An O ( n 2 ) bound for the relaxation time of a Markov chain oncladogramsby
نویسنده
چکیده
A cladogram is an unrooted tree with labeled leaves and unlabeled internal branchpoints of degree 3. Aldous has studied a Markov chain on the set of n-leaf cladograms in which each transition consists of removing a random leaf and its incident edge from the tree and then reattaching the leaf to a random edge of the remaining tree. Using coupling methods, Aldous has shown that a mixing-time parameter for this chain known as the relaxation time is O(n 3). Here, we use a Poincar e inequality to prove an O(n 2) bound for the relaxation time, establishing a conjecture of Aldous.
منابع مشابه
Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002 [10]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form ...
متن کاملAn O(n2) bound for the relaxation time of a Markov chain on cladograms
A cladogram is an unrooted tree with labeled leaves and unlabeled internal branchpoints of degree 3. Aldous has studied a Markov chain on the set of n-leaf cladograms in which each transition consists of removing a random leaf and its incident edge from the tree and then reattaching the leaf to a random edge of the remaining tree. Using coupling methods, Aldous showed that the relaxation time (...
متن کاملThe Rate of Rényi Entropy for Irreducible Markov Chains
In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.
متن کاملThe mixing time for simple exclusion
We obtain a tight bound of O(L log k) for the mixing time of the exclusion process in Z/LZ with k ≤ 1 2 L particles. Previously the best bound, based on the log Sobolev constant determined by Yau, was not tight for small k. When dependence on the dimension d is considered, our bounds are an improvement for all k. We also get bounds for the relaxation time that are lower-order in d than previous...
متن کاملRobust Mixing
In this paper, we develop a new“robust mixing” framework for reasoning about adversarially modified Markov Chains (AMMC). Let P be the transition matrix of an irreducible Markov Chain with stationary distribution π. An adversary announces a sequence of stochastic matrices {At}t>0 satisfying πAt = π. An AMMC process involves an application of P followed by At at time t. The robust mixing time of...
متن کامل