# Scaling limits of loop-erased random walks and uniform spanning trees

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## Abstract

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane.

The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Löwner differential equation, with boundary values

$$\frac{{\partial f}}{{\partial t}} = z\frac{{\zeta (t) + z}}{{\zeta (t) - z}}\frac{{\partial f}}{{\partial z}}$$

(1)

*f(z,0)=z*, in the range*z*∈U= {*w*∈ ℂ : •*w*• < 1},*t*≤0. We choose ζ(*t*):=B(−2*t*), where B(*t*) is Brownian motion on ∂\( \mathbb{U} \) starting at a random-uniform point in ∂\( \mathbb{U} \). Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to ∂\( \mathbb{U} \) has the same law as that of the path*f*(ζ*(t),t)*(where*f(z,t)*is extended continuously to ∂\( \mathbb{U} \)) ×(−∞,0]). We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.## Keywords

Span Tree Conformal Invariance Scaling Limit Simple Path Simple Random Walk
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- [Aiz] M. Aizenman,
*Continuum limits for critical percolation and other stochastic geometric models*, Preprint. http://xxx.lanl.gov/abs/math-ph/9806004.Google Scholar - [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.Google Scholar - [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.Google Scholar - [Ald90] D. J. Aldous,
*The random walk construction of uniform spanning trees and uniform labelled trees*, SIAM Journal on Discrete Mathematics**3**(1990), 450–465.zbMATHCrossRefMathSciNetGoogle Scholar - [Ben] I. Benjamini,
*Large scale degrees and the number of spanning clusters for the uniform spanning tree*, in*Perplexing Probability Problems: Papers in Honor of Harry Kesten*(M. Bramson and R. Durrett, eds.), Boston, Birkhäuser, to appear.Google Scholar - [BLPS98] I. Benjamini, R. Lyons, Y. Peres and O. Schramm,
*Uniform spanning forests*, Preprint. http://www.wisdom.weizmann.ac.il/≈schramm/papers/usf/.Google Scholar - [BJPP97] C. J. Bishop, P. W. Jones, R. Pemantle and Y. Peres,
*The dimension of the Brownian frontier is greater than 1*, Journal of Functional Analysis**143**(1997), 309–336.zbMATHCrossRefMathSciNetGoogle Scholar - [Bow] B. H. Bowditch,
*Treelike structures arising from continua and convergence groups*, Memoirs of the American Mathematical Society, to appear.Google Scholar - [Bro89] A. Broder,
*Generating random spanning trees*, in*30th Annual Symposium on Foundations of Computer Science*, IEEE, Research Triangle Park, NC, 1989, pp. 442–447.CrossRefGoogle Scholar - [BP93] R. Burton and R. Pemantle,
*Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances*, The Annals of Probability**21**(1993), 1329–1371.zbMATHCrossRefMathSciNetGoogle Scholar - [Car92] J. L. Cardy,
*Critical percolation in finite geometries*, Journal of Physics A**25**(1992), L201-L206.zbMATHCrossRefMathSciNetGoogle Scholar - [DD88] B. Duplantier and F. David,
*Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice*, Journal of Statistical Physics**51**(1988), 327–434.zbMATHCrossRefMathSciNetGoogle Scholar - [Dur83] P. L. Duren,
*Univalent Functions*, Springer-Verlag, New York, 1983.zbMATHGoogle Scholar - [Dur84] R. Durrett,
*Brownian Motion and Martingales in Analysis*, Wadsworth International Group, Belmont, California, 1984.zbMATHGoogle Scholar - [Dur91] R. Durrett,
*Probability*, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1991.zbMATHGoogle Scholar - [EK86] S. N. Ethier and T. G. Kurtz,
*Markov Processes*, Wiley, New York, 1986.zbMATHGoogle Scholar - [Gri89] G. Grimmett,
*Percolation*, Springer-Verlag, New York, 1989.zbMATHGoogle Scholar - [Häg95] O. Häggström,
*Random-cluster measures and uniform spanning trees*, Stochastic Processes and their Applications**59**(1995), 267–275.zbMATHCrossRefMathSciNetGoogle Scholar - [Itô61] K. Itô,
*Lectures on Stochastic Processes*, Notes by K. M. Rao, Tata Institute of Fundamental Research, Bombay, 1961.Google Scholar - [Jan12] Janiszewski, Journal de l'Ecole Polytechnique
