88 research outputs found
Fast simulation of large-scale growth models
We give an algorithm that computes the final state of certain growth models
without computing all intermediate states. Our technique is based on a "least
action principle" which characterizes the odometer function of the growth
process. Starting from an approximation for the odometer, we successively
correct under- and overestimates and provably arrive at the correct final
state.
Internal diffusion-limited aggregation (IDLA) is one of the models amenable
to our technique. The boundary fluctuations in IDLA were recently proved to be
at most logarithmic in the size of the growth cluster, but the constant in
front of the logarithm is still not known. As an application of our method, we
calculate the size of fluctuations over two orders of magnitude beyond previous
simulations, and use the results to estimate this constant.Comment: 27 pages, 9 figures. To appear in Random Structures & Algorithm
Rotor-router aggregation on the layered square lattice
In rotor-router aggregation on the square lattice Z^2, particles starting at
the origin perform deterministic analogues of random walks until reaching an
unoccupied site. The limiting shape of the cluster of occupied sites is a disk.
We consider a small change to the routing mechanism for sites on the x- and
y-axes, resulting in a limiting shape which is a diamond instead of a disk. We
show that for a certain choice of initial rotors, the occupied cluster grows as
a perfect diamond.Comment: 11 pages, 3 figures
Stable interaction-induced Anderson-like localization embedded in standing waves
We uncover the interaction-induced \emph{stable self-localization} of bosons
in disorder-free superlattices. In these nonthermalized multi-particle states,
one of the particles forms a superposition of multiple standing waves, so that
it provides a quasirandom potential to localize the other particles. We derive
effective Hamiltonians for self-localized states and find their energy level
spacings obeying the Poisson statistics for Anderson-like localization.
Surprisingly, we find that the correlated self-localization can be solely
induced by interaction in the well-studied nonintegrable Bose-Hubbard models,
which has been overlooked for a long time. We propose a dynamical scheme to
detect self-localization, where long-time quantum walks of a single particle
form a superposition of multiple standing waves for trapping the subsequently
loaded particles. Our work provides an experimentally feasible way to realize
stable Anderson-like localization in translation-invariant disorder-free
systems
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Information dissemination via random walks
Information dissemination is a fundamental task in distributed computing:
How to deliver a piece of information from a node of a network to some or all other nodes?
In the face of large and still growing modern networks, it is imperative that dissemination algorithms are decentralised and can operate under unreliable conditions.
In the past decades, randomised rumour spreading algorithms
have addressed these challenges.
In these algorithms, a message is initially placed at a source node of a network, and, at regular intervals, each node contacts a randomly selected neighbour.
A message may be transmitted in one or both directions during each of these communications, depending on the exact protocol.
The main measure of performance for these algorithms is their broadcast time, which is the time until a message originating from a source node is disseminated to all nodes of the network.
Apart from being extremely simple and robust to failures, randomised rumour spreading achieves theoretically optimal broadcast time in many common network topologies.
In this thesis, we propose an agent-based information dissemination algorithm, called Visit-Exchange.
In our protocol, a number of agents perform independent random walks in the network.
An agent becomes informed when it visits a node that has a message, and later informs all future nodes it visits.
Visit-Exchange shares many of the properties of randomised rumour spreading, namely, it is very simple and uses the same amount of communication in a unit of time.
Moreover, the protocol can be used as a simple model of non-recoverable epidemic processes.
We investigate the broadcast time of Visit-Exchange on a variety of network topologies, and compare it to traditional rumour spreading.
On dense regular networks we show that the two types of protocols are equivalent, which means that in this setting the vast literature on randomised rumour spreading applies in our model as well.
Since many networks of interest, including real-world ones, are very sparse, we also study agent-based broadcast for sparse networks.
Our results include almost optimal or optimal bounds for sparse regular graphs, expanders, random regular graphs, balanced trees and grids.
We establish that depending on the network topology, Visit-Exchange may be either slower or faster than traditional rumour spreading.
In particular, in graphs consisting of hubs that are not well connected, broadcast using agents can be significantly faster.
Our conclusion is that a combined broadcasting protocol that simultaneously uses both traditional rumour spreading and agent-based dissemination can be fast on a larger range of topologies than each of its components separately.Gates Cambridge Trust, St John's College Benefactors' Scholarshi
Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms
The celebrated Cheeger's Inequality \cite{am85,a86} establishes a bound on
the expansion of a graph via its spectrum. This inequality is central to a rich
spectral theory of graphs, based on studying the eigenvalues and eigenvectors
of the adjacency matrix (and other related matrices) of graphs. It has remained
open to define a suitable spectral model for hypergraphs whose spectra can be
used to estimate various combinatorial properties of the hypergraph.
In this paper we introduce a new hypergraph Laplacian operator (generalizing
the Laplacian matrix of graphs)and study its spectra. We prove a Cheeger-type
inequality for hypergraphs, relating the second smallest eigenvalue of this
operator to the expansion of the hypergraph. We bound other hypergraph
expansion parameters via higher eigenvalues of this operator. We give bounds on
the diameter of the hypergraph as a function of the second smallest eigenvalue
of the Laplacian operator. The Markov process underlying the Laplacian operator
can be viewed as a dispersion process on the vertices of the hypergraph that
might be of independent interest. We bound the {\em Mixing-time} of this
process as a function of the second smallest eigenvalue of the Laplacian
operator. All these results are generalizations of the corresponding results
for graphs.
We show that there can be no linear operator for hypergraphs whose spectra
captures hypergraph expansion in a Cheeger-like manner. For any , we give a
polynomial time algorithm to compute an approximation to the smallest
eigenvalue of the operator. We show that this approximation factor is optimal
under the SSE hypothesis (introduced by \cite{rs10}) for constant values of
.
Finally, using the factor preserving reduction from vertex expansion in
graphs to hypergraph expansion, we show that all our results for hypergraphs
extend to vertex expansion in graphs
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