Giant component and vacant set for random walk on a discrete torus
نویسندگان
چکیده
منابع مشابه
Giant Component and Vacant Set for Random Walk on a Discrete Torus
We consider random walk on a discrete torus E of side-length N , in sufficiently high dimension d. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time uNd. We show that when u is chosen small, as N tends to infinity, there is with overwhelming probability a unique connected component in the vacan...
متن کاملLogarithmic components of the vacant set for random walk on a discrete torus
This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus (Z/NZ)d up to time uNd in high dimension d. If u > 0 is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments o...
متن کاملGIANT VACANT COMPONENT LEFT BY A RANDOM WALK IN A RANDOM d-REGULAR GRAPH
We study the trajectory of a simple random walk on a d-regular graph with d ≥ 3 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. We sho...
متن کاملRandom Walk on a Discrete Torus and Ran- Dom Interlacements
We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ), d ≥ 3, until uN time steps, u > 0, and the model of random interlacements recently introduced by Sznitman [9]. 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 t...
متن کاملVacant Sets and Vacant Nets: Component Structures Induced by a Random Walk
Given a discrete random walk on a finite graph G, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step t. Let Γ(t) be the subgraph of G induced by the vacant set of the walk at step t. Similarly, let Γ̂(t) be the subgraph of G induced by the edges of the vacant net. For random r-regular graphs Gr, it was previously est...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of the European Mathematical Society
سال: 2008
ISSN: 1435-9855
DOI: 10.4171/jems/106