## Random walk on the incipient infinite cluster on trees

Venue: | Illinois J. Math |

Citations: | 26 - 7 self |

### BibTeX

@ARTICLE{Barlow_randomwalk,

author = {Martin T. Barlow and Takashi Kumagai},

title = {Random walk on the incipient infinite cluster on trees},

journal = {Illinois J. Math},

year = {},

pages = {2247823}

}

### OpenURL

### Abstract

Abstract. Let G be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree n0 + 1. We obtain estimates for the transition density of the the continuous time simple random walk Y on G; the process satisfies anomalous diffusion and has spectral dimension 4

### Citations

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Citation Context ...ff(A, B) −1 = inf{E(f, f) : f ∈ H 2 , f|A = 1, f|B = 0}. (2.12) Let Reff(x, y) = Reff({x}, {y}), and Reff(x, x) = 0. For general facts on effective resistance and its connection with random walks see =-=[2, 15, 33]-=-. We recall some basic properties of Reff(·, ·). Lemma 2.2. Let Γ = (G, E) be an infinite connected graph. (a) Reff is a metric on G. (b) If A ′ ⊂ A, B ′ ⊂ B then Reff(A ′ , B ′ ) ≥ Reff(A, B). (c) Re... |

369 |
The Theory of Branching Processes
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Citation Context ...e well known, but as we did not find them anywhere in exactly the form we needed, we give the proofs (which are quite short) here. 4Let f be the generator of the offspring distribution, so that From =-=[Har]-=- p. 21 we have Let Yn = n∑ k=0 f(s) = E(s X1 ) = n −n0 0 (s + n0 − 1) n0 . (2.1) P(Xn > 0) ∼ 2 nf ′′ (1) = 2n0 . (2.2) (n0 − 1)n Xk, gn(s) = E(s Yn ), fn(s) = Es Xn . Then conditioning on X1 we obtain... |

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Citation Context ...me n, and let Sn = µ(Wn) = ∑ x∈Wn µx. (1.7) We write Reff(0, B(R) c ) for the effective resistance between 0 and B(R) c in the electric network obtained by making each edge of Γ a unit resistor – see =-=[15]-=-. A precise mathematical definition of Reff(·, ·) will be given in Section 2. We now consider a probability space (Ω, F, P) carrying a family of random graphs Γ(ω) = (G(ω), E(ω), ω ∈ Ω). We assume tha... |

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Citation Context ...y1∈Zd · · · ∑ xl,yl∈Z d σ (l+1) n1,...,nl (x1, . . .,xl, y1, . . .,yl), (4.31) with a corresponding upper bound if the summations over the ni’s extend to R − 1. Lower bound. The Harris–FKG inequality =-=[16, 18]-=- implies that for increasing events A and B we have Qn(A ∩ B) ≥ Qn(A)P(B). If A and B are cylinder events, then by passing to the limit, we have Q∞(A ∩ B) ≥ Q∞(A)P(B). Hence With (4.27), this gives EZ... |

84 | Probability on Trees and Networks
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Citation Context ...ff(A, B) −1 = inf{E(f, f) : f ∈ H 2 , f|A = 1, f|B = 0}. (2.12) Let Reff(x, y) = Reff({x}, {y}), and Reff(x, x) = 0. For general facts on effective resistance and its connection with random walks see =-=[2, 15, 33]-=-. We recall some basic properties of Reff(·, ·). Lemma 2.2. Let Γ = (G, E) be an infinite connected graph. (a) Reff is a metric on G. (b) If A ′ ⊂ A, B ′ ⊂ B then Reff(A ′ , B ′ ) ≥ Reff(A, B). (c) Re... |

83 | An invariance principle for reversible Markov processes. Applications to random motions in random environments - Masi, Ferrari, et al. - 1989 |

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Citation Context ...i + 1)} \ F⃗m(⃗x,⃗y)). (4.34) The BK inequality implies that for increasing events A and B that depend on only finitely many edges we have P(A ◦ B) ≤ P(A)P(B), where A ◦ B denotes disjoint occurrence =-=[8, 18]-=-. We will bound the first term by passing to the limit in the BK inequality. Let A⃗m,n(⃗x) = {(0, 0) −→ n, (0, 0) −→ (xi, mi), i = 1, . . .,l}, 29and define F⃗m,n(⃗x,⃗y) analogously, by replacing A⃗m... |

