Growth exponent for loop-erased random walk in three dimensions

Shiraishi, Daisuke

doi:10.1214/16-aop1165

Cited by 23 publications

(98 citation statements)

References 39 publications

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“…Considering the time reversal of LE S[0, σ x ] and translating the path, we can relate the probability of the second condition (ii) to the non-intersection probability Es(n) defined as in ( 1.7). In fact, it is known that the probability that γ hits x is comparable to n −1 Es(n) if x is not too close to the origin and the boundary of B(n) (see [21] for this). Thus, loosely speaking, the proof of Theorem 1.1 boils down to that of Theorem 1.2.…”

Section: Some Words About the Proofmentioning

confidence: 99%

“…It suffices to show that b n = c 1 + O 2 −δn (1.11) for some c > 0 and δ > 0. Note that it is proved in [21] that c 1 ≤ b n ≤ c 2 for some constants c 1 , c 2 > 0.…”

Section: Some Words About the Proofmentioning

confidence: 99%

“…In this case, observation i) above is still easily verifiable and observation ii) follows thanks to an auxiliary result generally known as the "separation lemma". For the form that satisfies our setup, see Theorem 6.1.5 of [21] or Claim 3.4 of [19].…”

Section: Some Words About the Proofmentioning

confidence: 99%

“…In [8], it is shown thatfor some c, C, > 0. The existence of the critical exponent for p x 3 is established in [21]. Namely, it is proved that there exists α ∈ [ 1 3 , 1) such that p x 3 ≈ |x| −1−α .(1.2)This allows us to show that the dimension of LERW or its scaling limit is equal to 2 − α (see [21] and [22]).…”

mentioning

confidence: 98%

“…The existence of the critical exponent for p x 3 is established in [21]. Namely, it is proved that there exists α ∈ [ 1 3 , 1) such that p x 3 ≈ |x| −1−α .(1.2)This allows us to show that the dimension of LERW or its scaling limit is equal to 2 − α (see [21] and [22]). Numerical experiments and field-theoretical prediction suggest that 2 − α = 1.62 ± 0.01 (see [2], [25] and [23]).The main purpose of this paper is to improve ( 1.2).…”

mentioning

confidence: 98%

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One-point function estimates for loop-erased random walk in three dimensions

Shiraishi¹

2019

Electron. J. Probab.

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In this work, we consider loop-erased random walk (LERW) in three dimensions and give an asymptotic estimate on the one-point function for LERW and the non-intersection probability of LERW and simple random walk in three dimensions for dyadic scales. These estimates will be crucial to the characterization of the convergence of LERW to its scaling limit in natural parametrization. As a step in the proof, we also obtain a coupling of two pairs of LERW and SRW with different starting points conditioned to avoid each other.Loop-erased random walk (LERW) is a random simple path obtained by erasing all loops chronologically from a simple random walk path, which was originally introduced by Greg Lawler ([5]). Since his introduction of LERW, it has been studied extensively both in mathematics and physics literature. In two dimensions, it is proved that it has a conformally invariant scaling limit, which is charaterized by Schramm-Loewner evolution (SLE) (see [20] and [14]). LERW also has a strong connection with other models in statistical physics, e.g. the uniform spanning tree (UST) which arises in statistical physics in conjunction with the Potts model (see [18] and [24] for the relation between LERW and UST). In this paper, we consider the one-point function for LERW in three dimensions, i.e., we study the probability that LERW in Z 3 hits a given point and obtain an asymptotic bound with error estimate for dyadic scales.LERW in Z d enjoys a Gaussian behavior if d is large. In fact, it is known that the scaling limit of LERW is Brownian motion (see Theorem 7.7.6 of [10]) for d ≥ 4. Furthermore, the probability of LERW hitting a given point x ∈ Z d (we write p x d for this hitting probability) is of order |x| 2−d for d ≥ 5 and |x| −2 (log |x|) −1/3 for d = 4 assuming that LERW starts from the origin (see Section 11.5 of [13] for d ≥ 5 and [6] for d = 4).On the other hand, if d is small, the situation changes dramatically. In two dimensions, LERW converges to SLE 2 when the lattice spacing tends to 0 (see [20] and [14]). Furthermore, it is established by Rick Kenyon ([3]) that p x 2 ≈ |x| −3/4 (the notation ≈ means that the logarithm of both sides are asymptotic as |x| → ∞, see also [16] for estimates on p 2 x ). Recently, using SLE techniques, it is proved in [1] that p x 2 ∼ c|x| −3/4 for some constant c where the notation ∼ means that the both sides are asymptotic. In contrast to other dimensions, relatively little is known for LERW in three dimensions. One crucial reason for this is that we have no nice tool like SLE to describe the LERW scaling limit (the existence of the scaling limit is proved in [4] though). In [8], it is shown thatfor some c, C, > 0. The existence of the critical exponent for p x 3 is established in [21]. Namely, it is proved that there exists α ∈ [ 1 3 , 1) such that p x 3 ≈ |x| −1−α .(1.2)This allows us to show that the dimension of LERW or its scaling limit is equal to 2 − α (see [21] and [22]). Numerical experiments and field-theoretical prediction suggest that 2 − α = 1.62 ± 0.01 (see [2...

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Section: Some Words About the Proofmentioning

confidence: 99%

“…It suffices to show that b n = c 1 + O 2 −δn (1.11) for some c > 0 and δ > 0. Note that it is proved in [21] that c 1 ≤ b n ≤ c 2 for some constants c 1 , c 2 > 0.…”

Section: Some Words About the Proofmentioning

confidence: 99%