627 research outputs found
Percolation-like Scaling Exponents for Minimal Paths and Trees in the Stochastic Mean Field Model
In the mean field (or random link) model there are points and inter-point
distances are independent random variables. For and in the
limit, let (maximum number of steps
in a path whose average step-length is ). The function
is analogous to the percolation function in percolation theory:
there is a critical value at which becomes
non-zero, and (presumably) a scaling exponent in the sense
. Recently developed probabilistic
methodology (in some sense a rephrasing of the cavity method of Mezard-Parisi)
provides a simple albeit non-rigorous way of writing down such functions in
terms of solutions of fixed-point equations for probability distributions.
Solving numerically gives convincing evidence that . A parallel
study with trees instead of paths gives scaling exponent . The new
exponents coincide with those found in a different context (comparing optimal
and near-optimal solutions of mean-field TSP and MST) and reinforce the
suggestion that these scaling exponents determine universality classes for
optimization problems on random points.Comment: 19 page
Alan’s transitional processes: Exploring the importance of embodying cultural artefacts within BTEC sports qualifications
Multicritical continuous random trees
We introduce generalizations of Aldous' Brownian Continuous Random Tree as
scaling limits for multicritical models of discrete trees. These discrete
models involve trees with fine-tuned vertex-dependent weights ensuring a k-th
root singularity in their generating function. The scaling limit involves
continuous trees with branching points of order up to k+1. We derive explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with fractional
derivative couplings. We prove universality by rederiving the same results
within a purely continuous axiomatic approach based on the resolution of a set
of consistency relations for the multi-point correlations. The average profile
is shown to obey a fractional differential equation whose solution involves
hypergeometric functions and matches the integral formula of the discrete
approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps
Exact calculations of first-passage quantities on recursive networks
We present general methods to exactly calculate mean-first passage quantities
on self-similar networks defined recursively. In particular, we calculate the
mean first-passage time and the splitting probabilities associated to a source
and one or several targets; averaged quantities over a given set of sources
(e.g., same-connectivity nodes) are also derived. The exact estimate of such
quantities highlights the dependency of first-passage processes with respect to
the source-target distance, which has recently revealed to be a key parameter
to characterize transport in complex media. We explicitly perform calculations
for different classes of recursive networks (finitely ramified fractals,
scale-free (trans)fractals, non-fractals, mixtures between fractals and
non-fractals, non-decimable hierarchical graphs) of arbitrary size. Our
approach unifies and significantly extends the available results in the field.Comment: 16 pages, 10 figure
The Melbourne Shuffle: Improving Oblivious Storage in the Cloud
We present a simple, efficient, and secure data-oblivious randomized shuffle
algorithm. This is the first secure data-oblivious shuffle that is not based on
sorting. Our method can be used to improve previous oblivious storage solutions
for network-based outsourcing of data
Optimal spatial transportation networks where link-costs are sublinear in link-capacity
Consider designing a transportation network on vertices in the plane,
with traffic demand uniform over all source-destination pairs. Suppose the cost
of a link of length and capacity scales as for fixed
. Under appropriate standardization, the cost of the minimum cost
Gilbert network grows essentially as , where on and on . This quantity is an upper bound in
the worst case (of vertex positions), and a lower bound under mild regularity
assumptions. Essentially the same bounds hold if we constrain the network to be
efficient in the sense that average route-length is only times
average straight line length. The transition at corresponds to
the dominant cost contribution changing from short links to long links. The
upper bounds arise in the following type of hierarchical networks, which are
therefore optimal in an order of magnitude sense. On the large scale, use a
sparse Poisson line process to provide long-range links. On the medium scale,
use hierachical routing on the square lattice. On the small scale, link
vertices directly to medium-grid points. We discuss one of many possible
variant models, in which links also have a designed maximum speed and the
cost becomes .Comment: 13 page
Cutoff for the East process
The East process is a 1D kinetically constrained interacting particle system,
introduced in the physics literature in the early 90's to model liquid-glass
transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that
its mixing time on sites has order . We complement that result and show
cutoff with an -window.
The main ingredient is an analysis of the front of the process (its rightmost
zero in the setup where zeros facilitate updates to their right). One expects
the front to advance as a biased random walk, whose normal fluctuations would
imply cutoff with an -window. The law of the process behind the
front plays a crucial role: Blondel showed that it converges to an invariant
measure , on which very little is known. Here we obtain quantitative
bounds on the speed of convergence to , finding that it is exponentially
fast. We then derive that the increments of the front behave as a stationary
mixing sequence of random variables, and a Stein-method based argument of
Bolthausen ('82) implies a CLT for the location of the front, yielding the
cutoff result.
Finally, we supplement these results by a study of analogous kinetically
constrained models on trees, again establishing cutoff, yet this time with an
-window.Comment: 33 pages, 2 figure
Random tree growth by vertex splitting
We study a model of growing planar tree graphs where in each time step we
separate the tree into two components by splitting a vertex and then connect
the two pieces by inserting a new link between the daughter vertices. This
model generalises the preferential attachment model and Ford's -model
for phylogenetic trees. We develop a mean field theory for the vertex degree
distribution, prove that the mean field theory is exact in some special cases
and check that it agrees with numerical simulations in general. We calculate
various correlation functions and show that the intrinsic Hausdorff dimension
can vary from one to infinity, depending on the parameters of the model.Comment: 47 page
Empires and Percolation: Stochastic Merging of Adjacent Regions
We introduce a stochastic model in which adjacent planar regions merge
stochastically at some rate , and observe analogies with the
well-studied topics of mean-field coagulation and of bond percolation. Do
infinite regions appear in finite time? We give a simple condition on
for this {\em hegemony} property to hold, and another simple condition for it
to not hold, but there is a large gap between these conditions, which includes
the case . For this case, a non-rigorous analytic
argument and simulations suggest hegemony.Comment: 13 page
Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
We develop a scaling theory for KPZ growth in one dimension by a detailed
study of the polynuclear growth (PNG) model. In particular, we identify three
universal distributions for shape fluctuations and their dependence on the
macroscopic shape. These distribution functions are computed using the
partition function of Gaussian random matrices in a cosine potential.Comment: 4 pages, 3 figures, 1 table, RevTeX, revised version, accepted for
publication in PR
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