Loop-erased random walk

Lawler, Gregory F.; Limic, Vlada

doi:10.1017/cbo9780511750854.012

Cited by 4 publications

(7 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…See [25]. A simpler proof follows immediately from the symmetry of (12.2.3) in [26]. This result (and the proofs) also holds if we condition I' to exit aD at a prescribed u E Va(D), which corresponds to the event {y n aD = {uH = {f3 n aD = {u}} (assuming this has positive probability).…”

Section: Loop-erased Random Walk Backgroundmentioning

confidence: 77%

“…Many striking properties of UST and LERW have been discovered. See [32] for a survey of UST's and [26] for a survey of properties ofLERWin Zd, d > 2.…”

Section: Recent Progressmentioning

confidence: 99%

“…This can be done either by using Wilson's algorithm or, equivalently, by giving to each tree T a probability that is proportional to the product of the transition probabilities along the edges of T. In other words, P[T] = Z-l fleET Pe, where Z is a normalizing constant. (The equivalence is proved in [44]; see also [26].) We call this the UST corresponding to the walk S (even if this probability measure is not uniform).…”

Section: Zezmentioning

confidence: 99%

See 2 more Smart Citations

Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees

Lawler

Schramm

Werner

2011

Selected Works of Oded Schramm

289

477

View full text Add to dashboard Cite

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain D ~ tC 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 invariant. Assuming that aD is a C 1 -simp1e 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 c aD, is the chordal SLEg path in I5 joining the endpoints of A. A by-product of this result is that SLEg is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

show abstract

Section: Loop-erased Random Walk Backgroundmentioning

confidence: 77%

“…Many striking properties of UST and LERW have been discovered. See [32] for a survey of UST's and [26] for a survey of properties ofLERWin Zd, d > 2.…”

Section: Recent Progressmentioning

confidence: 99%

Section: Zezmentioning

confidence: 99%

See 1 more Smart Citation

Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees

Lawler

Schramm

Werner

2011

Selected Works of Oded Schramm

289

477

View full text Add to dashboard Cite

show abstract

“…This was considered in [99]. Our approach is slightly different as we emphasize the connection with statistical mechanics.…”

Section: The Brownian Loop Soupmentioning

confidence: 99%

2D growth processes: SLE and Loewner chains

Bauer¹,

Bernard²

2006

Physics Reports

195

357

View full text Add to dashboard Cite

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject. U, D = (planar) domain, ie. connected and simply connected open subset of the complex plane C. K = hulls, ie. connected compact subset of a domain D such that D \ K is a domain. Key words: PACS: * Member of CNRSγ [0,t] = the SLE curve with tip γ t at time t. K t = the SLE hull at time t. D t ≡ D \ K t , the domain D with the hull K t removed. g t : D t → D, the SLE Loewner map and f t : D → D t , its inverse. U t = g t (γ t )= image of the tip of the SLE curve. h t : D t → D, mapping the tip of the curve back to its starting point. vir = the Virasoro algebra. g h = a group element associated to a map h. G h = representation of g h in CFT Hilbert spaces. h r;s = [(rκ − 4s) 2 − (κ − 4) 2 ]/16κ for c = 1 − 6(κ − 4) 2 /4κ. |ψ r;s = highest weight vector with dimension h r;s . ϕ δ (x) = boundary primary field with dimension δ. ψ r;s (x) = degenerate boundary primary field with dimension h r;s . Φ(z,z) = bulk primary fields. 1 IntroductionThe main subject of this report is stochastic Loewner evolutions, and its interplay with statistical mechanics and conformal field theory.Stochastic Loewner evolutions are growth processes, and as such they fall in the more general category of growth phenomena. These are ubiquitous in the physical world at many scales, from crystals to plants to dunes and larger. They can be studied in many frameworks, deterministic of probabilistic, in discrete or continuous space and time. Understanding growth is usually a very difficult task. This is true even in two dimensions, the case we concentrate on in these notes. Yet two dimensions is a highly favorable situation because it allows to make use of the power of complex analysis in one variable. In many interesting cases, the growing object in two dimensions can be seen as a domain, i.e. a contractile open subset of the Riemann sphere (the complex plane with a point at infinity added) leading to a description by so-called Loewner chains.Stochastic Loewner evolution is a simple but particularly interesting example of growth process f...

show abstract

“…We briefly review the definition of the loop-erased random walk; see [10,Chapter 7] and [11] for more details. Since simple random walk in Z 2 is recurrent, it is not possible to construct loop-erased random walk by erasing loops from an infinite walk.…”

Section: Fomin's Identity For Loop-erased Walkmentioning

confidence: 99%

Estimates of Random Walk Exit Probabilities and Application to Loop-Erased Random Walk

Kozdron

Lawler

2005

Electron. J. Probab.

View full text Add to dashboard Cite

We prove an estimate for the probability that a simple random walk in a simply connected subset A ⊂ Z 2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius. We apply these estimates to prove a conjecture of S. Fomin [4] in 2001 concerning a relationship between crossing probabilities of loop-erased random walk and Brownian motion.

show abstract

Loop-erased random walk

Cited by 4 publications

References 6 publications

Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees

Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees

2D growth processes: SLE and Loewner chains

Estimates of Random Walk Exit Probabilities and Application to Loop-Erased Random Walk

Contact Info

Product

Resources

About