# 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…

## 17 Citations

### High-Dimensional Incipient Infinite Clusters Revisited

- Computer Science
- 2011

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

- Mathematics
- 2013

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

- MathematicsBrazilian 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

- MathematicsCommunications on Pure and Applied Mathematics
- 2019

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

- Mathematics
- 2015

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

- MathematicsJournal 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

- Mathematics
- 2017

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

- MathematicsCommunications on Pure and Applied Mathematics
- 2020

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

- MathematicsAnnales 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

- MathematicsCommunications on Pure and Applied Mathematics
- 2020

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…

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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.

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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

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- 2013

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…

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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…

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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…