper the case of \(G=\mathbb Z^2\) is considered throughout. There are intimate connections between the LERW and the UST. Two of the main open questions about these two models (and many related ones) are the following. Assume that the above models are defined on the lattice \(\delta \mathbb Z^2\) rather than on \(\mathbb Z^2\), is there a natural limiting random process for a properly scaled version of the above process as the mesh \(\delta>0\) tends to zero, and how can this limit be described? The present paper does not answer the first question, but (assuming the answer yes to the first one) the second.
The stochastic Löwner evolution is defined as follows. Let \((B(t))_{t\in[0,\infty)}\) be a standard Brownian motion on the boundary of the unit disc \(\mathbb U=\{z\in\mathbb C\colon |z|<1\}\), and fix a parameter \(\kappa\geq 0\). With \(\zeta(t)= B(-\kappa t)\) for \(t\leq 0\), solve the so-called Löwner differential equation \[ \frac{\partial f}{\partial t}=z f_t'(z)\frac{\zeta(t)+z}{\zeta(t)-z},\qquad z\in\mathbb U, \quad t\leq 0, \] with the boundary value \(f_0(z)=z\). Then \(f_t\) is a conformal mapping from \(\mathbb U\) into some domain \(D_t\). The process \((\mathbb U\setminus D_t)_{t\leq 0}\) is called the SLE. Assuming that the scaling limit of the LERW exists and is conformal invariant, it is proven that, for the choice \(\kappa =2\), this scaling limit has the same distribution as \((f_t(\zeta(t)))_{t\leq 0}\). An analogous assertion is proved for the UST. The choices \(\kappa=6\) and \(\kappa=8\) also lead to interesting processes in terms of which the conjectured scaling limit of critical site percolation and the Peano curve winding around the scaling limit of UST may be described in future work, respectively.
Reviewer: W.König (Berlin)


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
30C35 General theory of conformal mappings
Full Text: DOI arXiv


[1] [Aiz] M. Aizenman,Continuum limits for critical percolation and other stochastic geometric models, Preprint. http://xxx.lanl.gov/abs/math-ph/9806004. · Zbl 1253.82044
[2] [ABNW] M. Aizenman, A. Burchard, C. M. Newman and D. B. Wilson,Scaling limits for minimal and random spanning trees in two dimensions, Preprint. http://xxx.lanl.gov/abs/math/9809145. · Zbl 0939.60031
[3] [ADA] M. Aizenman, B. Duplantier and A. Aharony,Path crossing exponents and the external perimeter in 2D percolation, Preprint. http://xxx.lanl.gov/abs/cond-mat/9901018.
[4] Aldous, D. J., The random walk construction of uniform spanning trees and uniform labelled trees, SIAM Journal on Discrete Mathematics, 3, 450-465 (1990) · Zbl 0717.05028 · doi:10.1137/0403039
[5] [Ben] I. Benjamini,Large scale degrees and the number of spanning clusters for the uniform spanning tree, inPerplexing Probability Problems: Papers in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.), Boston, Birkhäuser, to appear.
[6] [BLPS98] I. Benjamini, R. Lyons, Y. Peres and O. Schramm,Uniform spanning forests, Preprint. http://www.wisdom.weizmann.ac.il/≈schramm/papers/usf/. · Zbl 1016.60009
[7] Bishop, C. J.; Jones, P. W.; Pemantle, R.; Peres, Y., The dimension of the Brownian frontier is greater than 1, Journal of Functional Analysis, 143, 309-336 (1997) · Zbl 0870.60077 · doi:10.1006/jfan.1996.2928
[8] [Bow] B. H. Bowditch,Treelike structures arising from continua and convergence groups, Memoirs of the American Mathematical Society, to appear. · Zbl 0961.20034
[9] Broder, A., Generating random spanning trees, 30th Annual Symposium on Foundations of Computer Science, 442-447 (1989), Research Triangle Park, NC: IEEE, Research Triangle Park, NC · doi:10.1109/SFCS.1989.63516
[10] Burton, R.; Pemantle, R., Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, The Annals of Probability, 21, 1329-1371 (1993) · Zbl 0785.60007 · doi:10.1214/aop/1176989121
[11] Cardy, J. L., Critical percolation in finite geometries, Journal of Physics A, 25, L201-L206 (1992) · Zbl 0965.82501 · doi:10.1088/0305-4470/25/4/009
[12] Duplantier, B.; David, F., Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice, Journal of Statistical Physics, 51, 327-434 (1988) · Zbl 1086.82501 · doi:10.1007/BF01028464
[13] Duren, P. L., Univalent Functions (1983), New York: Springer-Verlag, New York · Zbl 0514.30001
[14] Durrett, R., Brownian Motion and Martingales in Analysis (1984), Belmont, California: Wadsworth International Group, Belmont, California · Zbl 0554.60075
[15] Durrett, R., Probability (1991), Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA · Zbl 0709.60002
[16] Ethier, S. N.; Kurtz, T. G., Markov Processes (1986), New York: Wiley, New York · Zbl 0592.60049
[17] Grimmett, G., Percolation (1989), New York: Springer-Verlag, New York · Zbl 0691.60089
[18] Häggström, O., Random-cluster measures and uniform spanning trees, Stochastic Processes and their Applications, 59, 267-275 (1995) · Zbl 0840.60089 · doi:10.1016/0304-4149(95)00042-6
[19] Itô, K., Lectures on Stochastic Processes (1961), Bombay: Tata Institute of Fundamental Research, Bombay
[20] Janiszewski, Journal de l’Ecole Polytechnique, 16, 76-170 (1912)
[21] [Ken98a] R. Kenyon,Conformal invariance of domino tiling, Preprint. http://topo.math.u-psud.fr/≈kenyon/confinv.ps.Z. · Zbl 1043.52014
[22] [Ken98b] R. Kenyon,The asymptotic determinant of the discrete laplacian, Preprint. http://topo.math.u-psud.fr/≈kenyon/asymp.ps.Z. · Zbl 0982.05013
[23] [Ken99] R. Kenyon,Long-range properties of spanning trees, Preprint. · Zbl 0977.82011
[24] [Ken] R. Kenyon, in preparation.
