136 research outputs found
A fixed-point approximation for a routing model in equilibrium
We use a method of Luczak (arXiv:1212.3231) to investigate the equilibrium
distribution of a dynamic routing model on a network. In this model, there are
nodes, each pair joined by a link of capacity . For each pair of nodes,
calls arrive for this pair of endpoints as a Poisson process with rate
. A call for endpoints is routed directly onto the link
between the two nodes if there is spare capacity; otherwise two-link paths
between and are considered, and the call is routed along a path with
lowest maximum load, if possible. The duration of each call is an exponential
random variable with unit mean. In the case , it was suggested by Gibbens,
Hunt and Kelly in 1990 that the equilibrium of this process is related to the
fixed points of a certain equation. We show that this is indeed the case, for
every , provided the arrival rate is either sufficiently
small or sufficiently large. In either regime, we show that the equation has a
unique fixed point, and that, in equilibrium, for each , the proportion of
links at each node with load is strongly concentrated around the th
coordinate of the fixed point.Comment: 33 page
The structure of random discrete spacetime
The usual picture of spacetime consists of a continuous manifold, together with a metric of Lorentzian signature which imposes a causal structure on the spacetime. A model, first suggested by Bombelli et al., is considered in which spacetime consists of a discrete set of points taken at random from a manifold, with only the causal structure on this set remaining. This structure constitutes a partially ordered set (or poset). Working from the poset alone, it is shown how to construct a metric on the space which closely approximates the metric on the original spacetime manifold, how to define the effective dimension of the spacetime, and how such quantities may depend on the scale of measurement. Possible desirable features of the model are discussed
A 2D model of Causal Set Quantum Gravity: The emergence of the continuum
Non-perturbative theories of quantum gravity inevitably include
configurations that fail to resemble physically reasonable spacetimes at large
scales. Often, these configurations are entropically dominant and pose an
obstacle to obtaining the desired classical limit. We examine this "entropy
problem" in a model of causal set quantum gravity corresponding to a
discretisation of 2D spacetimes. Using results from the theory of partial
orders we show that, in the large volume or continuum limit, its partition
function is dominated by causal sets which approximate to a region of 2D
Minkowski space. This model of causal set quantum gravity thus overcomes the
entropy problem and predicts the emergence of a physically reasonable geometry.Comment: Corrections and clarifications. Conclusions unchange
Long-term concentration of measure and cut-off
We present new concentration of measure inequalities for Markov chains that
enjoy a contractive property. We apply our discrete-time inequality to the
well-studied Bernoulli-Laplace model of diffusion, and give a probabilistic
proof of cut-off, recovering the optimal bounds of Diaconis and Shahshahani. We
also extend the notion of cut-off to chains with an infinite state space, and
illustrate this in a second example, of a two-host model of disease in
continuous time.Comment: 36 page
Reorientations of covering graphs
AbstractThe aim of this paper is to construct a graph G on n vertices which is a connected covering graph but has 2o(n) diagram orientations. This provides a negative answer to a question of I. Rival
Asymptotic Enumeration of Labelled Interval Orders
Building on work by Zagier, Bousquet-M\'elou et al., and Khamis, we give an
asymptotic formula for the number of labelled interval orders on an -element
set.Comment: 6 page
The Reversal Ratio of a Poset
Felsner and Reuter introduced the linear extension diameter of a partially
ordered set , denoted \mbox{led}(\mathbf{P}), as the maximum
distance between two linear extensions of , where distance is
defined to be the number of incomparable pairs appearing in opposite orders
(reversed) in the linear extensions. In this paper, we introduce the reversal
ratio of as the ratio of the linear extension
diameter to the number of (unordered) incomparable pairs. We use probabilistic
techniques to provide a family of posets on at most
elements for which the reversal ratio , where
is a constant. We also examine the questions of bounding the reversal ratio
in terms of order dimension and width.Comment: 10 pages, 2 figures; Accepted for publication in ORDE
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