**16**(1912), 76–170.Google Scholar - [Ken98a] R. Kenyon,
*Conformal invariance of domino tiling*, Preprint. http://topo.math.u-psud.fr/≈kenyon/confinv.ps.Z.Google Scholar - [Ken98b] R. Kenyon,
*The asymptotic determinant of the discrete laplacian*, Preprint. http://topo.math.u-psud.fr/≈kenyon/asymp.ps.Z.Google Scholar - [Ken99] R. Kenyon,
*Long-range properties of spanning trees*, Preprint.Google Scholar - [Ken] R. Kenyon, in preparation.Google Scholar
- [Kes87] H. Kesten,
*Hitting probabilities of random walks on ℤ*_{d}, Stochastic Processes and their Applications**25**(1987), 165–184.zbMATHCrossRefMathSciNetGoogle Scholar - [Kuf47] P. P. Kufarev,
*A remark on integrals of Löwner's equation*, Doklady Akademii Nauk SSSR (N.S.)**57**(1947), 655–656.zbMATHMathSciNetGoogle Scholar - [LPSA94] R. Langlands, P. Pouliot and Y. Saint-Aubin,
*Conformal invariance in twodimensional percolation*, Bulletin of the American Mathematical Society (N.S.)**30**(1994), 1–61.zbMATHMathSciNetGoogle Scholar - [Law93] G. F. Lawler,
*A discrete analogue of a theorem of Makarov*, Combinatorics, Probability and Computing**2**(1993), 181–199.zbMATHMathSciNetCrossRefGoogle Scholar - [Law] G. F. Lawler,
*Loop-erased random walk*, in*Perplexing Probability Problems: Papers in Honor of Harry Kesten*(M. Bramson and R. Durrett, eds.), Boston, Birkhäuser, to appear.Google Scholar - [Löw23] K. Löwner,
*Untersuchungen über schlichte konforme abbildungen des einheitskreises, I*, Mathematische Annalen**89**(1923), 103–121.CrossRefMathSciNetzbMATHGoogle Scholar - [Lyo98] R. Lyons, A bird's-eye view of uniform spanning trees and forests, in
*Microsurveys in Discrete Probability (Princeton, NJ, 1997)*, American Mathematical Society, Providence, RI, 1998, pp. 135–162.Google Scholar - [MR] D. E. Marshall and S. Rohde, in preparation.Google Scholar
- [MMOT92] J. C. Mayer, L. K. Mohler, L. G. Oversteegen and E. D. Tymchatyn,
*Characterization of separable metric ℝ-trees*, Proceedings of the American Mathematical Society**115**(1992), 257–264.zbMATHCrossRefMathSciNetGoogle Scholar - [MO90] J. C. Mayer and L. G. Oversteegen,
*A topological characterization of ℝ-trees*, Transactions of the American Mathematical Society**320**(1990), 395–415.zbMATHCrossRefMathSciNetGoogle Scholar - [New92] M. H. A. Newman,
*Elements of the Topology of Plane Sets of Points*, second edition, Dover, New York, 1992.zbMATHGoogle Scholar - [Pem91] R. Pemantle,
*Choosing a spanning tree for the integer lattice uniformly*, The Annals of Probability**19**(1991), 1559–1574.zbMATHCrossRefMathSciNetGoogle Scholar - [Pom66] C. Pommerenke,
*On the Loewner differential equation*, The Michigan Mathematical Journal**13**(1966), 435–443.zbMATHCrossRefMathSciNetGoogle Scholar - [Rus78] L. Russo,
*A note on percolation*, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete**43**(1978), 39–48.zbMATHCrossRefGoogle Scholar - [SD87] H. Saleur and B. Duplantier,
*Exact determination of the percolation hull exponent in two dimensions*, Physical Review Letters**58**(1987), 2325–2328.CrossRefMathSciNetGoogle Scholar - [Sch] O. Schramm, in preparation.Google Scholar
- [Sla94] G. Slade,
*Self-avoiding walks*, The Mathematical Intelligencer**16**(1994), 29–35.zbMATHMathSciNetGoogle Scholar - [SW78] P. D. Seymour and D. J. A. Welsh,
*Percolation probabilities on the square lattice*, in*Advances in Graph Theory (Cambridge Combinatorial Conference, Trinity College, Cambridge, 1977*), Annals of Discrete Mathematics**3**(1978), 227–245.Google Scholar - [TW98] B. Tóth and W. Werner,
*The true self-repelling motion*, Probability Theory and Related Fields**111**(1998), 375–452.zbMATHCrossRefMathSciNetGoogle Scholar - [Wil96] 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)*, ACM, New York, 1996, pp. 296–303.CrossRefGoogle Scholar

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