[25] Kesten, H., Hitting probabilities of random walks on ℤ_d, Stochastic Processes and their Applications, 25, 165-184 (1987) · Zbl 0626.60067 · doi:10.1016/0304-4149(87)90196-7
[26] Kufarev, P. P., A remark on integrals of Löwner’s equation, Doklady Akademii Nauk SSSR (N.S.), 57, 655-656 (1947) · Zbl 0029.03702
[27] Langlands, R.; Pouliot, P.; Saint-Aubin, Y., Conformal invariance in twodimensional percolation, Bulletin of the American Mathematical Society (N.S.), 30, 1-61 (1994) · Zbl 0794.60109
[28] Lawler, G. F., A discrete analogue of a theorem of Makarov, Combinatorics, Probability and Computing, 2, 181-199 (1993) · Zbl 0799.60062 · doi:10.1017/S0963548300000584
[29] [Law] G. F. Lawler,Loop-erased random walk, inPerplexing Probability Problems: Papers in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.), Boston, Birkhäuser, to appear. · Zbl 0947.60055
[30] Löwner, K., Untersuchungen über schlichte konforme abbildungen des einheitskreises, I, Mathematische Annalen, 89, 103-121 (1923) · JFM 49.0714.01 · doi:10.1007/BF01448091
[31] Lyons, R., A bird’s-eye view of uniform spanning trees and forests, Microsurveys in Discrete Probability (Princeton, NJ, 1997), 135-162 (1998), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0909.60016
[32] [MR] D. E. Marshall and S. Rohde, in preparation.
[33] Mayer, J. C.; Mohler, L. K.; Oversteegen, L. G.; Tymchatyn, E. D., Characterization of separable metric ℝ-trees, Proceedings of the American Mathematical Society, 115, 257-264 (1992) · Zbl 0754.54026 · doi:10.2307/2159595
[34] Mayer, J. C.; Oversteegen, L. G., A topological characterization of ℝ-trees, Transactions of the American Mathematical Society, 320, 395-415 (1990) · Zbl 0729.54008 · doi:10.2307/2001765
[35] Newman, M. H. A., Elements of the Topology of Plane Sets of Points (1992), New York: Dover, New York · Zbl 0845.54002
[36] Pemantle, R., Choosing a spanning tree for the integer lattice uniformly, The Annals of Probability, 19, 1559-1574 (1991) · Zbl 0758.60010 · doi:10.1214/aop/1176990223
[37] Pommerenke, C., On the Loewner differential equation, The Michigan Mathematical Journal, 13, 435-443 (1966) · Zbl 0163.31801 · doi:10.1307/mmj/1028999601
[38] Russo, L., A note on percolation, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 43, 39-48 (1978) · Zbl 0363.60120 · doi:10.1007/BF00535274
[39] Saleur, H.; Duplantier, B., Exact determination of the percolation hull exponent in two dimensions, Physical Review Letters, 58, 2325-2328 (1987) · doi:10.1103/PhysRevLett.58.2325
[40] [Sch] O. Schramm, in preparation.
[41] Slade, G., Self-avoiding walks, The Mathematical Intelligencer, 16, 29-35 (1994) · Zbl 0795.60065
[42] [SW78] P. D. Seymour and D. J. A. Welsh,Percolation probabilities on the square lattice, inAdvances in Graph Theory (Cambridge Combinatorial Conference, Trinity College, Cambridge, 1977), Annals of Discrete Mathematics3 (1978), 227-245. · Zbl 0405.60015
[43] Tóth, B.; Werner, W., The true self-repelling motion, Probability Theory and Related Fields, 111, 375-452 (1998) · Zbl 0912.60056 · doi:10.1007/s004400050172
[44] Wilson, D. B., 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 (1996), New York: ACM, New York · Zbl 0946.60070 · doi:10.1145/237814.237880
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.