Scaling Limits of Loop-Erased Random Walks and Uniform Spanning Trees

doi:10.1007/BF02803524

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Scaling Limits of Loop-Erased Random Walks and Uniform Spanning Trees

Journal Article•DOI•

Scaling Limits of Loop-Erased Random Walks and Uniform Spanning Trees

Oded Schramm¹•Institutions (1)

Weizmann Institute of Science¹

01 Dec 2000-Israel Journal of Mathematics (Springer, New York, NY)-Vol. 118, Iss: 1, pp 221-288

TL;DR: In this paper, the authors considered the limits of the uniform spanning tree and the loop-erased random walk (LERW) on a fine grid in the plane, as the mesh goes to zero.

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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.

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Book Chapter•DOI•

Conformal invariance of planar loop-erased random walks and uniform spanning trees

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Gregory F. Lawler¹, Oded Schramm², Wendelin Werner³•Institutions (3)

Cornell University¹, Microsoft², University of Paris-Sud³

01 Jan 2004-Annals of Probability

TL;DR: In this article, it was shown that the scaling limit of a loop-erased random walk in a simply connected domain is equal to the radial SLE2 path, and that the limit exists and is conformally invariant.

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Abstract: This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain $D\mathop \subset \limits_ e \mathbb{C} $ is equal to the radial SLE2 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 invarint. Assuming that ∂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 are A ⊂ ∂ D, is the chordal SLE8 path in $\overline D $ joining the endpoints of A. A by-product of this result is that SLE8 is almost surely generated by a continuous path. The result and proofs are not restricted to particular choice of Iattice.

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633 citations

Journal Article•DOI•

A survey of random processes with reinforcement

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Robin Pemantle¹•Institutions (1)

University of Pennsylvania¹

12 Feb 2007-Probability Surveys

TL;DR: The models surveyed in this paper include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes, with a focus on methods and results, with sketches provided of some proofs.

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Abstract: The models surveyed include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.

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617 citations

Cites background from "Scaling Limits of Loop-Erased Rando..."

...voiding if and only if κ ≤ 4, and is Robin Pemantle/Random processes with reinforcement 68 space-ﬁlling when κ ≥ 8. Regarding the question of whether SLE is the scaling limit of LERW, it was shown in [Sch00] that if LERW has a scaling limit and this is conformally invariant, then this limit is SLE 2. The conformally invariant limit was conﬁrmed just a few years later: Theorem 6.15 ([LSW04, Theorem 1.3])....
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...otion conditioned immediately to enter the interior of the disk and stay there until it hits the origin. If we could compute in these coordinates, such conditioning would be routine. In 2000, Schramm [Sch00] observed that such a conformal map may be computed via the classical Lo¨wner equation. This is a diﬀerential equation satisﬁed by the conformal maps between a disk and the complement of a growing pat...
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Book Chapter•DOI•

Basic properties of SLE

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Steffen Rohde¹, Oded Schramm²•Institutions (2)

University of Washington¹, Microsoft²

01 Mar 2005-Annals of Mathematics

TL;DR: SLEκ as mentioned in this paper is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motions.

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Abstract: SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions.

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550 citations

Posted Content•

Basic properties of SLE

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Steffen Rohde¹, Oded Schramm²•Institutions (2)

University of Washington¹, Microsoft²

05 Jun 2001-arXiv: Probability

TL;DR: In this article, it was shown that the SLE trace is a path, a simple path, and a self-intersecting path for all values of 8, 4, and 8.

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Abstract: SLE is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed $\kappa$. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions. The present paper attempts a first systematic study of SLE. It is proved that for all $\kappa e 8$ the SLE trace is a path; for $\kappa\in[0,4]$ it is a simple path; for $\kappa\in(4,8)$ it is a self-intersecting path; and for $\kappa>8$ it is space-filling. It is also shown that the Hausdorff dimension of the SLE trace is a.s. at most $1+\kappa/8$ and that the expected number of disks of size $\eps$ needed to cover it inside a bounded set is at least $\eps^{-(1+\kappa/8)+o(1)}$ for $\kappa\in[0,8)$ along some sequence $\eps\to 0$. Similarly, for $\kappa\ge 4$, the Hausdorff dimension of the outer boundary of the SLE hull is a.s. at most $1+2/\kappa$, and the expected number of disks of radius $\eps$ needed to cover it is at least $\eps^{-(1+2/\kappa)+o(1)}$ for a sequence $\eps\to 0$.

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429 citations

Journal Article•DOI•

Values of Brownian intersection exponents, I: Half-plane exponents

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Gregory F. Lawler¹, Oded Schramm², Oded Schramm³, Wendelin Werner⁴•Institutions (4)

Duke University¹, Microsoft², Weizmann Institute of Science³, University of Paris-Sud⁴

01 Jan 2001-Acta Mathematica

TL;DR: Theoretical physics predicts that conformal invariance plays a crucial role in the macroscopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g., the authors ).

