5,960 research outputs found

### Biased random-to-top shuffling

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 $k=k(n)=o(n)$ cards, and we find the lower
bound $(n^3/(4\pi^2k(k-1)))\log n$. 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 $n-k$. Here the lower bound is again of
order $(n^3/k^2)\log n$, but in this case this does not seem to be tight unless
$k=O(1)$. What the correct order of mixing is in this case is an open question.
We show that when $k=n/2$, it is at least $\Theta(n^2)$.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

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

This paper is an up-to-date introduction to the problem of uniqueness versus
non-uniqueness of infinite clusters for percolation on ${\mathbb{Z}}^d$ and,
more generally, on transitive graphs. For iid percolation on ${\mathbb{Z}}^d$,
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

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 $L^2$ 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

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 $\ell^{-\alpha}$ where $\alpha
\ge0$ is a parameter. For the model with Brownian motions, a special case of
our results is that if the length of the $n$th interval is $c/n$, then there
are times at which a fixed point is not covered if and only if $c<2$ and there
are times at which the circle is not fully covered if and only if $c<3$. For
the Poisson updating model, we obtain analogous results with $c<\alpha$ and
$c<\alpha+1$ 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

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

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

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