210 research outputs found

    Biased random-to-top shuffling

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    Recently Wilson [Ann. Appl. Probab. 14 (2004) 274--325] introduced an important new technique for lower bounding the mixing time of a Markov chain. In this paper we extend Wilson's technique to find lower bounds of the correct order for card shuffling Markov chains where at each time step a random card is picked and put at the top of the deck. Two classes of such shuffles are addressed, one where the probability that a given card is picked at a given time step depends on its identity, the so-called move-to-front scheme, and one where it depends on its position. For the move-to-front scheme, a test function that is a combination of several different eigenvectors of the transition matrix is used. A general method for finding and using such a test function, under a natural negative dependence condition, is introduced. It is shown that the correct order of the mixing time is given by the biased coupon collector's problem corresponding to the move-to-front scheme at hand. For the second class, a version of Wilson's technique for complex-valued eigenvalues/eigenvectors is used. Such variants were presented in [Random Walks and Geometry (2004) 515--532] and [Electron. Comm. Probab. 8 (2003) 77--85]. Here we present another such variant which seems to be the most natural one for this particular class of problems. To find the eigenvalues for the general case of the second class of problems is difficult, so we restrict attention to two special cases. In the first case the card that is moved to the top is picked uniformly at random from the bottom k=k(n)=o(n)k=k(n)=o(n) cards, and we find the lower bound (n3/(4π2k(k1)))logn(n^3/(4\pi^2k(k-1)))\log n. Via a coupling, an upper bound exceeding this by only a factor 4 is found. This generalizes Wilson's [Electron. Comm. Probab. 8 (2003) 77--85] result on the Rudvalis shuffle and Goel's [Ann. Appl. Probab. 16 (2006) 30--55] result on top-to-bottom shuffles. In the second case the card moved to the top is, with probability 1/2, the bottom card and with probability 1/2, the card at position nkn-k. Here the lower bound is again of order (n3/k2)logn(n^3/k^2)\log n, but in this case this does not seem to be tight unless k=O(1)k=O(1). What the correct order of mixing is in this case is an open question. We show that when k=n/2k=n/2, it is at least Θ(n2)\Theta(n^2).Comment: Published at http://dx.doi.org/10.1214/10505160600000097 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Uniqueness and non-uniqueness in percolation theory

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    This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on Zd{\mathbb{Z}}^d and, more generally, on transitive graphs. For iid percolation on Zd{\mathbb{Z}}^d, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models -- most prominently the Fortuin--Kasteleyn random-cluster model -- and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.Comment: Published at http://dx.doi.org/10.1214/154957806000000096 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

    The spectrum and convergence rates of exclusion and interchange processes on the complete graph

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    We give a short and completely elementary method to find the full spectrum of the exclusion process and a nicely limited superset of the spectrum of the interchange process (a.k.a.\ random transpositions) on the complete graph. In the case of the exclusion process, this gives a simple closed form expression for all the eigenvalues and their multiplicities. This result is then used to give an exact expression for the distance in L2 L^2 from stationarity at any time and upper and lower bounds on the convergence rate for the exclusion process. In the case of the interchange process, upper and lower bounds are similarly found. Our results strengthen or reprove all known results of the mixing time for the two processes in a very simple way.Comment: 16 page

    Dynamical models for circle covering: Brownian motion and Poisson updating

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    We consider two dynamical variants of Dvoretzky's classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent Brownian motions and in the second model, the positions of the intervals are updated according to independent Poisson processes where an interval of length \ell is updated at rate α\ell^{-\alpha} where α0\alpha \ge0 is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the nnth interval is c/nc/n, then there are times at which a fixed point is not covered if and only if c<2c<2 and there are times at which the circle is not fully covered if and only if c<3c<3. For the Poisson updating model, we obtain analogous results with c<αc<\alpha and c<α+1c<\alpha+1 instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.Comment: Published in at http://dx.doi.org/10.1214/07-AOP340 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Visibility to infinity in the hyperbolic plane, despite obstacles

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    Suppose that ZZ is a random closed subset of the hyperbolic plane \H^2, whose law is invariant under isometries of \H^2. We prove that if the probability that ZZ contains a fixed ball of radius 1 is larger than some universal constant p<1p<1, then there is positive probability that ZZ contains (bi-infinite) lines. We then consider a family of random sets in \H^2 that satisfy some additional natural assumptions. An example of such a set is the covered region in the Poisson Boolean model. Let f(r)f(r) be the probability that a line segment of length rr is contained in such a set ZZ. We show that if f(r)f(r) decays fast enough, then there are almost surely no lines in ZZ. We also show that if the decay of f(r)f(r) is not too fast, then there are almost surely lines in ZZ. In the case of the Poisson Boolean model with balls of fixed radius RR we characterize the critical intensity for the almost sure existence of lines in the covered region by an integral equation. We also determine when there are lines in the complement of a Poisson process on the Grassmannian of lines in \H^2

    Coupling and Bernoullicity in random-cluster and Potts models

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    An explicit coupling construction of random-cluster measures is presented. As one of the applications of the construction, the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity

    Explicit isoperimetric constants and phase transitions in the random-cluster model

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    The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter q1q\geq 1. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value, and examples of planar regular graphs with regular dual where \pc^\f (q) > \pu^\w (q) for qq large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove non-robust phase transition for the Potts model on nonamenable regular graphs

    The overhand shuffle mixes in Θ(n2logn)\Theta(n^2\log n) steps

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    The overhand shuffle is one of the ``real'' card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37--49] who showed that the mixing time with respect to variation distance is at least of order n2n^2 and at most of order n2lognn^2\log n. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274--325] to establish a lower bound of order n2lognn^2\log n, thereby showing that n2lognn^2\log n is indeed the correct order of the mixing time. It is our hope that the extension of Wilson's lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n3logn)\Theta(n^3\log n) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77--85] for the Rudvalis shuffle.Comment: Published at http://dx.doi.org/10.1214/105051605000000692 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org
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