Given a finite, connected graph @math , the lamplighter chain on @math is the lazy random walk @math on the associated lamplighter graph @math . The mixing time of the lamplighter chain on the torus @math is known to have a cutoff at a time asymptotic to the cover time of @math if @math , and to half the cover time if @math . We show that the mixing time of the lamplighter chain on @math has a cutoff at @math times the cover time of @math as @math , where @math is an explicit weakly decreasing map from @math onto @math . In particular, as @math varies, the threshold continuously interpolates between the known thresholds for @math and @math . Perhaps surprisingly, we find a phase transition (non-smoothness of @math ) at the point @math , where high dimensional behavior ( @math for all @math ) commences. Here @math is the effective resistance from @math to @math in @math .
By combining the results of @cite_7 and @cite_5 , it is observed in @cite_4 that @math has a threshold at @math . Thus, gives the best bounds, since @math attains the lower bound and @math attains the upper bound. In @cite_3 , it is shown that @math when @math and more generally that @math whenever @math is a sequence of graphs with @math satisfying certain uniform local transience assumptions. This prompted the question [Section 7] Miller-Peres of whether for each @math there exists a (natural) family of graphs @math such that @math as @math . In this work we give an affirmative answer to this question by analyzing the lamplighter chain on a thin @math D torus.
@cite_5 @cite_3 @cite_4 @cite_7
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In this paper, we prove tight lower bounds on the smallest degree of a nonzero polynomial in the ideal generated by @math or @math in the polynomial ring @math , @math are coprime, which is called over @math . The immunity of @math is lower bounded by @math , which is achievable when @math is a multiple of @math ; the immunity of @math is exactly @math for every @math and @math . Our result improves the previous bound @math by Green. We observe how immunity over @math is related to @math circuit lower bound. For example, if the immunity of @math over @math is lower bounded by @math , and @math , then @math requires @math circuit of exponential size to compute. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math . Let @math be a 3-uniform hypergraph of order @math and size @math , and let @math be a subset of vertices of @math . The set @math is a strong transversal in @math if @math contains at least two vertices from every edge of @math . The strong transversal number @math of @math is the minimum size of a strong transversal in @math . We show that @math , and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph @math of order @math and size @math that achieves equality in this bound. A set @math of vertices in a graph @math is a double total dominating set of @math if every vertex of @math is adjacent to at least two vertices in @math . The minimum cardinality of a double total dominating set of @math is the double total domination number @math of @math . Let @math be a connected graph of order @math with minimum degree at least three. As an application of our hypergraph results, we show that @math with equality if and only if @math is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if @math is not the Heawood graph, we show that @math , while if @math is a cubic graph different from the Heawood graph, we show that @math , and this bound is sharp. Let @math be an induced subgraph of the torus @math . We show that when @math is even and @math divides some power of @math , then for sufficiently large @math the torus @math has a perfect vertex-packing with induced copies of @math . On the other hand, disproving a conjecture of Gruslys, we show that when @math is odd and not a prime power, then there exists @math such that @math divides some power of @math , but there is no @math such that @math has a perfect vertex-packing with copies of @math . We also disprove a conjecture of Gruslys, Leader and Tan by exhibiting a subgraph @math of the @math -dimensional hypercube @math , such that there is no @math for which @math has a perfect edge-packing with copies of @math .