Nayantara Bhatnagar

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 nonreconstruction regime on the k-regular tree showing nonreconstruction when λ (e+o(1)) ln2 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.

The computational complexity of estimating MCMC 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. There does not seem to be a general technique for upper bounding the convergence time that gives sufficiently sharp (useful in practice) bounds in all cases of interest. 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 computationally: - Given aMarkov 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 ∩ coAM. - Given a Markov chain that mixes rapidly it is coNP-hard to distinguish from an arbitrary starting state 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. - It is PSPACE-complete to distinguish whether the Markov chain is close to stationarity by time t or still far from stationarity at time ct for c ≥ 1.

Decay of correlations for the hardcore model on the

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

d

-regular random graph

-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.

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

Polynomials that sign represent parity and Descartes rule of signs

n

On the Diaconis-Gangolli Markov chain for sampling contingency tables with cell-bounded entries

cards divided over

The effect of boundary conditions on mixing rates of markov chains

k

A computational method for bounding the probability of reconstruction on trees

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