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Citation Context ...es Z d ×Z+ and directed bonds ((x, n), (y, n+1)), for n ≥ 0 and x, y ∈ Z d with 0 ≤ ‖x−y‖∞ ≤ L. Here L is a fixed positive integer and ‖x‖∞ = maxi=1,...,d |xi| for x = (x1, . . ., xd) ∈ Z d . Let p ∈ =-=[0, 1]-=-. We associate to each directed bond ((x, n), (y, n + 1)) an independent random variable taking the value 1 with probability p and 0 with probability 1 − p. We say a bond is occupied when the correspo... |

57 |
On tail probabilities for martingales
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Citation Context ...(2.19) we have Px,b(WNi + Nip ≥ m, Nip ≤ m/2) ≤ Px,b(WNi ≥ m/2, 〈W 〉Ni ≤ m(1 − p)/2) (m/2) ≤ exp(− 2 2((m/2) + m(1 − p)/2) ) ≤ e−cm , where we used an exponential martingale inequality – see (1.6) in =-=[F]-=-. For the second term, note that Ni � (X r/4[r/4])[2] and so using Lemma 2.2 we deduce that Px,b(Nip > m/2) ≤ ce −c3m . Combining these bounds completes the proof. Definition 2.11. Let x ∈ B, r ≥ 1, λ... |

55 |
Percolation (2nd edition
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Citation Context ...el on the lattice Z d : each bond is open with probability p ∈ (0, 1), independently of all the others. Let C(x) be the open cluster containing x; then if θ(p) = Pp(|C(x)| = +∞) it is well known (see =-=[Gm]-=-) that there exists pc = pc(d) such that θ(p) = 0 if p < pc and θ(p) > 0 if p > pc. If d = 2 or d ≥ 19 (or d > 6 for ‘spread out’ models) it is known (see [Gm], [HS]) that θ(pc) = 0, and it is conject... |

53 |
Mean-field critical behaviour for percolation in high dimensions
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Citation Context ...|C(x)| = +∞) it is well known (see [Gm]) that there exists pc = pc(d) such that θ(p) = 0 if p < pc and θ(p) > 0 if p > pc. If d = 2 or d ≥ 19 (or d > 6 for ‘spread out’ models) it is known (see [Gm], =-=[HS]-=-) that θ(pc) = 0, and it is conjectured that this holds for all d ≥ 2. At the critical probability p = pc it is believed that in any box of side n there exist with high probability open clusters of di... |

53 | Analysis on fractals - Kigami - 2001 |

51 |
Subdiffusive behavior of random walk on a random cluster
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Citation Context ...e construction of the IIC is much easier than for lattices, and there is a close connection between the IIC and a critical Bienaymé-Galton-Watson branching processes conditioned on non-extinction. In =-=[Ke2]-=- Kesten gave the construction of the IIC G for critical branching processes. This is an infinite subtree, which contains only one path from the root to infinity. This tree is quite sparse, and has pol... |

46 |
Diffusions on fractals. In Lectures on probability theory and statistics (Saint-Flour
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Citation Context ...i = Si+1 − Ti, i ≥ 1. Then by Proposition 4.6 there exists p = p(λ1) < 1 and c3 = c3(λ) > 0 such that P x( ξi ≤ s|σ(Yu, 0 ≤ u ≤ Ti) ) ≤ p + c3R −3 s. (4.14) Lemma 1.1 of [BB1] (see also Lemma 3.14 of =-=[B1]-=-) gives that, writing a = c3/R3 , (4.14) implies that log P x N/8 ∑ ( ξi ≤ t) ≤ −1 ( aNt ) 1/2 N log(1/p) + 2 . 8 8p i=1 Substituting for a we deduce that log P x ( (τB(x,NR) ≤ t) ≤ −N 2c4 − c5(t/(R 3... |

43 |
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Citation Context ...nt in the labyrinth’ [17] — has received much attention both in the physics and the mathematics literature. Recently, several papers have considered random walk on a supercritical percolation cluster =-=[5, 9, 34, 35]-=-. Roughly speaking, supercritical percolation clusters on Z d are d-dimensional, and these papers prove, in various ways, that a random walk on a supercritical percolation cluster behaves in a diffusi... |

