Loop-erased random walk and Poisson kernel on planar graphs

Yadin, Ariel; Yehudayoff, Amir

doi:10.1214/10-aop579

Cited by 20 publications

(51 citation statements)

References 17 publications

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“…Here we note that, for instance, in d = 2 the scaling limit of the loop-erasure of the simple random walk on the supercritical percolation cluster coincides with that for the ordinary simple random walk -namely SLE 2 (Yadin and Yehudayoff [35]). The question is thus whether one can extend this remarkable result to other dimensions (obviously, with a different scaling limit) and other conductance laws.…”

Section: Recentmentioning

confidence: 52%

Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

Biskup

Boukhadra²

2012

Journal of the London Mathematical Society

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ABSTRACT. We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on Z 4 ) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability P 2n ω (0, 0) after 2n steps is at most C(ω)n −2 log n, but the best lower bound till now has been C(ω)n −2 . Here we will show that the log n term marks a real phenomenon by constructing an environment, for each sequence λ n → ∞, such thatwith C(ω) > 0 a.s., along a deterministic subsequence of n's. Notably, this holds simultaneously with a (non-degenerate) quenched invariance principle. As for the d ≥ 5 cases studied earlier, the source of the anomalous decay is a trapping phenomenon although the contribution is in this case collected from a whole range of spatial scales.

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Section: Recentmentioning

confidence: 52%

Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

Biskup

Boukhadra²

2012

Journal of the London Mathematical Society

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“…We do not expect that there is a natural and more general condition for Temperleyan graphs. Among several equivalent specific formulations, we choose the following (essentially as in [34]).…”

Section: 2mentioning

confidence: 99%

“…Since spanning trees can be derived from random walks (Wilson's algorithm, [33]), a natural (and essentially minimal) criticality condition is that the underlying random walk (RW) on Γ converges, up to time change, to Brownian motion (BM). Under this assumption, Yadin and Yehudayoff [34] showed convergence of the Loop-Erased Random Walk (LERW) to SLE 2 (Schramm-Loewner Evolution with κ = 2), extending celebrated work of Lawler, Schramm, Werner on regular lattices [25].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of Height Change on Toroidal Temperleyan Dimer Models

Dubédat

Gheissari

2015

J Stat Phys

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The dimer model is an exactly solvable model of planar statistical mechanics. In its critical phase, various aspects of its scaling limit are known to be described by the Gaussian free field. For periodic graphs, criticality is an algebraic condition on the spectral curve of the model, determined by the edge weights [21]; isoradial graphs provide another class of critical dimer models, in which the edge weights are determined by the local geometry.In the present article, we consider another class of graphs: general Temperleyan graphs, i.e. graphs arising in the (generalized) Temperley bijection between spanning trees and dimer models. Building in particular on Forman's formula and representations of Laplacian determinants in terms of Poisson operators, and under a minimal assumption -viz. that the underlying random walk converges to Brownian motion -we show that the natural topological observable on macroscopic tori converges in law to its universal limit, i.e. the law of the periods of the dimer height function converges to that of the periods of a compactified free field. arXiv:1407.6227v2 [math.PR]

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“…For obtaining the result as stated above we first prove the convergence of the driving function of the loop erasure (Theorem 5.1). The proof is made in a way similar to [3], [10] and [7]. In [7] the harmonic explorer, an evolution of a self avoiding random curve, is introduced and proved to converge to a chordal SLE 4 curve.…”

Section: Introductionmentioning

confidence: 99%

“…By the way, Proposition 4.1 provides an improvement of the convergence to a radial SLE 2 . In [10] the loop erasure is unti-chronological (loops are discarded in the reverse order). The reason is that one wants to consider the loop erasure determined from the boundary.…”

Section: Introductionmentioning

confidence: 99%

Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve

Suzuki¹

2014

Kodai Math. J.

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In this paper we consider the 'natural' random walk on a planar graph and scale it by a small positive number δ. Given a simply connected domain D and its two boundary points a and b, we start the scaled walk at a vertex of the graph nearby a and condition it on its exiting D through a vertex nearby b, and prove that the loop erasure of the conditioned walk converges, as δ ↓ 0, to the chordal SLE 2 that connects a and b in D, provided that an invariance principle is valid for both the random walk and the dual walk of it. Our result is an extension of one due to Dapeng Zhan [12] where the problem is considered on the square lattice. A convergence to the radial SLE 2 has been obtained by Lawler, Schramm and Werner [3] for the square and triangular lattices and by Yadin and Yehudayoff [10] for a wide class of planar graphs. Our proof, though an adaptation of that of [3] and [10], involves some new ingredients that arise from two sources: one for dealing with a martingale observable that is different from that used in [3] and [10] and the other for estimating the harmonic measures of the random walk started at a boundary point of a domain. MSC2010 subject classifications. 60F17, 60J67, 82B41.

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Loop-erased random walk and Poisson kernel on planar graphs

Cited by 20 publications

References 17 publications

Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

Asymptotics of Height Change on Toroidal Temperleyan Dimer Models

Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve

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