{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:6PCYSP37UYFCAJ5G6GSI4W2ZTH","short_pith_number":"pith:6PCYSP37","schema_version":"1.0","canonical_sha256":"f3c5893f7fa60a2027a6f1a48e5b5999f6bb04c7e7815b27bfaa4d260538a11f","source":{"kind":"arxiv","id":"1012.5117","version":1},"attestation_state":"computed","paper":{"title":"Giant vacant component left by a random walk in a random d-regular graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Augusto Teixeira, David Windisch, Jiri Cerny","submitted_at":"2010-12-22T22:21:58Z","abstract_excerpt":"We study the trajectory of a simple random walk on a d-regular graph with d>2 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 show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* such that, with high probability as n grows, if u2 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 show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* such that, with high probability as n grows, if u