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Abstract: Theoretical physics predicts that conformal invariance plays a crucial role in the macroscopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g., [4], [6]). For instance, by making the assumption that critical planar percolation behaves in a conformally invariant way in the scaling limit, and using ideas involving conformal field theory, Cardy [7] produced an exact formula for the limit, as N → ∞, of the probability that, in two-dimensional critical percolation, there exists a cluster crossing the rectangle [0, aN] × [0, bN]. Also, Duplantier and Saleur [13] predicted the “fractal dimension” of the hull of a very large percolation cluster. These are just two examples among many such predictions.

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418 citations

Cites background from "Scaling Limits of Loop-Erased Rando..."

...In [39], the focus is on the case κ = 2, which is conjectured there to correspond to the scaling limit of loop-erased random walks, but the conjecture that SLE6 corresponds to the scaling limit of critical percolation cluster boundaries is also mentioned....
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...Independently, Schramm [39] defined a new class of conformally invariant stochastic processes indexed by a real parameter κ ≥ 0, called SLEκ (for stochastic Löwner evolution process with parameter κ)....
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...We will say that the path γ is nice if it is a continuous simple path γ : [0, 1] → H, such that γ(0), γ(1) ∈ R \ {0} and γ(0, 1) ⊂ H....
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...Although, at present, a proof of the conjecture that SLE6 is the scaling limit of critical percolation cluster boundaries seems out of reach, this conjecture does lead one to believe that SLE6 must satisfy a “locality” property, namely, it is not affected by the boundary of a domain when it is in…...
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...Let f : D → H be a conformal homeomorphism from a domain D ⊂ C onto H. Suppose that N is a nice neighborhood of 0 in H. Define D∗ = f−1(N) and let f ∗ be the conformal homeomorphism ψN ◦ f from D∗ onto H....
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References

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Proceedings Article•DOI•

Generating random spanning trees more quickly than the cover time

[...]

David B. Wilson¹•Institutions (1)

Massachusetts Institute of Technology¹

01 Jul 1996

TL;DR: This paper gives a new algorithm for generating random spanning trees of an undirected graph that is easy to code up, has small running time constants, and has a nice proof that it generates trees with the right probabilities.

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Abstract: It is widely known how to generate random spanning trees of an undirected graph. Broder showed how at FOCS [6], and Aldous too found the algorithm [2]. Start at any vertex and do a simple random walk on the graph. Each time a vertex is first encountered, mark the edge from which it was discovered. When all the vertices are discovered, the marked edges form a random spanning tree. This algorithm is easy to code up, has small running time constants, and has a nice proof that it generates trees with the right probabilities. This paper gives a new algorithm for generating random spanning trees. It too is simple, easy to code up, and has nice proofs. The new algorithm also has the following advantages:

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577 citations

Journal Article•DOI•

Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I.

[...]

Karl Löwner

01 Mar 1923-Mathematische Annalen

564 citations

Journal Article•DOI•

Critical percolation in finite geometries

[...]

John Cardy

21 Feb 1992-Journal of Physics A

TL;DR: In this paper, the methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold.

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Abstract: The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent.

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486 citations

"Scaling Limits of Loop-Erased Rando..." refers background in this paper

...One can, in fact, show that κ = 8, by deriving an appropriate analogue of Cardy’s [ Car92 ]...
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...γ. From this, one can derive Cardy’s [ Car92 ] conjectured formula for the limiting crossing...
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...[ Car92 ] conjectured formula for the limiting crossing probabilities of critical percolation,...
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Journal Article•DOI•

Exact Determination of the Percolation Hull Exponent in Two Dimensions

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Hubert Saleur, Bertrand Duplantier

01 Jun 1987-Physical Review Letters

TL;DR: It is argued finally that the different fractal dimensions observed recently by Grossman and Aharony, who modified the definition of the hull, are all equal to ${D}_{e}=\frac{4}{3}$.

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Abstract: By mapping the two-dimensional percolation problem on a Coulomb gas, we obtain the exact fractal dimension of the external perimeter (or "hull") of the infinite percolation cluster: ${D}_{H}=\frac{7}{4}$, in agreement with numerical estimates and a recent conjecture. We also determine an infinite set of exact exponents associated with various topologies of this hull. We argue finally that the different fractal dimensions observed recently by Grossman and Aharony, who modified the definition of the hull, are all equal to ${D}_{e}=\frac{4}{3}$.

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402 citations

Proceedings Article•DOI•

Generating random spanning trees

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Andrei Z. Broder

30 Oct 1989

TL;DR: It is shown that the Markov chain on the space of all spanning trees of a given graph where the basic step is an edge swap is rapidly mixing.

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Abstract: The author describes a probabilistic algorithm that, given a connected, undirected graph G with n vertices, produces a spanning tree of G chosen uniformly at random among the spanning trees of G. The expected running time is O(n log n) per generated tree for almost all graphs, and O(n/sup 3/) for the worst graphs. Previously known deterministic algorithms are much more complicated and require O(n/sup 3/) time per generated tree. A Markov chain is called rapidly mixing if it gets close to the limit distribution in time polynomial in the log of the number of states. Starting from the analysis of the above algorithm, it is shown that the Markov chain on the space of all spanning trees of a given graph where the basic step is an edge swap is rapidly mixing. >

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395 citations