2 A Rapidly Mixing Circulant Markov

Mixing in Continuous Quantum Walks on Graphs

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs th...

Conductance and Rapidly Mixing Markov Chains

Conductance is a measure of a Markov chain that quantifies its tendency to circulate around its states. A Markov chain with low conductance will tend to get ‘stuck’ in a subset of its states whereas one with high conductance will jump around its state space more freely. The mixing time of a Markov chain is the number of steps required for the chain to approach its stationary distribution. There...

Rapidly Mixing Markov Chains for Sampling Contingency Tables with a Constant Number of Rows

We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. Cryan and Dyer [3] have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which employs Markov chain methods indirectly. They leave open the question as to whether a natural Markov...

Very rapidly mixing Markov Chains for 2Δ-colourings and for independent sets in a graph with maximum degree 4

A Generalization of Circulant Matrices for Non-Abelian Groups

A circulant matrix of order n is the matrix of convolution by a fixed element of the group algebra of the cyclic group Zn. Replacing Zn by an arbitrary finite group G gives the class of matrices that we call G-circulant. We determine the eigenvalues of such matrices with the tools of representation theory and the non-abelian Fourier transform. Definition 1 An n by n matrix C is circulant if the...