489 research outputs found
Dispersion processes
We study a synchronous dispersion process in which particles are
initially placed at a distinguished origin vertex of a graph . At each time
step, at each vertex occupied by more than one particle at the beginning of
this step, each of these particles moves to a neighbour of chosen
independently and uniformly at random. The dispersion process ends once the
particles have all stopped moving, i.e. at the first step at which each vertex
is occupied by at most one particle.
For the complete graph and star graph , we show that for any
constant , with high probability, if , then the
process finishes in steps, whereas if , then
the process needs steps to complete (if ever). We also show
that an analogous lazy variant of the process exhibits the same behaviour but
for higher thresholds, allowing faster dispersion of more particles.
For paths, trees, grids, hypercubes and Cayley graphs of large enough sizes
(in terms of ) we give bounds on the time to finish and the maximum distance
traveled from the origin as a function of the number of particles
An All-But-One Entropic Uncertainty Relation, and Application to Password-based Identification
Entropic uncertainty relations are quantitative characterizations of
Heisenberg's uncertainty principle, which make use of an entropy measure to
quantify uncertainty. In quantum cryptography, they are often used as
convenient tools in security proofs. We propose a new entropic uncertainty
relation. It is the first such uncertainty relation that lower bounds the
uncertainty in the measurement outcome for all but one choice for the
measurement from an arbitrarily large (but specifically chosen) set of possible
measurements, and, at the same time, uses the min-entropy as entropy measure,
rather than the Shannon entropy. This makes it especially suited for quantum
cryptography. As application, we propose a new quantum identification scheme in
the bounded quantum storage model. It makes use of our new uncertainty relation
at the core of its security proof. In contrast to the original quantum
identification scheme proposed by Damg{\aa}rd et al., our new scheme also
offers some security in case the bounded quantum storage assumption fails hold.
Specifically, our scheme remains secure against an adversary that has unbounded
storage capabilities but is restricted to non-adaptive single-qubit operations.
The scheme by Damg{\aa}rd et al., on the other hand, completely breaks down
under such an attack.Comment: 33 pages, v
Ordering of Energy Levels in Heisenberg Models and Applications
In a recent paper we conjectured that for ferromagnetic Heisenberg models the
smallest eigenvalues in the invariant subspaces of fixed total spin are
monotone decreasing as a function of the total spin and called this property
ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture
for the Heisenberg model with arbitrary spins and coupling constants on a
chain. In this paper we give a pedagogical introduction to this result and also
discuss some extensions and implications. The latter include the property that
the relaxation time of symmetric simple exclusion processes on a graph for
which FOEL can be proved, equals the relaxation time of a random walk on the
same graph. This equality of relaxation times is known as Aldous' Conjecture.Comment: 20 pages, contribution for the proceedings of QMATH9, Giens,
September 200
On the ideals of equivariant tree models
We introduce equivariant tree models in algebraic statistics, which unify and
generalise existing tree models such as the general Markov model, the strand
symmetric model, and group based models. We focus on the ideals of such models.
We show how the ideals for general trees can be determined from the ideals for
stars. The main novelty is our proof that this procedure yields the entire
ideal, not just an ideal defining the model set-theoretically. A corollary of
theoretical importance is that the ideal for a general tree is generated by the
ideals of its flattenings at vertices.Comment: 23 pages. Greatly improved exposition, in part following suggestions
by a referee--thanks! Also added exampl
Quantum information reclaiming after amplitude damping
We investigate the quantum information reclaim from the environment after
amplitude damping has occurred. In particular we address the question of
optimal measurement on the environment to perform the best possible correction
on two and three dimensional quantum systems. Depending on the dimension we
show that the entanglement fidelity (the measure quantifying the correction
performance) is or is not the same for all possible measurements and uncover
the optimal measurement leading to the maximum entanglement fidelity
Generalized Ensemble and Tempering Simulations: A Unified View
From the underlying Master equations we derive one-dimensional stochastic
processes that describe generalized ensemble simulations as well as tempering
(simulated and parallel) simulations. The representations obtained are either
in the form of a one-dimensional Fokker-Planck equation or a hopping process on
a one-dimensional chain. In particular, we discuss the conditions under which
these representations are valid approximate Markovian descriptions of the
random walk in order parameter or control parameter space. They allow a unified
discussion of the stationary distribution on, as well as of the stationary flow
across each space. We demonstrate that optimizing the flow is equivalent to
minimizing the first passage time for crossing the space, and discuss the
consequences of our results for optimizing simulations. Finally, we point out
the limitations of these representations under conditions of broken ergodicity.Comment: 11 pages Latex, 2 eps figures, revised version, typos corrected, PRE
in pres
Spectral Analysis of the Supreme Court
The focus of this paper is the linear algebraic framework in which the spectral analysis of voting data like that above is carried out. As we will show, this framework can be used to pinpoint voting coalitions in small voting bodies like the United States Supreme Court. Our goal is to show how simple ideas from linear algebra can come together to say something interesting about voting. And what could be more simple than where our story begins— with counting
One-sided Cauchy-Stieltjes Kernel Families
This paper continues the study of a kernel family which uses the
Cauchy-Stieltjes kernel in place of the celebrated exponential kernel of the
exponential families theory. We extend the theory to cover generating measures
with support that is unbounded on one side. We illustrate the need for such an
extension by showing that cubic pseudo-variance functions correspond to
free-infinitely divisible laws without the first moment. We also determine the
domain of means, advancing the understanding of Cauchy-Stieltjes kernel
families also for compactly supported generating measures
Winning quick and dirty: the greedy random walk
As a strategy to complete games quickly, we investigate one-dimensional
random walks where the step length increases deterministically upon each return
to the origin. When the step length after the kth return equals k, the
displacement of the walk x grows linearly in time. Asymptotically, the
probability distribution of displacements is a purely exponentially decaying
function of |x|/t. The probability E(t,L) for the walk to escape a bounded
domain of size L at time t decays algebraically in the long time limit, E(t,L)
~ L/t^2. Consequently, the mean escape time ~ L ln L, while ~
L^{2n-1} for n>1. Corresponding results are derived when the step length after
the kth return scales as k^alpha$ for alpha>0.Comment: 7 pages, 6 figures, 2-column revtext4 forma
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
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