661 research outputs found

### TASEP hydrodynamics using microscopic characteristics

The convergence of the totally asymmetric simple exclusion process to the
solution of the Burgers equation is a classical result. In his seminal 1981
paper, Herman Rost proved the convergence of the density fields and local
equilibrium when the limiting solution of the equation is a rarefaction fan. An
important tool of his proof is the subadditive ergodic theorem. We prove his
results by showing how second class particles transport the rarefaction-fan
solution, as characteristics do for the Burgers equation, avoiding
subadditivity. In the way we show laws of large numbers for tagged particles,
fluxes and second class particles, and simplify existing proofs in the shock
cases. The presentation is self contained.Comment: 20 pages, 13 figures. This version is accepted for publication in
Probability Surveys, February 20 201

### Regularity of quasi-stationary measures for simple exclusion in dimension d >= 5

We consider the symmetric simple exclusion process on Z^d, for d>= 5, and
study the regularity of the quasi-stationary measures of the dynamics
conditionned on not occupying the origin. For each \rho\in ]0,1[, we establish
uniqueness of the density of quasi-stationary measures in L^2(d\nur), where
\nur is the stationary measure of density \rho. This, in turn, permits us to
obtain sharp estimates for P_{\nur}(\tau>t), where \tau is the first time the
origin is occupied.Comment: 18 pages. Corrections after referee report. To be published in Ann
Proba

### Hitting times for independent random walks on $\mathbb{Z}^d$

We consider a system of asymmetric independent random walks on
$\mathbb{Z}^d$, denoted by $\{\eta_t,t\in{\mathbb{R}}\}$, stationary under the
product Poisson measure $\nu_{\rho}$ of marginal density $\rho>0$. We fix a
pattern $\mathcal{A}$, an increasing local event, and denote by $\tau$ the
hitting time of $\mathcal{A}$. By using a loss network representation of our
system, at small density, we obtain a coupling between the laws of $\eta_t$
conditioned on $\{\tau>t\}$ for all times $t$. When $d\ge3$, this provides
bounds on the rate of convergence of the law of $\eta_t$ conditioned on
$\{\tau>t\}$ toward its limiting probability measure as $t$ tends to infinity.
We also treat the case where the initial measure is close to $\nu_{\rho}$
without being product.Comment: Published at http://dx.doi.org/10.1214/009117906000000106 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues

In the Hammersley-Aldous-Diaconis process infinitely many particles sit in R
and at most one particle is allowed at each position. A particle at x$ whose
nearest neighbor to the right is at y, jumps at rate y-x to a position
uniformly distributed in the interval (x,y). The basic coupling between
trajectories with different initial configuration induces a process with
different classes of particles. We show that the invariant measures for the
two-class process can be obtained as follows. First, a stationary M/M/1 queue
is constructed as a function of two homogeneous Poisson processes, the arrivals
with rate \lambda and the (attempted) services with rate \rho>\lambda. Then put
the first class particles at the instants of departures (effective services)
and second class particles at the instants of unused services. The procedure is
generalized for the n-class case by using n-1 queues in tandem with n-1
priority-types of customers. A multi-line process is introduced; it consists of
a coupling (different from Liggett's basic coupling), having as invariant
measure the product of Poisson processes. The definition of the multi-line
process involves the dual points of the space-time Poisson process used in the
graphical construction of the system. The coupled process is a transformation
of the multi-line process and its invariant measure the transformation
described above of the product measure.Comment: 21 pages, 6 figure

### Fleming-Viot selects the minimal quasi-stationary distribution: The Galton-Watson case

Consider N particles moving independently, each one according to a
subcritical continuous-time Galton-Watson process unless it hits 0, at which
time it jumps instantaneously to the position of one of the other particles
chosen uniformly at random. The resulting dynamics is called Fleming-Viot
process. We show that for each N there exists a unique invariant measure for
the Fleming-Viot process, and that its stationary empirical distribution
converges, as N goes to infinity, to the minimal quasi-stationary distribution
of the Galton-Watson process conditioned on non-extinction.Comment: 25 page

### No phase transition for Gaussian fields with bounded spins

Let a<b, \Omega=[a,b]^{\Z^d} and H be the (formal) Hamiltonian defined on
\Omega by
H(\eta) = \frac12 \sum_{x,y\in\Z^d} J(x-y) (\eta(x)-\eta(y))^2 where
J:\Z^d\to\R is any summable non-negative symmetric function (J(x)\ge 0 for all
x\in\Z^d, \sum_x J(x)<\infty and J(x)=J(-x)). We prove that there is a unique
Gibbs measure on \Omega associated to H. The result is a consequence of the
fact that the corresponding Gibbs sampler is attractive and has a unique
invariant measure.Comment: 7 page

### Escape of mass in zero-range processes with random rates

We consider zero-range processes in ${\mathbb{Z}}^d$ with site dependent jump
rates. The rate for a particle jump from site $x$ to $y$ in ${\mathbb{Z}}^d$ is
given by $\lambda_xg(k)p(y-x)$, where $p(\cdot)$ is a probability in
${\mathbb{Z}}^d$, $g(k)$ is a bounded nondecreasing function of the number $k$
of particles in $x$ and $\lambda =\{\lambda_x\}$ is a collection of i.i.d.
random variables with values in $(c,1]$, for some $c>0$. For almost every
realization of the environment $\lambda$ the zero-range process has product
invariant measures $\{{\nu_{\lambda, v}}:0\le v\le c\}$ parametrized by $v$,
the average total jump rate from any given site. The density of a measure,
defined by the asymptotic average number of particles per site, is an
increasing function of $v$. There exists a product invariant measure ${\nu
_{\lambda, c}}$, with maximal density. Let $\mu$ be a probability measure
concentrating mass on configurations whose number of particles at site $x$
grows less than exponentially with $\|x\|$. Denoting by $S_{\lambda}(t)$ the
semigroup of the process, we prove that all weak limits of $\{\mu
S_{\lambda}(t),t\ge 0\}$ as $t\to \infty$ are dominated, in the natural partial
order, by ${\nu_{\lambda, c}}$. In particular, if $\mu$ dominates ${\nu
_{\lambda, c}}$, then $\mu S_{\lambda}(t)$ converges to ${\nu_{\lambda, c}}$.
The result is particularly striking when the maximal density is finite and the
initial measure has a density above the maximal.Comment: Published at http://dx.doi.org/10.1214/074921707000000300 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Detection of spatial pattern through independence of thinned processes

Let N, N' and N'' be point processes such that N' is obtained from N by
homogeneous independent thinning and N''= N- N'. We give a new elementary proof
that N' and N'' are independent if and only if N is a Poisson point process. We
present some applications of this result to test if a homogeneous point process
is a Poisson point process.Comment: 11 pages, one figur

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