# Random walks on discrete cylinders and random interlacements

@article{Sznitman2008RandomWO, title={Random walks on discrete cylinders and random interlacements}, author={A. S. Sznitman}, journal={Probability Theory and Related Fields}, year={2008}, volume={145}, pages={143-174} }

AbstractWe explore some of the connections between the local picture left by the trace of simple random walk on a cylinder $${(\mathbb {Z} / N\mathbb {Z})^d \times \mathbb {Z}}$$ , d ≥ 2, running for times of order N2d and the model of random interlacements recently introduced in Sznitman (
http://www.math.ethz.ch/u/sznitman/preprints). In particular, we show that for large N in the neighborhood of a point of the cylinder with vertical component of order Nd the complement of the set of points… Expand

#### 38 Citations

Random walks on discrete cylinders with large bases and random interlacements

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We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus $({\mathbb Z} / N{\mathbb Z})^d$, $d \geq 3$, until $uN^d$ time steps, $u > 0$, and… Expand

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We consider simple random walk on a discrete cylinder with base a large $d$-dimensional torus of side-length $N$, when $d$ is two or more. We develop a stochastic domination control on the local… Expand

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We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter… Expand

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We study the asymptotic behavior for large N of the disconnection time T N of a simple random walk on the discrete cylinder (ℤ/Nℤ) d x Z, when d > 2. We explore its connection with the model of… Expand

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We consider the interlacement Poisson point process on the space of doubly-infinite Z d -valued trajectories modulo time shift, tending to infinity at pos- itive and negative infinite times. The set… Expand

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We define two families of Poissonian soups of bidirectional trajectories on $${\mathbb {Z}}^2$$Z2, which can be seen to adequately describe the local picture of the trace left by a random walk on the… Expand

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#### References

SHOWING 1-10 OF 17 REFERENCES

Random walk on a discrete torus and random interlacements

- Mathematics
- 2008

We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus $({\mathbb Z} / N{\mathbb Z})^d$, $d \geq 3$, until $uN^d$ time steps, $u > 0$, and… Expand

Vacant Set of Random Interlacements and Percolation

- Mathematics, Physics
- 2007

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter… Expand

Upper bound on the disconnection time of discrete cylinders and random interlacements

- Mathematics
- 2009

We study the asymptotic behavior for large N of the disconnection time T N of a simple random walk on the discrete cylinder (ℤ/Nℤ) d x Z, when d > 2. We explore its connection with the model of… Expand

Percolation for the Vacant Set of Random Interlacements

- Mathematics, Physics
- 2008

We investigate random interlacements on Z^d, d bigger or equal to 3. This model recently introduced in arXiv:0704.2560 corresponds to a Poisson cloud on the space of doubly infinite trajectories… Expand

How universal are asymptotics of disconnection times in discrete cylinders

- Mathematics
- 2008

We investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base. In a recent article of A. Dembo and the author it was found that for large N… Expand

On the disconnection of a discrete cylinder by a random walk

- Mathematics
- 2006

We investigate the large N behavior of the time the simple random walk on the discrete cylinder needs to disconnect the discrete cylinder. We show that when d≥2, this time is roughly of order N2d and… Expand

On the uniqueness of the infinite cluster of the vacant set of random interlacements

- Mathematics
- 2009

We consider the model of random interlacements on $\mathbb{Z}^d$ introduced in Sznitman [Vacant set of random interlacements and percolation (2007) preprint]. For this model, we prove the uniqueness… Expand

Random Walks on Infinite Graphs and Groups

- Mathematics
- 2000

Part I. The Type Problem: 1. Basic facts 2. Recurrence and transience of infinite networks 3. Applications to random walks 4. Isoperimetric inequalities 5. Transient subtrees, and the classification… Expand

On Positive Solutions of the Equation $\mathfrak{A}U + Vu = 0$

- Mathematics
- 1959

Let $X_t $ be a path of the continuous Markov process in the domain D with boundary $\Gamma $ in a metric space, $\tau $ is the moment of reaching $\Gamma $; $\mathfrak{A}$ is the extended… Expand

Strong invariance for local times

- Mathematics
- 1983

SummaryLet Y1, Y2, ... be a sequence of i.i.d. random variables with distribution P(Y1 = k) = pk (k = ±1, ±2,...), E(Y1) = 0, E(Y12) = σ2<∞. Put Tn = Y1+...+Yn and N(x,n) = # {k:0<k≦n, Tk = x}.… Expand