The number of inversions of permutations with fixed shape

The Robinson-Schensted correspondence can be viewed as a map from permutations to partitions. In this work, we study the number of inversions of permutations corresponding to a fixed partition $\lambda$ under this map. Hohlweg characterized permutations having shape $\lambda$ with the minimum number of inversions. Here, we give the first results in this direction for higher numbers of inversions. We give explicit conjectures for both the structure and the number of permutations associated to $\lambda$ where the extra number of inversions is less than the length of the smallest column of $\lambda$. We prove the result when $\lambda$ has two columns.Comment: 19 pages, 2 figure

The Computational Complexity of Estimating Convergence Time

An important problem in the implementation of Markov Chain Monte Carlo algorithms is to determine the convergence time, or the number of iterations before the chain is close to stationarity. For many Markov chains used in practice this time is not known. Even in cases where the convergence time is known to be polynomial, the theoretical bounds are often too crude to be practical. Thus, practitioners like to carry out some form of statistical analysis in order to assess convergence. This has led to the development of a number of methods known as convergence diagnostics which attempt to diagnose whether the Markov chain is far from stationarity. We study the problem of testing convergence in the following settings and prove that the problem is hard in a computational sense: Given a Markov chain that mixes rapidly, it is hard for Statistical Zero Knowledge (SZK-hard) to distinguish whether starting from a given state, the chain is close to stationarity by time t or far from stationarity at time ct for a constant c. We show the problem is in AM intersect coAM. Second, given a Markov chain that mixes rapidly it is coNP-hard to distinguish whether it is close to stationarity by time t or far from stationarity at time ct for a constant c. The problem is in coAM. Finally, it is PSPACE-complete to distinguish whether the Markov chain is close to stationarity by time t or far from being mixed at time ct for c at least 1

Reconstruction Threshold for the Hardcore Model

In this paper we consider the reconstruction problem on the tree for the hardcore model. We determine new bounds for the non-reconstruction regime on the k-regular tree showing non-reconstruction when lambda < (ln 2-o(1))ln^2(k)/(2 lnln(k)) improving the previous best bound of lambda < e-1. This is almost tight as reconstruction is known to hold when lambda> (e+o(1))ln^2(k). We discuss the relationship for finding large independent sets in sparse random graphs and to the mixing time of Markov chains for sampling independent sets on trees.Comment: 14 pages, 2 figure

Decay of Correlations for the Hardcore Model on the dd-regular Random Graph

A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. We determine the local weak limit of the hardcore model on random regular graphs asymptotically until just below its condensation threshold, showing that it converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. As a consequence we show that the reconstruction threshold on the random graph, indicative of the onset of point to set spatial correlations, is equal to the reconstruction threshold on the $d$-regular tree for which we determine precise asymptotics. We expect that our methods will generalize to a wide range of spin systems for which the second moment method holds.Comment: 39 pages, 5 figures. arXiv admin note: text overlap with arXiv:1004.353

Analysis of top-swap shuffling for genome rearrangements

We study Markov chains which model genome rearrangements. These models are useful for studying the equilibrium distribution of chromosomal lengths, and are used in methods for estimating genomic distances. The primary Markov chain studied in this paper is the top-swap Markov chain. The top-swap chain is a card-shuffling process with $n$ cards divided over $k$ decks, where the cards are ordered within each deck. A transition consists of choosing a random pair of cards, and if the cards lie in different decks, we cut each deck at the chosen card and exchange the tops of the two decks. We prove precise bounds on the relaxation time (inverse spectral gap) of the top-swap chain. In particular, we prove the relaxation time is $\Theta(n+k)$. This resolves an open question of Durrett.Comment: Published in at http://dx.doi.org/10.1214/105051607000000177 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Lipschitz Constant of the RSK Correspondence

We view the RSK correspondence as associating to each permutation $\pi \in S_n$ a Young diagram $\lambda=\lambda(\pi)$, i.e. a partition of $n$. Suppose now that $\pi$ is left-multiplied by $t$ transpositions, what is the largest number of cells in $\lambda$ that can change as a result? It is natural refer to this question as the search for the Lipschitz constant of the RSK correspondence. We show upper bounds on this Lipschitz constant as a function of $t$. For $t=1$, we give a construction of permutations that achieve this bound exactly. For larger $t$ we construct permutations which come close to matching the upper bound that we prove.Comment: Updated presentation based on comments made by reviewers. Accepted for publication to JCT