{"id":357905,"date":"2017-01-25T14:30:13","date_gmt":"2017-01-25T22:30:13","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357905"},"modified":"2018-10-16T20:02:05","modified_gmt":"2018-10-17T03:02:05","slug":"uniform-mixing-time-random-walk-lamplighter-graphs","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/uniform-mixing-time-random-walk-lamplighter-graphs\/","title":{"rendered":"Uniform Mixing Time For Random Walk On Lamplighter Graphs"},"content":{"rendered":"

Suppose that $\\CG$<\/span><\/span><\/span> is a finite, connected graph and X<\/span><\/span><\/span><\/span> is a lazy random walk on $\\CG$<\/span><\/span><\/span>. The lamplighter chain X<\/span>\u22c4<\/span><\/span><\/span><\/span><\/span> associated with X<\/span><\/span><\/span><\/span> is the random walk on the wreath product $\\CG^\\diamond = \\Z_2 \\wr \\CG$<\/span><\/span><\/span>, the graph whose vertices consist of pairs (<\/span>f<\/span>,<\/span>x<\/span>)<\/span><\/span><\/span><\/span> where f<\/span><\/span><\/span><\/span> is a labeling of the vertices of $\\CG$<\/span><\/span><\/span> by elements of $\\Z_2$<\/span><\/span><\/span> and x<\/span><\/span><\/span><\/span> is a vertex in $\\CG$<\/span><\/span><\/span>. There is an edge between (<\/span>f<\/span>,<\/span>x<\/span>)<\/span><\/span><\/span><\/span> and (<\/span>g<\/span>,<\/span>y<\/span>)<\/span><\/span><\/span><\/span> in $\\CG^\\diamond$<\/span><\/span><\/span> if and only if x<\/span><\/span><\/span><\/span> is adjacent to y<\/span><\/span><\/span><\/span> in $\\CG$<\/span><\/span><\/span>and f<\/span>(<\/span>z<\/span>)<\/span>=<\/span>g<\/span>(<\/span>z<\/span>)<\/span><\/span><\/span><\/span> for all z<\/span>\u2260<\/span>x<\/span>,<\/span>y<\/span><\/span><\/span><\/span>. In each step, X<\/span>\u22c4<\/span><\/span><\/span><\/span><\/span> moves from a configuration (<\/span>f<\/span>,<\/span>x<\/span>)<\/span><\/span><\/span><\/span> by updating x<\/span><\/span><\/span><\/span> to y<\/span><\/span><\/span><\/span> using the transition rule of X<\/span><\/span><\/span><\/span> and then sampling both f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span> and f<\/span>(<\/span>y<\/span>)<\/span><\/span><\/span><\/span> according to the uniform distribution on $\\Z_2$<\/span><\/span><\/span>; f<\/span>(<\/span>z<\/span>)<\/span><\/span><\/span><\/span> for z<\/span>\u2260<\/span>x<\/span>,<\/span>y<\/span><\/span><\/span><\/span> remains unchanged. We give matching upper and lower bounds on the uniform mixing time of X<\/span>\u22c4<\/span><\/span><\/span><\/span><\/span> provided $\\CG$<\/span><\/span><\/span> satisfies mild hypotheses. In particular, when $\\CG$<\/span><\/span><\/span> is the hypercube $\\Z_2^d$<\/span><\/span><\/span>, we show that the uniform mixing time of X<\/span>\u22c4<\/span><\/span><\/span><\/span><\/span> is \u0398<\/span>(<\/span>d<\/span>2<\/span>d<\/span><\/span>)<\/span><\/span><\/span><\/span>. More generally, we show that when $\\CG$<\/span><\/span><\/span> is a torus $\\Z_n^d$<\/span><\/span><\/span> for d<\/span>\u2265<\/span>3<\/span><\/span><\/span><\/span>, the uniform mixing time of X<\/span>\u22c4<\/span><\/span><\/span><\/span><\/span> is \u0398<\/span>(<\/span>d<\/span>n<\/span>d<\/span><\/span>)<\/span><\/span><\/span><\/span> uniformly in n<\/span><\/span><\/span><\/span> and d<\/span><\/span><\/span><\/span>. A critical ingredient for our proof is a concentration estimate for the local time of random walk in a subset of vertices.<\/p>\n","protected":false},"excerpt":{"rendered":"

Suppose that $\\CG$ is a finite, connected graph and X is a lazy random walk on $\\CG$. The lamplighter chain X\u22c4 associated with X is the random walk on the wreath product $\\CG^\\diamond = \\Z_2 \\wr \\CG$, the graph whose vertices consist of pairs (f,x) where f is a labeling of the vertices of $\\CG$ […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Cornell University 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