416 research outputs found
Mixing of Permutations by Biased Transposition
Markov chains defined on the set of permutations of n elements have been studied widely by mathematicians and theoretical computer scientists. We consider chains in which a position i<n is chosen uniformly at random, and then sigma(i) and sigma(i+1) are swapped with probability depending on sigma(i) and sigma(i+1). Our objective is to identify some conditions that assure rapid mixing.
One case of particular interest is what we call the "gladiator chain," in which each number g is assigned a "strength" s_g and when g and g\u27 are swapped, g comes out on top with probability s_g / ( s_g + s_g\u27 ). The stationary probability of this chain is the same as that of the slow-mixing "move ahead one" chain for self-organizing lists, but an open conjecture of Jim Fill\u27s implies that all gladiator chains mix rapidly. Here we obtain some positive partial results by considering cases where the gladiators fall into only a few strength classes
Iterated Decomposition of Biased Permutations via New Bounds on the Spectral Gap of Markov Chains
The spectral gap of a Markov chain can be bounded by the spectral gaps of
constituent "restriction" chains and a "projection" chain, and the strength of
such a bound is the content of various decomposition theorems. In this paper,
we introduce a new parameter that allows us to improve upon these bounds. We
further define a notion of orthogonality between the restriction chains and
"complementary" restriction chains. This leads to a new Complementary
Decomposition theorem, which does not require analyzing the projection chain.
For -orthogonal chains, this theorem may be iterated
times while only giving away a constant multiplicative factor on the overall
spectral gap. As an application, we provide a -orthogonal decomposition of
the nearest neighbor Markov chain over -class biased monotone permutations
on [], as long as the number of particles in each class is at least . This allows us to apply the Complementary Decomposition theorem iteratively
times to prove the first polynomial bound on the spectral gap when is
as large as . The previous best known bound assumed was
at most a constant
Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement
This paper studies Markov chains on the chambers of real hyperplane
arrangements, a model that generalizes famous examples, such as the Tsetlin
library and riffle shuffles. We discuss cutoff for the Tsetlin library for
general weights, and we give an exact formula for the separation distance for
the hyperplane arrangement walk. We introduce lower bounds, which allow for the
first time to study cutoff for hyperplane arrangement walks under certain
conditions. Using similar techniques, we also prove a uniform lower bound for
the mixing time of Glauber dynamics on a monotone system.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1605.0833
Algorithmic Problems Arising in Posets and Permutations
Partially ordered sets and permutations are combinatorial structures having vast applications in theoretical computer science. In this thesis, we study various computational and algorithmic problems related to these structures. The first chapter of the thesis contains discussion about randomized fully polynomial approximation schemes obtained by employing Markov chain Monte Carlo. In this chapter we study various Markov chains that we call: the gladiator chain, the interval chain, and cube shuffling. Our objective is to identify some conditions that assure rapid mixing; and we obtain partial results. The gladiator chain is a biased random walk on the set of permutations. This chain is related to self organizing lists, and various versions of it have been studied. The interval chain is a random walk on the set of points in whose coordinates respect a partial order. Since the sample space of the interval chain is continuous, many mixing techniques for discrete chains are not applicable to it. The cube shuffle chain is a generalization of H\r{a}stad\u27s square shuffle. The importance of this chain is that it mixes in constant number of steps. In the second chapter, we are interested in calculating expected value of real valued function on a set of combinatorial structures , given a probability distribution on it. We first suggest a Markov chain Monte Carlo approach to this problem. We identify the conditions under which our proposed solution will be efficient, and present examples where it fails. Then, we study homomesy. Homomesy is a phenomenon introduced by Jim Propp and Tom Roby. We say the triple ( is a permutation mapping to itself) exhibits homomesy, if the average of along all -orbits of is a constant only depending on and . We study homomesy and obtain some results when is the set of ideals in a class of simply described lattices
Combinatorial Markov chains on linear extensions
We consider generalizations of Schuetzenberger's promotion operator on the
set L of linear extensions of a finite poset of size n. This gives rise to a
strongly connected graph on L. By assigning weights to the edges of the graph
in two different ways, we study two Markov chains, both of which are
irreducible. The stationary state of one gives rise to the uniform
distribution, whereas the weights of the stationary state of the other has a
nice product formula. This generalizes results by Hendricks on the Tsetlin
library, which corresponds to the case when the poset is the anti-chain and
hence L=S_n is the full symmetric group. We also provide explicit eigenvalues
of the transition matrix in general when the poset is a rooted forest. This is
shown by proving that the associated monoid is R-trivial and then using
Steinberg's extension of Brown's theory for Markov chains on left regular bands
to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in
terms of discrete time Markov chain
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