5,960 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

    Multiplication of solutions for linear overdetermined systems of partial differential equations

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    A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains the Cauchy-Riemann equations and the cofactor pair systems, included as special cases. The multiplication provides a method for generating, in a pure algebraic way, large classes of non-trivial solutions that can be constructed by forming convergent power series of trivial solutions.Comment: 27 page

    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

    Condition based maintenance of trains doors

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    As part of the project DUST financed by Vinnova, we have investigated whether event data generated on trains can be used for finding evidence of wear on train doors. We have compared the event data and maintenance reports relating to doors of Regina trains. Although some interesting relations were found, the overall result is that the information in event data about wear of doors is very limited

    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