5,630 research outputs found
Extra heads and invariant allocations
Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm
version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists
a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y
of \Pi such that translating \Pi by -Y yields a configuration whose law is that
of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of
\Pi, and prove that there is no shift coupling in which \Pi is a function of
\Pi^*. The key ingredient is a deterministic translation-invariant rule to
allocate sets of equal volume (forming a partition of R^d) to the points of
\Pi. The construction is based on the Gale-Shapley stable marriage algorithm
[Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random
element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A
shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show
that there exists an extra head scheme which is a function of \Gamma if and
only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law
of \Gamma is product measure and d\geq3, we prove that there exists an extra
head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of
Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].Comment: Published at http://dx.doi.org/10.1214/009117904000000603 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Optimal distinction between non-orthogonal quantum states
Given a finite set of linearly independent quantum states, an observer who
examines a single quantum system may sometimes identify its state with
certainty. However, unless these quantum states are orthogonal, there is a
finite probability of failure. A complete solution is given to the problem of
optimal distinction of three states, having arbitrary prior probabilities and
arbitrary detection values. A generalization to more than three states is
outlined.Comment: 9 pages LaTeX, one PostScript figure on separate pag
Infinite matrices may violate the associative law
The momentum operator for a particle in a box is represented by an infinite
order Hermitian matrix . Its square is well defined (and diagonal),
but its cube is ill defined, because . Truncating these
matrices to a finite order restores the associative law, but leads to other
curious results.Comment: final version in J. Phys. A28 (1995) 1765-177
Quenched exit times for random walk on dynamical percolation
We consider random walk on dynamical percolation on the discrete torus
. In previous work, mixing times of this process for
were obtained in the annealed setting where one averages
over the dynamical percolation environment. Here we study exit times in the
quenched setting, where we condition on a typical dynamical percolation
environment. We obtain an upper bound for all which for matches the
known lower bound
The most probable wave function of a single free moving particle
We develop the most probable wave functions for a single free quantum
particle given its momentum and energy by imposing its quantum probability
density to maximize Shannon information entropy. We show that there is a class
of solutions in which the quantum probability density is self-trapped with
finite-size spatial support, uniformly moving hence keeping its form unchanged.Comment: revtex, 4 page
Cutoff for the noisy voter model
Given a continuous time Markov Chain on a finite set , the
associated noisy voter model is the continuous time Markov chain on
, which evolves in the following way: (1) for each two sites and
in , the state at site changes to the value of the state at site
at rate ; (2) each site rerandomizes its state at rate 1. We show that
if there is a uniform bound on the rates and the corresponding
stationary distributions are almost uniform, then the mixing time has a sharp
cutoff at time with a window of order 1. Lubetzky and Sly proved
cutoff with a window of order 1 for the stochastic Ising model on toroids; we
obtain the special case of their result for the cycle as a consequence of our
result. Finally, we consider the model on a star and demonstrate the surprising
phenomenon that the time it takes for the chain started at all ones to become
close in total variation to the chain started at all zeros is of smaller order
than the mixing time.Comment: Published at http://dx.doi.org/10.1214/15-AAP1108 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Extendable self-avoiding walks
The connective constant mu of a graph is the exponential growth rate of the
number of n-step self-avoiding walks starting at a given vertex. A
self-avoiding walk is said to be forward (respectively, backward) extendable if
it may be extended forwards (respectively, backwards) to a singly infinite
self-avoiding walk. It is called doubly extendable if it may be extended in
both directions simultaneously to a doubly infinite self-avoiding walk. We
prove that the connective constants for forward, backward, and doubly
extendable self-avoiding walks, denoted respectively by mu^F, mu^B, mu^FB,
exist and satisfy mu = mu^F = mu^B = mu^FB for every infinite, locally finite,
strongly connected, quasi-transitive directed graph. The proofs rely on a 1967
result of Furstenberg on dimension, and involve two different arguments
depending on whether or not the graph is unimodular.Comment: Accepted versio
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