39 | Random walks on supercritical percolation clusters
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Citation Context ... of ˜ Cd in [HHS] and [HJ] are consistent with this holding for large d. (Or for any d above the critical dimension for spread-out models). Random walks on supercritical clusters in Zd are studied in =-=[B2]-=- (transition density estimates) and [SS] (invariance principle for d ≥ 4). In these cases the large scale behaviour of the random walk approximates that of the random walk on Zd , and the unique infin... |

38 |
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Citation Context ...t with this holding for large d. (Or for any d above the critical dimension for spread-out models). Random walks on supercritical clusters in Zd are studied in [B2] (transition density estimates) and =-=[SS]-=- (invariance principle for d ≥ 4). In these cases the large scale behaviour of the random walk approximates that of the random walk on Zd , and the unique infinite cluster has spectral dimension d. In... |

38 |
The critical contact process dies out
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Citation Context ... and that random walk on a large critical cluster behaves subdiffusively. Critical percolation clusters are believed to be finite in all dimensions, and are known to be finite in the oriented setting =-=[11]-=-. To avoid finite-size issues associated with random walk on a finite cluster, it is convenient to consider random walk on the incipient infinite cluster (IIC), which can be understood as a critical p... |

34 |
The construction of Brownian motion on the Sierpiński carpet
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Citation Context ... −5 rP x (Y2t ̸∈ B ′ ) = λ −5 r(1 − P x (Y2t ∈ B ′ )) ≥ 1 2 rλ−5 . 23 □Lemma 4.9. Suppose x satisfies G2(N, R). Then P x (τ B(x,NR) ≤ t) ≤ e −c1N provided N ≥ c2t/R 3 . Proof. We use the argument of =-=[BB1]-=-. Let A = {y ∈ G : B(y, R/2) is λ1–good}. Define stopping times (Ti), (Si) by taking T0 = min{t : Yt ∈ A}, and Sn = min{t ≥ Tn−1 : Yt ̸∈ B(YTn−1 , R/2)}, Tn = min{t ≥ Sn : Yt ∈ A}. Since x satisfies G... |

30 | Telcs A., Sub-Gaussian estimates of heat kernels on infinite graphs
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Citation Context ... why the n −1/3 scaling arises in (b) it is helpful to consider the behaviour of random walks on regular deterministic graphs with a large scale fractal structure – see for example [Jo], [BB2], [HK], =-=[GT1]-=-, [GT2] and [BCK]. Let df ≥ 1 give the volume growth, so that |B(x, r)| ∼ r df , and suppose that the effective electrical resistance R(x, B(x, r) c ) between x and the exterior of B(x, r) satisfies R... |

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Citation Context ...e n −1/3 scaling arises in (b) it is helpful to consider the behaviour of random walks on regular deterministic graphs with a large scale fractal structure – see for example [Jo], [BB2], [HK], [GT1], =-=[GT2]-=- and [BCK]. Let df ≥ 1 give the volume growth, so that |B(x, r)| ∼ r df , and suppose that the effective electrical resistance R(x, B(x, r) c ) between x and the exterior of B(x, r) satisfies R(x, B(x... |

29 | A generalised inductive approach to the lace expansion, Probab. Theory Related Fields 122
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Citation Context ...enotes an absolute constant. The values of the constants K and ¯ K may change from one occurrence to the next. 244.1 Preliminaries In this section, we recall and slightly extend various results from =-=[20, 21, 25, 26]-=-. These results isolate the necessary ingredients from other papers that will be used in the proof of Propositions 3.1–3.2. 4.1.1 Critical oriented percolation r-point functions The critical oriented ... |

26 | The birth of the infinite cluster: finite–size scaling in percolation
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Citation Context ... is conjectured that this holds for all d ≥ 2. At the critical probability p = pc it is believed that in any box of side n there exist with high probability open clusters of diameter of order n – see =-=[BCKS]-=-. For large n the local properties of these large finite clusters can, in certain circumstances, be captured by regarding them as subsets of an infinite cluster ˜ C, called the ‘incipient infinite clu... |

