190,230 research outputs found
Stationary distributions of the multi-type ASEP
We give a recursive construction of the stationary distribution of multi-type
asymmetric simple exclusion processes on a finite ring or on the infinite line
. The construction can be interpreted in terms of "multi-line diagrams" or
systems of queues in tandem. Let be the asymmetry parameter of the system.
The queueing construction generalises the one previously known for the totally
asymmetric () case, by introducing queues in which each potential service
is unused with probability when the queue-length is . The analysis is
based on the matrix product representation of Prolhac, Evans and Mallick.
Consequences of the construction include: a simple method for sampling exactly
from the stationary distribution for the system on a ring; results on common
denominators of the stationary probabilities, expressed as rational functions
of with non-negative integer coefficients; and probabilistic descriptions
of "convoy formation" phenomena in large systems.Comment: 54 pages, 4 figure
Limiting shape for directed percolation models
We consider directed first-passage and last-passage percolation on the
nonnegative lattice Z_+^d, d\geq2, with i.i.d. weights at the vertices. Under
certain moment conditions on the common distribution of the weights, the limits
g(x)=lim_{n\to\infty}n^{-1}T(\lfloor nx\rfloor) exist and are constant a.s. for
x\in R_+^d, where T(z) is the passage time from the origin to the vertex z\in
Z_+^d. We show that this shape function g is continuous on R_+^d, in particular
at the boundaries. In two dimensions, we give more precise asymptotics for the
behavior of g near the boundaries; these asymptotics depend on the common
weight distribution only through its mean and variance. In addition we discuss
growth models which are naturally associated to the percolation processes,
giving a shape theorem and illustrating various possible types of behavior with
output from simulations.Comment: Published at http://dx.doi.org/10.1214/009117904000000838 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Reconstruction thresholds on regular trees
We consider a branching random walk with binary state space and index set
, the infinite rooted tree in which each node has k children (also known
as the model of "broadcasting on a tree"). The root of the tree takes a random
value 0 or 1, and then each node passes a value independently to each of its
children according to a 2x2 transition matrix P. We say that "reconstruction is
possible" if the values at the d'th level of the tree contain non-vanishing
information about the value at the root as . Adapting a method of
Brightwell and Winkler, we obtain new conditions under which reconstruction is
impossible, both in the general case and in the special case . The
latter case is closely related to the "hard-core model" from statistical
physics; a corollary of our results is that, for the hard-core model on the
(k+1)-regular tree with activity , the unique simple invariant Gibbs
measure is extremal in the set of Gibbs measures, for any k.Comment: 12 page
The Neolithic transition in Europe: archaeological models and genetic evidence
The major pattern in the European gene pool is a southeast-northwest frequency gradient of classic genetic markers such as blood groups, which population geneticists initially attributed to the demographic impact of Neolithic farmers dispersing from the Near East. Molecular genetics has enriched this picture, with analyses of mitochondrial DNA and the Y chromosome allowing a more detailed exploration of alternative models for the spread of the Neolithic into Europe. This paper considers a range of possible models in the light of the detailed information now emerging from genetic studies
Batch queues, reversibility and first-passage percolation
We consider a model of queues in discrete time, with batch services and
arrivals. The case where arrival and service batches both have Bernoulli
distributions corresponds to a discrete-time M/M/1 queue, and the case where
both have geometric distributions has also been previously studied. We describe
a common extension to a more general class where the batches are the product of
a Bernoulli and a geometric, and use reversibility arguments to prove versions
of Burke's theorem for these models. Extensions to models with continuous time
or continuous workload are also described. As an application, we show how these
results can be combined with methods of Seppalainen and O'Connell to provide
exact solutions for a new class of first-passage percolation problems.Comment: 16 pages. Mostly minor revisions; some new explanatory text added in
various places. Thanks to a referee for helpful comments and suggestion
Heavy tails in last passage percolation
We consider last-passage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index . We prove scaling laws and asymptotic distributions, both for the passage times and for the shape of optimal paths; these are expressed in terms of a family (indexed by ) of ``continuous last-passage percolation'' models in the unit square. In the extreme case (corresponding to a distribution with slowly varying tail) the asymptotic distribution of the optimal path can be represented by a random self-similar measure on [0,1], whose multifractal spectrum we compute. By extending the continuous last-passage percolation model to we obtain a heavy-tailed analogue of the Airy process, representing the limit of appropriately scaled vectors of passage times to different points in the plane. We give corresponding results for a directed percolation problem based on -stable Levy processes, and indicate extensions of the results to higher dimensions
Quantum fields, dark matter and non-standard Wigner classes
The Elko field of Ahluwalia and Grumiller is a quantum field for massive
spin-1/2 particles. It has been suggested as a candidate for dark matter. We
discuss our attempts to interpret the Elko field as a quantum field in the
sense of Weinberg. Our work suggests that one should investigate quantum fields
based on representations of the full Poincar\'e group which belong to one of
the non-standard Wigner classes.Comment: 6 pages. Submitted to proceedings of Dark2009, Christchurch, New
Zealand, January 200
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