ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 1, pp. 65–73.
Pleiades Publishing, Inc., 2009.
Original Russian Text
V.A. Malyshev, V.A. Shvets, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 1, pp. 71–79.
On Decay of Correlations for an Exclusion
Process with Asymmetric Boundary Conditions
V. A. Malyshev
Faculty of Mathematics and Mechanics, Lomonosov Moscow State University
Pension Fund of the Russian Federation
Received September 11, 2008; in ﬁnal form, December 24, 2008
Abstract—We consider a symmetric exclusion process on a discrete interval of S points with
various boundary conditions at the endpoints. We study the asymptotic decay of correlations
as S →∞. The main result is proving asymptotic independence of a stationary distribution at
points of the interval that are far enough away. We do not use Derrida’s algebraic technique
but develop a new technique, which has a visual probabilistic sense.
Exclusion processes are a long-standing popular subject in both mathematics and theoretical
physics. They are a particular case of processes with local interaction on a lattice. The interest
to them is mainly due to the fact that exclusion processes are the simplest nontrivial model for
collisions in a multiparticle system. Using them, heat conduction , viscosity [2, 3], quantum fer-
romagnet , nonequilibrium process , etc., models are constructed. Even irrespective of physics,
these processes are natural probabilistic objects. The ﬁrst systematic exposition of the relevant
theory was given in the well-known monograph .
The simplest exclusion process is a symmetric exclusion process on the integer lattice Z.This
process is well known to possess a continuum of (spatially) uniform invariant measures, which are
mixtures of Bernoulli measures. The same holds for an exclusion process on a ﬁnite segment of the
lattice if boundary conditions are empty. For other boundary conditions, as a rule, it is easy to
show that there are no invariant Bernoulli measures. Using a matrix method (ansatz; see [5,7]), one
can obtain an explicit form of correlation functions for such a process. This would imply that the
invariant measure is asymptotically Bernoullian. It should be noted that this powerful algebraic
method, similar to the famous Bethe ansatz, is rather cumbersome, is not always mathematically
well-grounded, and has substantial restrictions on its range of applicability. Namely, we are un-
aware of any application where jumps of particles can be longer than 1. It is also important that
the probabilistic nature of the method is absolutely unclear.
In the present paper, we propose another—simple and natural from the probability theory
viewpoint—approach, which can be extended to jumps of lengths greater than 1, as well as to
other boundary conditions. Being quite diﬀerent from the Bethe–Derrida methods, our approach
has something in common with them, namely, a certain recursive procedure. Here we demonstrate
the idea of our approach in the simplest situation. Various generalizations will be considered in