1,654 research outputs found
Stochastic Flips on Two-letter Words
This paper introduces a simple Markov process inspired by the problem of
quasicrystal growth. It acts over two-letter words by randomly performing
\emph{flips}, a local transformation which exchanges two consecutive different
letters. More precisely, only the flips which do not increase the number of
pairs of consecutive identical letters are allowed. Fixed-points of such a
process thus perfectly alternate different letters. We show that the expected
number of flips to converge towards a fixed-point is bounded by in the
worst-case and by in the average-case, where denotes the
length of the initial word.Comment: ANALCO'1
Distances on Rhombus Tilings
The rhombus tilings of a simply connected domain of the Euclidean plane are
known to form a flip-connected space (a flip is the elementary operation on
rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi).
Motivated by the study of a quasicrystal growth model, we are here interested
in better understanding how "tight" rhombus tiling spaces are flip-connected.
We introduce a lower bound (Hamming-distance) on the minimal number of flips to
link two tilings (flip-distance), and we investigate whether it is sharp. The
answer depends on the number n of different edge directions in the tiling:
positive for n=3 (dimer tilings) or n=4 (octogonal tilings), but possibly
negative for n=5 (decagonal tilings) or greater values of n. A standard proof
is provided for the n=3 and n=4 cases, while the complexity of the n=5 case led
to a computer-assisted proof (whose main result can however be easily checked
by hand).Comment: 18 pages, 9 figures, submitted to Theoretical Computer Science
(special issue of DGCI'09
Glassy Phase of Optimal Quantum Control
We study the problem of preparing a quantum many-body system from an initial
to a target state by optimizing the fidelity over the family of bang-bang
protocols. We present compelling numerical evidence for a universal
spin-glass-like transition controlled by the protocol time duration. The glassy
critical point is marked by a proliferation of protocols with close-to-optimal
fidelity and with a true optimum that appears exponentially difficult to
locate. Using a machine learning (ML) inspired framework based on the manifold
learning algorithm t-SNE, we are able to visualize the geometry of the
high-dimensional control landscape in an effective low-dimensional
representation. Across the transition, the control landscape features an
exponential number of clusters separated by extensive barriers, which bears a
strong resemblance with replica symmetry breaking in spin glasses and random
satisfiability problems. We further show that the quantum control landscape
maps onto a disorder-free classical Ising model with frustrated nonlocal,
multibody interactions. Our work highlights an intricate but unexpected
connection between optimal quantum control and spin glass physics, and shows
how tools from ML can be used to visualize and understand glassy optimization
landscapes.Comment: Modified figures in appendix and main text (color schemes). Corrected
references. Added figures in SI and pseudo-cod
Markovian dynamics of concurrent systems
Monoid actions of trace monoids over finite sets are powerful models of
concurrent systems---for instance they encompass the class of 1-safe Petri
nets. We characterise Markov measures attached to concurrent systems by
finitely many parameters with suitable normalisation conditions. These
conditions involve polynomials related to the combinatorics of the monoid and
of the monoid action. These parameters generalise to concurrent systems the
coefficients of the transition matrix of a Markov chain.
A natural problem is the existence of the uniform measure for every
concurrent system. We prove this existence under an irreducibility condition.
The uniform measure of a concurrent system is characterised by a real number,
the characteristic root of the action, and a function of pairs of states, the
Parry cocyle. A new combinatorial inversion formula allows to identify a
polynomial of which the characteristic root is the smallest positive root.
Examples based on simple combinatorial tilings are studied.Comment: 35 pages, 6 figures, 33 reference
On the problem of the relation between phason elasticity and phason dynamics in quasicrystals
It has recently been claimed that the dynamics of long-wavelength phason
fluctuations has been observed in i-AlPdMn quasicrystals. We will show that the
data reported call for a more detailed development of the elasticity theory of
Jaric and Nelsson in order to determine the nature of small phonon-like atomic
displacements with a symmetry that follows the phason elastic constants. We
also show that a simple model with a single diffusing tile is sufficient to
produce a signal that (1) is situated at a "satellite position'' at a distance
q from each Bragg peak, that (2) has an intensity that scales with the
intensity of the corresponding Bragg peak, (3) falls off as 1/q-squared and (4)
has a time decay constant that is proportional to 1/(D q-squared). It is thus
superfluous to call for a picture of "phason waves'' in order to explain such
data, especially as such "waves'' violate many physical principles.Comment: 36 pages, 0 figures, discussion about vacancies, fluctuating Fourier
components, and difference between static and dynamical structure factors
added, other addition
Local energy approach to the dynamic glass transition
We propose a new class of phenomenological models for dynamic glass
transitions. The system consists of an ensemble of mesoscopic regions to which
local energies are allocated. At each time step, a region is randomly chosen
and a new local energy is drawn from a distribution that self-consistently
depends on the global energy of the system. Then, the transition is accepted or
not according to the Metropolis rule. Within this scheme, we model an energy
threshold leading to a mode-coupling glass transition as in the p-spin model.
The glassy dynamics is characterized by a two-step relaxation of the energy
autocorrelation function. The aging scaling is fully determined by the
evolution of the global energy and linear violations of the fluctuation
dissipation relation are found for observables uncorrelated with the energies.
Interestingly, our mean-field approach has a natural extension to finite
dimension, that we briefly discuss.Comment: 4 pages, 5 figure
Nonasymptotic noisy lossy source coding
This paper shows new general nonasymptotic achievability and converse bounds
and performs their dispersion analysis for the lossy compression problem in
which the compressor observes the source through a noisy channel. While this
problem is asymptotically equivalent to a noiseless lossy source coding problem
with a modified distortion function, nonasymptotically there is a noticeable
gap in how fast their minimum achievable coding rates approach the common
rate-distortion function, as evidenced both by the refined asymptotic analysis
(dispersion) and the numerical results. The size of the gap between the
dispersions of the noisy problem and the asymptotically equivalent noiseless
problem depends on the stochastic variability of the channel through which the
compressor observes the source.Comment: IEEE Transactions on Information Theory, 201
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