117,561 research outputs found
Percolation games, probabilistic cellular automata, and the hard-core model
Let each site of the square lattice be independently assigned
one of three states: a \textit{trap} with probability , a \textit{target}
with probability , and \textit{open} with probability , where
. Consider the following game: a token starts at the origin, and two
players take turns to move, where a move consists of moving the token from its
current site to either or . A player who moves the token
to a trap loses the game immediately, while a player who moves the token to a
target wins the game immediately. Is there positive probability that the game
is \emph{drawn} with best play -- i.e.\ that neither player can force a win?
This is equivalent to the question of ergodicity of a certain family of
elementary one-dimensional probabilistic cellular automata (PCA). These
automata have been studied in the contexts of enumeration of directed lattice
animals, the golden-mean subshift, and the hard-core model, and their
ergodicity has been noted as an open problem by several authors. We prove that
these PCA are ergodic, and correspondingly that the game on has
no draws.
On the other hand, we prove that certain analogous games \emph{do} exhibit
draws for suitable parameter values on various directed graphs in higher
dimensions, including an oriented version of the even sublattice of
in all . This is proved via a dimension reduction to a
hard-core lattice gas in dimension . We show that draws occur whenever the
corresponding hard-core model has multiple Gibbs distributions. We conjecture
that draws occur also on the standard oriented lattice for
, but here our method encounters a fundamental obstacle.Comment: 35 page
Dipolar Bose-Einstein condensate of Stationary-Light Dark-state Polaritons
We put forward and discuss in detail a scheme to achieve Bose-Einstein
condensation of stationary-light dark-state polaritons with dipolar
interaction. To this end we have introduced a diamond-like coupling scheme in a
vapor of Rydberg atoms under the frozen gas approximation. To determine the
system's dynamics we employ normal modes and identify the dark-state polariton
corresponding to one of the modes. We show that in contrast to atomic dipolar
ultra-cold vapors dark-state polariton Bose-Einstein condensates proposed here
can be stable for a negative dipolar interaction constant.Comment: 5 pages, 1 figur
ENVIRONMENTAL AND ECONOMIC ASPECTS OF RECYCLING LIVESTOCK WASTES--ALGAE PRODUCTION USING WASTE PRODUCTS
Environmental Economics and Policy, Livestock Production/Industries,
Exact and Fast Numerical Algorithms for the Stochastic Wave Equation
On the basis of integral representations we propose fast numerical methods to solve the Cauchy problem for the stochastic wave equation without boundaries and with the Dirichlet boundary conditions. The algorithms are exact in a probabilistic sense
A local moment approach to the degenerate Anderson impurity model
The local moment approach is extended to the orbitally-degenerate [SU(2N)]
Anderson impurity model (AIM). Single-particle dynamics are obtained over the
full range of energy scales, focussing here on particle-hole symmetry in the
strongly correlated regime where the onsite Coulomb interaction leads to
many-body Kondo physics with entangled spin and orbital degrees of freedom. The
approach captures many-body broadening of the Hubbard satellites, recovers the
correct exponential vanishing of the Kondo scale for all N, and its universal
scaling spectra are found to be in very good agreement with numerical
renormalization group (NRG) results. In particular the high-frequency
logarithmic decays of the scaling spectra, obtained here in closed form for
arbitrary N, coincide essentially perfectly with available numerics from the
NRG. A particular case of an anisotropic Coulomb interaction, in which the
model represents a system of N `capacitively-coupled' SU(2) AIMs, is also
discussed. Here the model is generally characterised by two low-energy scales,
the crossover between which is seen directly in its dynamics.Comment: 23 pages, 7 figure
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