## The scaling limit of loop-erased random walk in three dimensions

Citations: | 9 - 0 self |

### BibTeX

@MISC{Kozma_thescaling,

author = {Gady Kozma},

title = {The scaling limit of loop-erased random walk in three dimensions},

year = {}

}

### OpenURL

### Abstract

ABSTRACT. We show that the scaling limit exists and is invariant to dilations and rotations. We give some tools that might be useful to show universality.

### Citations

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Citation Context ... non-trivial only in dimensions 2 and 3. In dimension 1 it follows from the “gambler ruin problem” that ξ1(k, l) = k + l while in dimensions ≥ 4 Brownian motions never intersect [DEK50] so ξ ≡ 0. See =-=[L91]-=- for a more detailed explanation of these facts. Hence from now on we will only relate to dimensions 2 and 3. 2. In [BL90a] it was shows that the same ξ(k, l) hold for the equivalent problem for rando... |

122 | Conformal invariance of planar loop-erased random walks and uniform spanning trees
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Citation Context ... fractal dimensions, critical exponents and winding numbers [D92, M92]. Rigorously, 3 different approaches proved fruitful: the connection to random domino tilings [K00a, K00b], the connection to SLE =-=[LSW04a]-=-, and the approach we will pursue in this paper, [K]. In fact, SLE was discovered [S00] in the context of LERW. Attempts to understand LERW in dimension 3 focused mainly on the number of steps it take... |

99 | Generating random spanning trees more quickly than the cover time
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Citation Context ...panning trees of G: the path in the UST between two points is distributed like a LERW between them [P91], and further, the entire UST can be generated using repeated use of LERW by Wilson’s algorithm =-=[W96]-=-. Both models, and the connections between them are interesting on a general graph, but we shall be most interested in lattices on Rd and open subsets thereof, in which case these models arise natural... |

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Citation Context ...the uniform spanning tree (UST), a random spanning tree of a graph G selected uniformly between all spanning trees of G: the path in the UST between two points is distributed like a LERW between them =-=[P91]-=-, and further, the entire UST can be generated using repeated use of LERW by Wilson’s algorithm [W96]. Both models, and the connections between them are interesting on a general graph, but we shall be... |

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Citation Context ...tart from the same point then the nice symmetry argument of [L99] can show that this probability is > 1 − Cr −1/3 . Presumably, an equivalent argument would work in our case as well. The arguments of =-=[AB99]-=- should also give a usable estimate. 5. ISOTROPIC INTERPOLATION The purpose of this chapter is to compare random walks on two or more graphs all of which are isotropic, with uniformly bounded structur... |

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Citation Context ...iciently small. Using Harnack’s inequality (lemma 2.2) for the domains B(0, 1) \ B(0, λ) ⊂ B(0, 2) \ B(0, 1 2λ) proves (13). The two dimensional case follows from electrical resistance arguments. See =-=[S94]-=- for background on this topic. The maximum principle shows that G(v, w) ≤ G(v, v) and the latter is equal to the resistance between v and ∂S. The electrical resistance is preserved (up to a constant) ... |

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Citation Context ...(B(v, √ t)) e−Cδ(v,w)2 /t ≤ P v (R(t) = w) ≤ for all δ(v, w) ≤ t. C ω(B(v, √ t)) e−cδ(v,w)2 /t Further, for any two clauses, all constants in the first depend only on the constants in the second. See =-=[D99]-=-. One of the important consequences of this theorem is that the parabolic Harnack inequality is invariant to rough isometries: as already remarked, the combination of the volume doubling property and ... |

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Citation Context ...a graph roughly isometric to R d has a d-dimensional heat kernel decay follows essentially from Varopoulos [V85]. To get an estimate for the Green function one can apply e.g. Hebisch and Saloff-Coste =-=[HSC93]-=- which gives a square exponential decay upper bound. Lemma 2.4. Let G be a Euclidean net and let v ∈ G and r > 1. Then for some constant λ(G) sufficiently large, P w (R( ⌊ λr 2⌋ ) ∈ B(v, r)) ≤ 1 2 Pro... |

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Citation Context ... [BL90b] it was shown that the ξ(k, l) are strictly increasing, and in particular that ξ3(1, 1) < 1. The proof uses the Wiener shell test, somewhat like the techniques we will use in chapter 4. 5. In =-=[L96a]-=- the estimate (23) was improved to P(x, y, r) ≈ r −ξ(1,1) (26) i.e. the error was shown to be in a constant only (for better comparison, write P(x, y, r) = r −ξ(1,1)+O(1/ log r) ). Roughly, this follo... |

34 |
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Citation Context ...hat both f ◦ g and g ◦ f are rough identities. In this case we call both f and g rough isometries. The term was introduced by Kanai in [K85], though in more restricted settings it already appeared in =-=[G81]-=-. There are various equivalent definitions in the literature, but I prefer the above “categorical” one. A rough isometry completely ignores all local structure, and in fact R d is roughly isometric to... |

