Giant vacant component left by a random walk in a random d-regular graph
نویسندگان
چکیده
منابع مشابه
GIANT 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...
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We consider the simple random walk on a random d-regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. It was shown in [ČTW11] that this so-called vacant set exhibits a phase transition at u = u?: there is a giant component if u < u? and only small components when u > u?. In ...
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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...
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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...
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We consider random walks on two classes of random graphs and explore the likely structure of the vacant set viz. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set. We show that for random graphs Gn,p (above the connectivity threshold) and for random regular graphs Gr, r ≥ 3 there is a phase transition in the sense of the well-known Erdős-Renyi phase transition. T...
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ژورنال
عنوان ژورنال: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
سال: 2011
ISSN: 0246-0203
DOI: 10.1214/10-aihp407