Random walk on the incipient infinite cluster for oriented percolation in high dimensions
 44 Pages
 2006
 2.98 MB
 3833 Downloads
 English
Research Institute for Mathematical Sciences , Kyoto, Japan
Statement  by Martin T. Barlow ... [et al.]. 
Series  Sūri Kaiseki Kenkyūjo kōkyūroku  RIMS1563 
Contributions  Barlow, M. T. 
Classifications  

LC Classifications  MLCS 2007/43958 (Q) 
The Physical Object  
Pagination  44 p. : 
ID Numbers  
Open Library  OL16288873M 
LC Control Number  2007533241 



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We consider simple random walk on the incipient inﬁnite cluster for the spreadout model of oriented percolation on Zd × Z+. In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient inﬁnite cluster is 4 3, and thereby prove a version of the Alexander–.
We consider simple random walk on the incipient infinite cluster for the spreadout model of oriented percolation on $${\mathbb{Z}}^{d} \times {\mathbb{Z}}_+$$. In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is $$\frac {4}{3}$$, and thereby prove a Cited by: Random Walk on the Incipient Infinite Cluster for Oriented Percolation in High Dimensions Article (PDF Available) in Communications in Mathematical Physics (2) September with 40 Reads.
Abstract: We consider simple random walk on the incipient infinite cluster for the spreadout model of oriented percolation on $Z^d \times Z_+$. In dimensions $d>6$, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is 4/3, and thereby prove a version of the Alexander Author: Martin T.
Barlow, Antal A. Jarai, Takashi Kumagai, Gordon Slade. We consider simple random walk on the incipient infinite cluster for the spreadout model of oriented percolation on {mathbb{Z}} d × {mathbb{Z}}_+. In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is frac {4}{3}, and thereby prove a version of the Alexander Cited by: R.
van der Hofstad, F. den Hollander, and G. Slade, Construction of the incipient infinite cluster for spreadout oriented percolation above 4+1 dimensions, Commun. lation in high dimensions, the scaling limits of the incipient in nite cluster’s twopoint and threepoint Random walk on the incipient infinite cluster for oriented percolation in high dimensions book are those of integrated superBrownian excursion (ISE).
The proof uses an extension of the lace expansion for percolation. Introduction Percolation has received much attention in mathematics and in physics, as a simple model of a phase transition. Random walk on the incipient infinite cluster for oriented percolation in high dimensions.
Communications in Mathematical Physics– Bollobás, B. and Riordan, O. Percolation.
Download Random walk on the incipient infinite cluster for oriented percolation in high dimensions FB2
Cambridge: Cambridge University Press. Brézis, H. and Lieb, E. A relation between pointwise convergence of functions and convergence of. Van der Hofstad and Járai constructed the incipient inﬁnite cluster in high dimensions, and earlier, van der Hofstad, den Hollander and Slade constructed the IIC for highdimensional oriented percolation.
Both constructions were achieved by making use of the lace expansion. The lace expansion for percolation. TITLE = {Construction of the incipient infinite cluster for spreadout oriented percolation above {$4+1$} dimensions}, JOURNAL = {Comm.
Math. Phys.}, FJOURNAL = {Communications in Mathematical Physics}. We consider simple random walk on the incipient infinite cluster for the spreadout model of oriented percolation on $Z^d \times Z_+$. In dimensions $d>6$, we obtain bounds on exit times.
7) Random walk on the incipient infinite cluster for oriented percolation in high dimensions (AMS Sectional Meeting, Storrs, October ) PDF File (kb) 6) Stochastic Processes on Fractals (Stochastic Analysis and Related Topics, Marburg, July ) PDF File (kb) 5) A trace theorem for Dirichlet forms on fractals.
Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Comm. Math. Phys., (), no 2, PDF File (kb), Post Script File. (kb) 39) I. Fujii and T. Kumagai, Heat kernel estimates on the incipient infinite cluster for critical branching processes.
Proceedings of GermanJapanese symposium in. 'This book, written with great care, is a comprehensive course on random walks on graphs, with a focus on the relation between rough geometric properties of the underlying graph and the asymptotic behavior of the random walk on it.
It is accessible to graduate students but may also serve as a good reference for researchers. random walk on a critical percolation cluster moves significantly slower than diffusive.
Several critical exponents for the walk have been established rigorously in the context of highdimensional oriented and unoriented percolation [BJKS08,KN09]. Recent work has extended this to situations. out oriented percolation incipient inﬁnite cluster above 4 dimensions and of incipient inﬁnite branching random walk in any dimension.