34 |
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Citation Context ...ric if there exist rough morphisms f : X → Y and g : Y → X such that both f ◦ g and g ◦ f are rough identities. In this case we call both f and g rough isometries. The term was introduced by Kanai in =-=[K85]-=-, though in more restricted settings it already appeared in [G81]. There are various equivalent definitions in the literature, but I prefer the above “categorical” one. A rough isometry completely ign... |

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Citation Context ...h less is known. The discrete Gaussian free field converges to the continuum Gaussian free field and the Richardson model was shown to have a limit shape from subadditivity arguments in any dimension =-=[R73]-=-, and I cannot resist citing the beautiful work on branched polymers in dimension 3 [BI03]. But these examples are the exception, not the rule. 1.1. Sketch of the proof. The core of the argument is ve... |

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27 |
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Citation Context ...mS. Remark. The use of Delmotte’s theorem here is somewhat of an overkill. The fact that the a graph roughly isometric to R d has a d-dimensional heat kernel decay follows essentially from Varopoulos =-=[V85]-=-. To get an estimate for the Green function one can apply e.g. Hebisch and Saloff-Coste [HSC93] which gives a square exponential decay upper bound. Lemma 2.4. Let G be a Euclidean net and let v ∈ G an... |

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Citation Context ...ng self-loops it is not at all easy to construct examples of graphs satisfying the elliptic Harnack inequality without satisfying the parabolic one. See [BB99, GSC05] for some constructions (see also =-=[HSC01]-=-). Theorem (Delmotte). Let G be an infinite connected graph and assume ω(v, v) > c for all v ∈ G. Then the following are equivalent (i) G satisfies the volume doubling property and the weak Poincaré i... |

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Citation Context ... to SLE 4 [SS]. The case of the self-avoiding walk demonstrates the difficulties involved nicely: it has been proved that if the limit exists and is conformally invariant, it would be chordal SLE 8/3 =-=[LSW04b]-=-, but the existence of the limit is still open. In high dimensions lace expansion has been used to show that the scaling limit of the self avoiding walk is Brownian motion [BS85, HS92], and that the s... |

18 |
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Citation Context ...necessitates that a random walk that starts quite close to a loop-erased random walk will avoid hitting it for a long while. This, however, contradicts the discrete Beurling projection principle (see =-=[K87]-=-) that states that a random walk starting near any path has a high probability to intersect it 1 . See [S00, lemma 2.1] or [K, lemma 18]. Unfortunately this argument no longer holds in three dimension... |

15 |
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Citation Context ...understand LERW in dimension 3 focused mainly on the number of steps it takes to reach the distance r. Physicists conjecture that it is ≈ rξ and did numerical experiments to show that ξ = 1.62 ± 0.01 =-=[GB90]-=-. Rigorously the existence of ξ is not proved (so we must talk about an upper and lower exponents ξ ≤ ξ), and the best estimates known are 1 < ξ ≤ ξ ≤ 5/3 [L99]. LERW has no natural continuum equivale... |

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Citation Context ...ick as above. Remark. The fact that a graph roughly isometric to R d satisfies the elliptic Harnack inequality was known before [D99]. In the setting of graphs, it was proved concurrently by Delmotte =-=[D97]-=- and Holopainen-Soardi [HS97] (who proved it for the pLaplacian for any p). In the setting of manifolds this goes back to Kanai [K85], who j □THE SCALING LIMIT OF LOOP-ERASED RANDOM WALK IN THREE DIM... |

12 |
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Citation Context ...ct that a graph roughly isometric to R d satisfies the elliptic Harnack inequality was known before [D99]. In the setting of graphs, it was proved concurrently by Delmotte [D97] and Holopainen-Soardi =-=[HS97]-=- (who proved it for the pLaplacian for any p). In the setting of manifolds this goes back to Kanai [K85], who j □THE SCALING LIMIT OF LOOP-ERASED RANDOM WALK IN THREE DIMENSIONS 14 proved that a mani... |

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Citation Context ...g of Brownian motion and random walk on the same probability space so as to be quite close. The analog of [L96a] for general ξ(k, l) was proved in [L98] while the analog of [L96b] is [LP00]. See also =-=[LSW02b]-=-.THE SCALING LIMIT OF LOOP-ERASED RANDOM WALK IN THREE DIMENSIONS 27 6. Both [L96a] and [L96b] used the estimate (26) to prove the existence of many cut times or cut points for random walk, using rel... |

