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\\) by elements of \\(\\Z_2\\) and \\(x\\) is a vertex in \\(\\CG\\). There is an edge between \\((f,x)\\) and \\((g,y)\\) in \\(\\CG^\\diamond\\) if and only if \\(x\\) is adjacent to \\(y\\) in \\(\\CG\\)and \\(f(z)=g(z)\\) for all \\(z\\neq x,y\\). In each step, \\(X_\u22c4\\) moves from a configuration \\((f,x)\\) by updating \\(x\\) to \\(y\\) using the transition rule of \\(X\\) and then sampling both \\(f(x)\\) and \\(f(y)\\) according to the uniform distribution on \\(\\Z_2\\); \\(f(z)\\) for \\(z\\neq x,y\\) remains unchanged. We give matching upper and lower bounds on the uniform mixing time of \\(X_\u22c4\\) provided \\(\\CG\\) satisfies mild hypotheses. In particular, when \\(\\CG\\) is the hypercube \\(\\Z_2^d\\), we show that the uniform mixing time of \\(X_\u22c4\\) is \\(\\Theta (d2_d)\\). More generally, we show that when \\(\\CG\\) is a torus \\(\\Z_n^d\\) for \\(d\\geq 3\\), the uniform mixing time of \\(X_\u22c4\\) is \\(\\Theta (dn_d)\\) uniformly in \\(n\\) and \\(d\\). 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 is a finite, connected graph and is a lazy random walk on . The lamplighter chain associated with is the random walk on the wreath product , the graph whose vertices consist of pairs where is a labeling of the vertices of by elements of and is a vertex in . 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