We conjecture that it also arises as the scaling limit in various other models above the uppercritical dimension, such as the incipient inﬁ. Barlow & Kumagai (): random walk on the IIC on a tree (‘d = ∞’) has subGaussian heat kernel estimates.
Croydon (), the scaling limit is Brownian motion on the continuum random tree. Barlow, Jarai, Kumagai and Slade (), random walk on high dimensional spreadout oriented percolation.
Details Random walk on the incipient infinite cluster for oriented percolation in high dimensions PDF
Random walks onpercolation clusters – p. Chapter Random walks on percolation clusters Random walks on the in nite cluster Random walks on nite critical clusters Random walk on the incipient in nite cluster Chapter Related results Superprocess limits of percolation clusters Oriented percolation Scaling limit of.
We study the asymptotic behavior the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by. The incipient infinite cluster, with the random walk thereon, has attracted much attention in a variety of further contexts since Harry's work on Z 2.
CiteSeerX  Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We construct the incipient infinite cluster measure (IIC) for sufficiently spreadout oriented percolation on Zd × Z+, for d + 1> 4 + 1.
We consider two different constructions. For the first construction, we define Pn(E) by taking the probability of the intersection of an event E with the event that the origin is. The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite.
Using the lace expansion, we construct the IIC measure for highdimensional percolation models in three different ways, extending previous work by the secondnamed author and Járai.
We derive quenched subdiffusive lower bounds for the exit time τ (n) from a box of size n for the simple random walk on the planar invasion percolation cluster. The first part of the paper is devoted to proving an almost sure analogue of H.
Kesten’s subdiffusivity theorem for the random walk on the incipient infinite cluster and the invasion percolation cluster using ideas of M. Aizenman, A. Random walk on the incipient inﬁnite cluster for oriented percolation in high dimensions, (). Random walk on the incipient inﬁnite cluster on trees, ().
Critical twopoint functions and the lace expansion for spreadout highdimensional percolation and related models. Ann. Probab. (). PDF file; R. van der Hofstad, F. den Hollander and G.
Slade. Construction of the incipient infinite cluster for spreadout oriented percolation above 4+1 dimensions. Commun. Math. 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.
Construction of the incipient infinite cluster for spreadout Paginanavigatie: Main; Save publication. Save as MODS; Export to Mendeley; Save as EndNote. Progress in HighDimensional Percolation and Random Graphs Markus Heydenreich, Remco van der Hofstad (auth.) This text presents an engaging exposition of the active field of highdimensional percolation that will likely provide an impetus for future work.
We consider random walks on random graphs determined by a some kind of continuum percolation on $\mathbf{R}$. The vertex set of the random graph is given by the Poisson points conditioned that all points of $\mathbf{Z}$ are contained.
The edge set of the random graph is determined by the random radii of the spheres centered at each points. Random Walk on the Incipient Infinite Cluster for Oriented Percolation in High Dimensions.
Martin T. Barlow, Antal Járai, We show that the electrical resistance between the origin and generation n of the incipient infinite oriented branching random walk in dimensions d.
Random walk on the incipient infinite cluster for oriented percolation in high dimensions. Communications in Mathematical Physics, () arXiv:math/v2; A.A.
Járai and F. Redig: Infinite volume limit of the Abelian sandpile model in dimensions d ≥ 3.
Description Random walk on the incipient infinite cluster for oriented percolation in high dimensions FB2
Probability Theory and Related Fields, () Random walk on the incipient infinite cluster for oriented percolation in high dimensions We consider simple random walk on the incipient infinite cluster for the spreadout model of oriented percolation in d+1 dimensions. For d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that.
We consider oriented bond percolation on Z d × N, at the critical occupation density p c, for d>4. The model is a “spreadout” model having long range parameterised by L. We consider configurations in which the cluster of the origin survives to time n, and scale space by n 1/ prove that for L sufficiently large all the moment measures converge, as n→∞, to those of the canonical.


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