Celebratio Mathematica

E. D. Andjel, M. D. Bramson, and T. M. Liggett: “Shocks in the asymmetric exclusion process,” Probab. Theory Relat. Fields 78 : 2 (June 1988), pp. 231–247. MR 945111 Zbl 0632.60107 article

In this paper, we consider limit theorems for the asymmetric nearest neighbor exclusion process on the integers. The initial distribution is a product measure with asymptotic density \( \lambda \) at \( -\infty \) and \( \rho \) at \( +\infty \). Earlier results described the limiting behavior in all cases except for \( 0 < \lambda < 1/2 \), \( \lambda+\rho = 1 \). Here we treat the exceptional case, which is more delicate. It corresponds to the one in which a shock wave occurs in an associated partial differential equation. In the cases treated earlier, the limit was an extremal invariant measure. By contrast, in the present case the limit is a mixture of two invariant measures. Our theorem resolves a conjecture made by the third author in [1975]. The convergence proof is based on coupling and symmetry considerations.

Enrique Daniel Andjel	Related
Maury D. Bramson	Related

T. M. Liggett: “Branching random walks on finite trees,” pp. 315–330 in Perplexing problems in probability: Festschrift in honor of Harry Kesten. Edited by M. Bramson and R. Durrett. Progress in Probability 44. Birkhäuser (Boston), 1999. MR 1703138 Zbl 0948.60097 incollection

Richard Timothy Durrett	Related
Maury D. Bramson	Related
Harry Kesten	Related

M. Bramson, T. M. Liggett, and T. Mountford: “Characterization of stationary measures for one-dimensional exclusion processes,” Ann. Probab. 30 : 4 (2002), pp. 1539–1575. MR 1944000 Zbl 1039.60086 article

The product Bernoulli measures \( \nu_{\alpha} \) with densities \( \alpha \), \( \alpha\in [0,1] \), are the extremal translation invariant stationary measures for an exclusion process on \( \mathbb{Z} \) with irreducible random walk kernel \( p(\cdot) \). Stationary measures that are not translation invariant are known to exist for finite range \( p(\cdot) \) with positive mean. These measures have particle densities that tend to 1 as \( x\to\infty \) and tend to 0 as \( x\to -\infty \); the corresponding extremal measures form a one-parameter family and are translates of one another. Here, we show that for an exclusion process where \( p(\cdot) \) is irreducible and has positive mean, there are no other extremal stationary measures. When \[ \sum_{x < 0}x^2p(x) = \infty ,\] we show that any nontranslation invariant stationary measure is not a blocking measure; that is, there are always either an infinite number of particles to the left of any site or an infinite number of empty sites to the right of the site. This contrasts with the case where \( p(\cdot) \) has finite range and the above stationary measures are all blocking measures. We also present two results on the existence of blocking measures when \( p(\cdot) \) has positive mean, and \[ p(y)\leq p(x) \quad\text{and}\quad p(-y)\leq p(-x) \] for \( 1\leq x\leq y \). When the left tail of \( p(\cdot) \) has slightly more than a third moment, stationary blocking measures exist. When \( p(-x)\leq p(x) \) for \( x > 0 \) and \[ \sum_{x < 0}x^2p(x) = \infty ,\] stationary blocking measures also exist.

Maury D. Bramson	Related
Thomas Simon Mountford	Related

M. Bramson and T. M. Liggett: “Exclusion processes in higher dimensions: Stationary measures and convergence,” Ann. Probab. 33 : 6 (2005), pp. 2255–2313. MR 2184097 Zbl 1099.60067 article

There has been significant progress recently in our understanding of the stationary measures of the exclusion process on \( \mathbb{Z} \). The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on \( \mathbb{Z}^d \) and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on \( \mathbb{Z}^d \). In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as \( t\to\infty \) for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.

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Thomas Milton Liggett

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