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Citation Context ...e finite clusters can, in certain circumstances, be captured by regarding them as subsets of an infinite cluster ˜ C, called the ‘incipient infinite cluster’ (IIC). This was constructed when d = 2 in =-=[Ke1]-=-, by taking the limit as N → ∞ of the cluster C(0) conditioned to intersect the boundary of a box of side N with center at the origin. See [Ja1], [Ja2] for other constructions of the IIC in two dimens... |

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Citation Context ...|C(x)| = +∞) it is well known (see [Gm]) that there exists pc = pc(d) such that θ(p) = 0 if p < pc and θ(p) > 0 if p > pc. If d = 2 or d ≥ 19 (or d > 6 for ‘spread out’ models) it is known (see [Gm], =-=[HS]-=-) that θ(pc) = 0, and it is conjectured that this holds for all d ≥ 2. At the critical probability p = pc it is believed that in any box of side n there exist with high probability open clusters of di... |

25 |
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Citation Context ...cess. To understand why the n −1/3 scaling arises in (b) it is helpful to consider the behaviour of random walks on regular deterministic graphs with a large scale fractal structure – see for example =-=[Jo]-=-, [BB2], [HK], [GT1], [GT2] and [BCK]. Let df ≥ 1 give the volume growth, so that |B(x, r)| ∼ r df , and suppose that the effective electrical resistance R(x, B(x, r) c ) between x and the exterior of... |

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Citation Context ...To understand why the n −1/3 scaling arises in (b) it is helpful to consider the behaviour of random walks on regular deterministic graphs with a large scale fractal structure – see for example [Jo], =-=[BB2]-=-, [HK], [GT1], [GT2] and [BCK]. Let df ≥ 1 give the volume growth, so that |B(x, r)| ∼ r df , and suppose that the effective electrical resistance R(x, B(x, r) c ) between x and the exterior of B(x, r... |

19 | Directed percolation and random walk
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Citation Context ...all dimensions d ≥ 1 and for all L ≥ 1, there is a critical value pc = pc(d, L) ∈ (0, 1) such that θ(p) = 0 for p ≤ pc and θ(p) > 0 for p > pc. In particular, there is no infinite cluster when p = pc =-=[11, 19]-=-. For the remainder of this paper, we fix p = pc, so that P = Ppc. To define the IIC, some terminology is required. A cylinder event is an event that is determined by the occupation status of a finite... |

17 |
Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries
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Citation Context ...rstand why the n −1/3 scaling arises in (b) it is helpful to consider the behaviour of random walks on regular deterministic graphs with a large scale fractal structure – see for example [Jo], [BB2], =-=[HK]-=-, [GT1], [GT2] and [BCK]. Let df ≥ 1 give the volume growth, so that |B(x, r)| ∼ r df , and suppose that the effective electrical resistance R(x, B(x, r) c ) between x and the exterior of B(x, r) sati... |

17 | The speed of biased random walk on percolation clusters. Probability Theory and Related - Berger, Gantert, et al. - 2003 |

17 | Random Walks and Random Environments, volume 2: Random Environments - Hughes - 1996 |

15 |
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Citation Context ...ts transition density (see Section 3 for a precise definition). Define the spectral dimension of ˜ Cd by ds( ˜ Cd) = −2 lim t→∞ log qt(x, x) , (1.1) log t (if this limit exists). Alexander and Orbach =-=[AO]-=- conjectured that, for any d ≥ 2, ds( ˜ Cd) = 4/3. While it is now thought that this is unlikely to be true for small d, the results on the geometry of ˜ Cd in [HHS] and [HJ] are consistent with this ... |

14 |
Characterization of subGaussian heat kernel estimates on strongly recurrent graphs
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Citation Context ...caling arises in (b) it is helpful to consider the behaviour of random walks on regular deterministic graphs with a large scale fractal structure – see for example [Jo], [BB2], [HK], [GT1], [GT2] and =-=[BCK]-=-. Let df ≥ 1 give the volume growth, so that |B(x, r)| ∼ r df , and suppose that the effective electrical resistance R(x, B(x, r) c ) between x and the exterior of B(x, r) satisfies R(x, B(x, r) c ) ∼... |

14 | Construction of the incipient infinite cluster for spread-out oriented percolation above 4
- Hofstad, Hollander, et al.
(Show Context)
Citation Context ... between x and y with probability pL −d whenever y is in a cube side L with center x, and the parameter L is large enough. Rather more is known about the IIC for oriented percolation on Z+ × Z d (see =-=[HHS]-=-, [HS]), but in this discussion, which mainly concerns what is conjectured rather than what is known, we specialize to the case of Z d . We write ˜ Cd for the IIC in Z d . It is believed that the glob... |