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9 |
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Citation Context ...nalyzed using the nonintersections of simple random walk directly [L91, chapter 7] giving an easy proof that the scaling limit is Brownian motion. In 4 dimensions a logarithmic correction is required =-=[L95]-=-, and that too has been proved with no use of the difficult technique of lace expansion (see [BS85, HvdHS03] for lace expansion). Borrowing a term from physics we might say that the upper critical dim... |

9 |
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Citation Context ...periments to show that ξ = 1.62 ± 0.01 [GB90]. Rigorously the existence of ξ is not proved (so we must talk about an upper and lower exponents ξ ≤ ξ), and the best estimates known are 1 < ξ ≤ ξ ≤ 5/3 =-=[L99]-=-. LERW has no natural continuum equivalent in dimensions smaller than 4 — Brownian motion has a dense set of loops and therefore it is not clear how to remove them in chronological order. In two dimen... |

8 |
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Citation Context ...while in dimensions ≥ 4 Brownian motions never intersect [DEK50] so ξ ≡ 0. See [L91] for a more detailed explanation of these facts. Hence from now on we will only relate to dimensions 2 and 3. 2. In =-=[BL90a]-=- it was shows that the same ξ(k, l) hold for the equivalent problem for random walks (see also [CM91]). Other relevant variations consider using Brownian motions (or random walks) W ∗ i with fixed len... |

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8 | Cut times for simple random walk
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(Show Context)
Citation Context ... up to t and from t on as two random walks, and reversing the first part, we see that it is important to estimate the probability that two random walks of length t will not intersect. This is ≈ t −2ξ =-=[L96b]-=- where ξ is the famous non-intersection exponent of Brownian motion. See section 3.2 for a description of this topic. Heuristically speaking, a set is “hittable” if its Hausdorff dimension is > 1, and... |

8 | The intersection exponent for simple random walk
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Citation Context ...edding, a coupling of Brownian motion and random walk on the same probability space so as to be quite close. The analog of [L96a] for general ξ(k, l) was proved in [L98] while the analog of [L96b] is =-=[LP00]-=-. See also [LSW02b].THE SCALING LIMIT OF LOOP-ERASED RANDOM WALK IN THREE DIMENSIONS 27 6. Both [L96a] and [L96b] used the estimate (26) to prove the existence of many cut times or cut points for ran... |

7 | Exact fractal dimension of the loop-erased self-avoiding walk in two dimensions, Phys. Rev - Majumdar |

6 | Scaling limit of loop-erased random walk - a naive approach
- Kozma
(Show Context)
Citation Context ...rs [D92, M92]. Rigorously, 3 different approaches proved fruitful: the connection to random domino tilings [K00a, K00b], the connection to SLE [LSW04a], and the approach we will pursue in this paper, =-=[K]-=-. In fact, SLE was discovered [S00] in the context of LERW. Attempts to understand LERW in dimension 3 focused mainly on the number of steps it takes to reach the distance r. Physicists conjecture tha... |

6 |
G.F.: Loop-erased self-avoiding random walk and the Laplacian random walk
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(Show Context)
Citation Context ...obability that LE(R[0, T1]) will, if conditioned to start from γ, will have as its next vertex x. Indeed, this is the well known “Laplacian random walk” representation of loop-erased random walk, see =-=[L87]-=-. □ Lemma 1.3. Lemma 1.2 holds also when the random walk is conditioned not to hit a given B ⊂ G, w ̸∈ B. In a formula, LE(R[0, T1]) | R[1, T1] ∩ B = ∅ ∼ LE(R[0, Tn]) | R[1, Tn] ∩ B = ∅ ∀n = 2, 3, . .... |

5 | Branched polymers and dimensional reduction
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Citation Context ...e field and the Richardson model was shown to have a limit shape from subadditivity arguments in any dimension [R73], and I cannot resist citing the beautiful work on branched polymers in dimension 3 =-=[BI03]-=-. But these examples are the exception, not the rule. 1.1. Sketch of the proof. The core of the argument is very similar to that of [K], so let us recall the argumentation there. Let R be a random wal... |

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Citation Context ...(1) . (25) For example, notice that, if τR is the stopping time when W exits ∂B(0, r), then the probability that either τR > Cr 2 log r or τR < cr 2 / log r are negligible, which explains (25). 3. In =-=[L89]-=- it was shown that ξd(2, 1) = 4 − d. Very roughly, the proof uses the fact that two random walks starting from the same point can be thought of as one bi-directional walk, which allows to “reduce one ... |

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3 |
Scaling limits of random walks
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(Show Context)
Citation Context ...erent approaches proved fruitful: the connection to random domino tilings [K00a, K00b], the connection to SLE [LSW04a], and the approach we will pursue in this paper, [K]. In fact, SLE was discovered =-=[S00]-=- in the context of LERW. Attempts to understand LERW in dimension 3 focused mainly on the number of steps it takes to reach the distance r. Physicists conjecture that it is ≈ rξ and did numerical expe... |