Random Walk on the High-Dimensional IIC

@article{Heydenreich2012RandomWO,
  title={Random Walk on the High-Dimensional IIC},
  author={Markus Heydenreich and Remco van der Hofstad and Tim Hulshof},
  journal={Communications in Mathematical Physics},
  year={2012},
  volume={329},
  pages={57-115}
}
We study the asymptotic behavior of the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by Kumagai and Misumi (J Theor Probab 21:910–935, 2008). We do this by getting bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the… 

High-Dimensional Incipient Infinite Clusters Revisited

TLDR
This work constructs the incipient infinite cluster measure for high-dimensional percolation models in three different ways, extending previous work by the second-named author and Járai, and shows that each construction yields the same measure, indicating that the IIC is a robust object.

BACKBONE SCALING LIMIT OF THE HIGH-DIMENSIONAL IIC

We identify the scaling limit of the backbone of the high-dimensional incipient infinite cluster (IIC), both in the long- as well as in the finite-range setting. In the finite-range setting, this

Unions of random walk and percolation on infinite graphs

  • K. Okamura
  • Mathematics
    Brazilian Journal of Probability and Statistics
  • 2019
We consider a random object that is associated with both random walks and random media, specifically, the superposition of a configuration of subcritical Bernoulli percolation on an infinite

Scaling Limit for the Ant in High‐Dimensional Labyrinths

We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e., the behavior of the “ant in the labyrinth.” It is natural to conjecture that the scaling

On the chemical distance in critical percolation

We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the

Some Results for Range of Random Walk on Graph with Spectral Dimension Two

  • K. Okamura
  • Mathematics
    Journal of Theoretical Probability
  • 2020
We consider the range of the simple random walk on graphs with spectral dimension two. We give a form of strong law of large numbers under a certain uniform condition, which is satisfied by not only

Progress in High-Dimensional Percolation and Random Graphs

Preface -- 1. Introduction and motivation -- 2. Fixing ideas: Percolation on a tree and branching random walk -- 3. Uniqueness of the phase transition -- 4. Critical exponents and the triangle

Strict Inequality for the Chemical Distance Exponent in Two‐Dimensional Critical Percolation

We provide the first nontrivial upper bound for the chemical distance exponent in two‐dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal

Scaling limits of stochastic processes associated with resistance forms

  • D. Croydon
  • Mathematics
    Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
  • 2018
We establish that if a sequence of spaces equipped with resistance metrics and measures converge with respect to the Gromov-Hausdorff-vague topology, and a certain non-explosion condition is

Restricted Percolation Critical Exponents in High Dimensions

Despite great progress in the study of critical percolation on ℤd for d large, properties of critical clusters in high‐dimensional fractional spaces and boxes remain poorly understood, unlike the

References

SHOWING 1-10 OF 44 REFERENCES

High-Dimensional Incipient Infinite Clusters Revisited

TLDR
This work constructs the incipient infinite cluster measure for high-dimensional percolation models in three different ways, extending previous work by the second-named author and Járai, and shows that each construction yields the same measure, indicating that the IIC is a robust object.

Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤd in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the

BACKBONE SCALING LIMIT OF THE HIGH-DIMENSIONAL IIC

We identify the scaling limit of the backbone of the high-dimensional incipient infinite cluster (IIC), both in the long- as well as in the finite-range setting. In the finite-range setting, this

Incipient infinite percolation clusters in 2D

We study several kinds of large critical percolation clusters in two dimensions. We show that from the microscopic (lattice scale) perspective these clusters can be described by Kesten's incipient

Random walks on supercritical percolation clusters

We obtain Gaussian upper and lower bounds on the transition density qt(x;y) of the continuous time simple random walk on a supercritical percolation cluster C1 in the Euclidean lattice. The bounds,

The Alexander-Orbach conjecture holds in high dimensions

AbstractWe examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and

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 spread-out model of oriented percolation on $${\mathbb{Z}}^{d} \times {\mathbb{Z}}_+$$. In dimensions d > 6, we obtain bounds

The Incipient Infinite Cluster for High-Dimensional Unoriented Percolation

AbstractWe consider bond percolation on $$\mathbb{Z}^d$$ at the critical occupation density pc for d>6 in two different models. The first is the nearest-neighbor model in dimension d≫6. The second

Cycle structure of percolation on high-dimensional tori

Abstract In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The

Random graph asymptotics on high-dimensional tori II: volume, diameter and mixing time

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich