Surface order large deviations for high-density percolation

Deuschel, Jean–Dominique; Pisztora, Ágoston

doi:10.1007/s004400050031

Cited by 51 publications

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“…In Sect. 4 we prove (1.2) using these results. Consider the following sequence of scales L κ = L 0 (80L) κ , for κ ≥ 0, (1.…”

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confidence: 83%

On the size of a finite vacant cluster of random interlacements with small intensity

Teixeira

2010

Probab. Theory Relat. Fields

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In this paper, we establish some properties of percolation for the vacant set of random interlacements, for d ≥ 5 and small intensity u. The model of random interlacements was first introduced by Sznitman in (Ann Math, arXiv:0704.2560, 2010. It is known that, for small u, almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u, see Sidoravicius and Sznitman (Commun Pure Appl Math 62 (6): 2009) and Teixeira (Ann Appl Probab 19(1):454-466, 2009). We estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin, given that it is finite. This comes as a by-product of our main theorem, which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube. As another application, we show that with high probability, the unique infinite connected component of the vacant set is "ubiquitous" in large neighborhoods of the origin.

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“…In Sect. 4 we prove (1.2) using these results. Consider the following sequence of scales L κ = L 0 (80L) κ , for κ ≥ 0, (1.…”

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confidence: 83%

On the size of a finite vacant cluster of random interlacements with small intensity

Teixeira

2010

Probab. Theory Relat. Fields

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“…Since u(y ) ≥ Then with a similar argument as in (2.31) and (A.6) of Deuschel-Pisztora [8], we see that for some i(z) ∈ {1, 2, 3},…”

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confidence: 95%

“…In view of (2.38) and the isoperimetric controls in (A.3) of Deuschel-Pisztora [8] (here d = 2), we have that…”

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confidence: 99%

On the disconnection of a discrete cylinder by a random walk

Dembo

Sznitman

2005

Probab. Theory Relat. Fields

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Abstract. We investigate the large N behavior of the time the simple random walk on the discrete cylinder (Z Z/NZ Z) d × Z Z needs to disconnect the discrete cylinder. We show that when d ≥ 2, this time is roughly of order N 2d and comparable to the cover time of the slice (Z Z/NZ Z) d × {0}, but substantially larger than the cover timer of the base by the projection of the walk. IntroductionConsider simple random walk on an infinite discrete cylinder having a base modeled on a d-dimensional discrete torus of side length N . In this note we investigate the following question of H.J. Hilhorst: what is the asymptotic behavior for large N of the time needed by the walk to disconnect the cylinder? When d = 1, it is straightforward to argue that this time is roughly of order N 2 and comparable to the time for the projection of the process to cover the base. We show here that things behave differently when d ≥ 2, and that in a suitable sense a massive clogging occurs inside the cylinder by the time the disconnection happens.Before discussing our results any further, we describe the model more precisely. For integer N ≥ 1, we consider the state spacethat we tacitly endow with its natural graph structure. We say that a finite subsetare contained in two distinct connected components of E\S. We denote with P x , x ∈ E, the canonical law on E l N of the simple random walk on E starting at x, and with (X n ) n≥0 the canonical process. We are principally interested in the disconnection time of E:(0.2)Under P x , x ∈ E, the Markov chain X . is irreducible, recurrent, and it is plain that T N < ∞, P x -a.s., for all x ∈ E . (0.3)A. Dembo:

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“…(16) In other words, the density of the crossing cluster determined by Theorem 2.1 lies between θ f and θ w . Yet in most cases these two quantities coincide thanks to our last result, which generalizes those of Lebowitz [24] and Grimmett [19]: …”

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confidence: 96%

A coarse graining for the Fortuin–Kasteleyn measure in random media

Wouts

2008

Stochastic Processes and their Applications

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By means of a multi-scale analysis we describe the typical geometrical structure of the clusters under the FK measure in random media. Our result holds in any dimension d 2 provided that slab percolation occurs under the averaged measure, which should be the case for the whole supercritical phase. This work extends that of Pisztora [A. Pisztora, Surface order large deviations for Ising, Potts and percolation models, Probab. Theory Related Fields 104 (4) (1996) 427-466] and provides an essential tool for the analysis of the supercritical regime in disordered FK models and in the corresponding disordered Ising and Potts models.

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Surface order large deviations for high-density percolation

Cited by 51 publications

References 0 publications

On the size of a finite vacant cluster of random interlacements with small intensity

On the size of a finite vacant cluster of random interlacements with small intensity

On the disconnection of a discrete cylinder by a random walk

A coarse graining for the Fortuin–Kasteleyn measure in random media

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