18,355 research outputs found
Stochastic Domination and Comb Percolation
There exists a Lipschitz embedding of a d-dimensional comb graph (consisting
of infinitely many parallel copies of Z^{d-1} joined by a perpendicular copy)
into the open set of site percolation on Z^d, whenever the parameter p is close
enough to 1 or the Lipschitz constant is sufficiently large. This is proved
using several new results and techniques involving stochastic domination, in
contexts that include a process of independent overlapping intervals on Z, and
first-passage percolation on general graphs.Comment: 21 page
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
Counselling and psychotherapy orientations in Australia: Responses from 24 Australian psychotherapists
The Jammed Phase of the Biham-Middleton-Levine Traffic Model
Initially a car is placed with probability p at each site of the
two-dimensional integer lattice. Each car is equally likely to be East-facing
or North-facing, and different sites receive independent assignments. At odd
time steps, each North-facing car moves one unit North if there is a vacant
site for it to move into. At even time steps, East-facing cars move East in the
same way. We prove that when p is sufficiently close to 1 traffic is jammed, in
the sense that no car moves infinitely many times. The result extends to
several variant settings, including a model with cars moving at random times,
and higher dimensions.Comment: 15 pages, 5 figures; revised journal versio
The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity
We give the full supersymmetric and kappa-symmetric actions for the Dirichlet
p-branes, including their coupling to background superfields of ten-dimensional
type IIA and IIB supergravity.Comment: 24 pages, plain TeX. Replacement corrects text positioning in
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