Random walk on a discrete torus and random interlacements

By David Windisch. 2008

58 citations recorded by the source corpus.

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 the model of random interlacements recently introduced by Sznitman. In particular, we show that for large $N$, the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time $uN^d$ converges to independent copies of the random interlacement at level $u$.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f3f356f6f5c0e2441eabd5e29a5da8049cd3233a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Interlacement percolation on transient weighted graphs (2009, background): “In a heuristic sense, the random interlacements describe for these time scales the microscopic limiting “texture in the bulk” created by the walk, see [20], [15].”
On the fragmentation of a torus by random walk (2010, background): “2 in [24]), λ1 − λ2 ≥ 1 − λ2 − cǫN −2−ǫ(d−2)/2 ≥ cǫN, proving Lemma A.”
Gumbel fluctuations for cover times in the discrete torus (2012, background): “Random interlacements are intimitaly related to random walk in the torus; intuitively speaking, the trace of Y· in a “local box” of the torus, when run up to time uNd, in some sense “looks like” Iu intersected with a box (see [26], [23]).”
Vacant Set of Random Interlacements and Percolation (2007, background): “The results presented here have triggered some progress on the questions mentioned above; see in particular [15], [20], [21], [22], and [25].”
On the size of a finite vacant cluster of random interlacements with small intensity (2010, unknown): “Mathematics Subject Classification (2000) 60K35 · 82C41”
Soft local times and decoupling of random interlacements (2012, background): “Recently, great effort has been devoted to studying this model [18], [19], [31], [24], [25], [14], [3] as well as to establishing rigorously the relation between random interlacements and the trace left by random walks on large graphs [20], [34], [32], [4].”
Connectivity properties of random interlacement and intersection of random walks (2010, methodology): “The model of random interlacements was introduced by Sznitman in [16] in order to describe the local picture left by the trajectory of a random walk on the discrete torus (Z/NZ)d, d ≥ 3 when it runs up to times of order Nd, or on the discrete cylinder (Z/NZ)d × Z , d ≥ 2, when it runs up to times of”