14 |
The lace expansion and its applications
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- 2006
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Citation Context ...nfinite. The IIC has been constructed so far only when d = 2 [29], when d > 6 (in the spread-out case) [24], and when d > 4 for oriented percolation on Zd ×Z+ (again in the spread-out case) [21]. See =-=[36]-=- for a summary of the high-dimensional results. Also, it is not difficult to construct the IIC on a tree [7, 30]. Random walk on the IIC has been proved to be subdiffusive on Z2 [30] and on a tree [7,... |

11 | Volume and time doubling of graphs and random walk, the strongly recurrent case
- Telcs
(Show Context)
Citation Context ...to give much control of the random walk. However, the graphs considered here have strong recurrence properties, and are therefore simpler to handle than general graphs. We use techniques developed in =-=[6, 7, 37, 38, 39]-=-. We will prove in Theorem 1.7 that Assumption 1.2 holds for the IIC for sufficiently spread-out oriented percolation on Z d × Z+ when d > 6. As the reader of Sections 4–5 will see, obtaining volume a... |

10 | Heat kernel estimates for strongly recurrent random walk on random
- Kumagai, Misumi
(Show Context)
Citation Context ...ve degree less than c0. Then log |Wn| 2 lim = n→∞ log n 3 , P x ω-a.s. (1.24) See Example 1.8 for a graph with unbounded degree which satisfies Assumption 1.2, but for which (1.24) fails. Remark. See =-=[32]-=- for results which generalise the above theorems to the situation where there exist indices α < β such that V (R) is comparable to R α and Reff(0, B(R) c ) is comparable to R β−α . Our case is α = 2, ... |

9 |
Analysis on fractals, Cambridge univ
- Kigami
- 2001
(Show Context)
Citation Context ...of Γ. The effective resistance between A and B is defined R(A, B) −1 = inf{E(f, f) : f ∈ H 2 , f|A = 1, f|B = 0}. (3.3) Let R(x, y) = R({x}, {y}), and R(x, x) = 0. In general R is a metric on Γ – see =-=[Kig]-=- Section 2.3. If (Γ, µ) has natural weights then R(x, y) ≤ d(x, y), and if in addition Γ is a tree then R(x, y) = d(x, y). The following is an easy consequence of (3.3). Lemma 3.1. For all f ∈ R Γ and... |

8 |
Local Sub-Gaussian Estimates on Graphs: The Strongly Recurrent Case, Electronic
- Telcs
- 2001
(Show Context)
Citation Context ...to give much control of the random walk. However, the graphs considered here have strong recurrence properties, and are therefore simpler to handle than general graphs. We use techniques developed in =-=[6, 7, 37, 38, 39]-=-. We will prove in Theorem 1.7 that Assumption 1.2 holds for the IIC for sufficiently spread-out oriented percolation on Z d × Z+ when d > 6. As the reader of Sections 4–5 will see, obtaining volume a... |

5 |
Gennes, “La percolation : un concept unificateur
- de
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(Show Context)
Citation Context ...ical oriented percolation obtained previously via the lace expansion. 1 Introduction and main results 1.1 Introduction The problem of random walk on a percolation cluster — the ‘ant in the labyrinth’ =-=[17]-=- — has received much attention both in the physics and the mathematics literature. Recently, several papers have considered random walk on a supercritical percolation cluster [5, 9, 34, 35]. Roughly s... |

5 |
The survival probability for critical spread-out oriented percolation above 4
- Hofstad, Hollander, et al.
(Show Context)
Citation Context ... 5.2 and the BK inequality [8] to bound Ppc[AJ(n, w, x, y)]. For this, we need the following preliminaries. The critical survival probability is defined by i=0 θN = Ppc(0 −→ N). (5.19) The two papers =-=[22, 23]-=- show that for d > 4 and L ≥ L0(d), we have θN ∼ cN −1 as N → ∞, for some c = c(d, L) = 2 + O(L −d ). Moreover, θN ≤ K′ N , N ≥ 0, L ≥ L0, (5.20) with the constant K ′ = 5 which is of course independe... |

5 |
The incipient infinite cluster for high-dimensional unoriented percolation
- Hofstad, Járai
- 2004
(Show Context)
Citation Context ...nite cluster (IIC), which can be understood as a critical percolation cluster conditioned to be infinite. The IIC has been constructed so far only when d = 2 [29], when d > 6 (in the spread-out case) =-=[24]-=-, and when d > 4 for oriented percolation on Zd ×Z+ (again in the spread-out case) [21]. See [36] for a summary of the high-dimensional results. Also, it is not difficult to construct the IIC on a tre... |

5 | Invasion percolation on regular trees
- Angel, Goodman, et al.
(Show Context)
Citation Context ....5–1.6 and Theorems 1.7–1.8 hold for random walk on this IIC. The results of [7] go beyond Theorem 1.7(a) and (b) in this context, but Theorem 1.7(c) and Theorem 1.8 here are new. (ii) It is shown in =-=[4]-=- that the invasion percolation cluster on a regular tree is stochastically dominated by the IIC for the binomial tree. Consequently, upper bounds on the volume and lower bounds on the effective resist... |

4 |
Incipient infinite percolation clusters in 2D
- Járai
- 2003
(Show Context)
Citation Context ...nite cluster’ (IIC). This was constructed when d = 2 in [Ke1], by taking the limit as N → ∞ of the cluster C(0) conditioned to intersect the boundary of a box of side N with center at the origin. See =-=[Ja1]-=-, [Ja2] for other constructions of the IIC in two dimensions. For large d a construction of the IIC in Z d is given in [HJ], using the lace expansion. It is believed that the results there will hold f... |

4 |
Invasion percolation and the incipient infinite cluster in 2D
- Járai
- 2003
(Show Context)
Citation Context ...uster’ (IIC). This was constructed when d = 2 in [Ke1], by taking the limit as N → ∞ of the cluster C(0) conditioned to intersect the boundary of a box of side N with center at the origin. See [Ja1], =-=[Ja2]-=- for other constructions of the IIC in two dimensions. For large d a construction of the IIC in Z d is given in [HJ], using the lace expansion. It is believed that the results there will hold for any ... |

4 |
Volume growth and heat kernel estimates for the continuum random tree, Probab. Theory Relat
- Croydon
(Show Context)
Citation Context ...of the high-dimensional results. Also, it is not difficult to construct the IIC on a tree [7, 30]. Random walk on the IIC has been proved to be subdiffusive on Z2 [30] and on a tree [7, 30]. See also =-=[13, 14]-=- for related results in the continuum limit. In this paper, we prove several estimates for random walk on the IIC for spread-out oriented percolation on Zd × Z+ in dimensions d > 6. These estimates, w... |

3 | Infinite canonical super-Brownian motion and scaling limits
- Hofstad
(Show Context)
Citation Context ...connected subsets of B containing 0. Let G ′ be a rooted labeled tree chosen with the distribution P: we call this the incipient infinite cluster (IIC) on B. For more information on G ′ see [Ke2] and =-=[vH]-=- but we remark that P–a.s. G ′ has exactly one infinite descending path from 0, which we call the backbone, and denote H. 9 □It will be useful to give another construction of the IIC, obtained by mod... |

3 | Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree
- Croydon
- 2008
(Show Context)
Citation Context ...of the high-dimensional results. Also, it is not difficult to construct the IIC on a tree [7, 30]. Random walk on the IIC has been proved to be subdiffusive on Z2 [30] and on a tree [7, 30]. See also =-=[13, 14]-=- for related results in the continuum limit. In this paper, we prove several estimates for random walk on the IIC for spread-out oriented percolation on Zd × Z+ in dimensions d > 6. These estimates, w... |

3 | The field theory approach to percolation processes
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- 2005
(Show Context)
Citation Context ...dimensions d > 4 [21]. For simplicity, we will consider only the most basic example of a spread-out model. (In the physics literature, oriented percolation is usually called directed percolation; see =-=[28]-=-.) The spread-out oriented percolation model is defined as follows. Consider the graph with vertices Z d ×Z+ and directed bonds ((x, n), (y, n+1)), for n ≥ 0 and x, y ∈ Z d with 0 ≤ ‖x−y‖∞ ≤ L. Here L... |