We study two different versions of the simple exclusion process on augmented Galton-Watson trees, the constant speed model and the varying speed model. In both cases, the simple exclusion process starts from an equilibrium distribution with non-vanishing particle density. Moreover, we assume to have initially a particle in the root, the tagged particle. We show for both models that the tagged particle has a positive linear speed and we give explicit formulas for the speeds.
Studying the behavior of the tagged particle in an exclusion process is a classical problem @cite_2 . When the underlying graph is @math , many results were obtained. In the case of translation invariant transition probabilities, a law of large numbers for the position of the tagged particle is known in all regimes @cite_15 @cite_17 . For the @math -dimensional ladder graph, the speed of a tagged particle was studied by Zhang @cite_0 . For the exclusion process on regular trees, @ established a law of large numbers @cite_5 . We obtain their results as a special case. For random environments of the exclusion process, less results are known. In @cite_10 , Chayes and Liggett consider the case of the exclusion process in a one-dimensional i.i.d. random environment. For @math when the transition probabilities are symmetric and not concentrated on the nearest neighbors in the one-dimensional case, Kipnis and Varadhan established a central limit theorem for the tagged particle in their famous paper @cite_18 . Their result was the starting point for a sequence of papers showing central limit theorems for the position of the tagged particle, see @cite_7 @cite_16 for an overview.
@cite_18 @cite_7 @cite_0 @cite_2 @cite_5 @cite_15 @cite_16 @cite_10 @cite_17
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We investigate a tagged particle in the exclusion processes on 1, ..., N ×ℤ d , with different densities in different levels k ×ℤ d , ∀ k. Ignoring the level the tagged particle lying in, we only concern its position in ℤ d , denoted by X t . Note that the whole space is not homogeneous. We define the environment process viewed from the tagged particle, of which X t can be expressed as a functional. It is called the tagged particle process. We show the ergodicity of the tagged particle process, then prove the strong law of large numbers. Furthermore, we show the central limit theorem of X t provided the zero-mean condition. By symmetry, P has eigenvalues 1 = I03 > I381 > ?> I 31xI- 1 2 -1. This paper develops methods for getting upper and lower bounds on 8i3 by comparison with a second reversible chain on the same state space. This extends the ideas introduced in Diaconis and Saloff-Coste (1993), where random walks on finite groups were considered. The bounds involve geometric properties such as the diameter and covering number of an associated graph along the lines of Diaconis and Stroock (1991). The main application gives a sharp upper bound on the second eigenvalue of the symmetric exclusion process. Thus, let S0 be a connected undirected graph with n vertices. For simplicity, we assume in this introduction that SW is regular. To start, r unlabelled particles are placed in an initial configuration, 1 < r < n. At each step, a particle is chosen at random; then one of the neighboring sites of this particle is chosen at random. If the neighboring site is unoccupied, the chosen particle is moved there; if the neighboring site is occupied, the system stays as it was. This is a reversible Markov chain on the r-sets of 1, 2, . . ., n with uniform stationary distribution. Liggett (1985) gives background and motivation (he focuses on infinite systems). Fill (1991) gives bounds on the second eigenvalue of the labeled exclusion process on the finite circle ZZn 1 We study this chain by comparison with a second Markov chain on r-sets that proceeds by picking a particle at random, picking an unoccupied site at random (not necessarily a neighboring site) and moving the particle to the unoccupied site. This is a well studied chain (the Bernoulli-Laplace model for diffusion). Its eigenvalues are known. We show that the comparison techniques apply to give upper bounds on the eigenvalues of the exclusion Abstract At the heart of this article will be the study of a branching Brownian motion (BBM) with killing , where individual particles move as Brownian motions with drift − ρ , perform dyadic branching at rate β and are killed on hitting the origin. Firstly, by considering properties of the right-most particle and the extinction probability, we will provide a probabilistic proof of the classical result that the ‘one-sided’ FKPP travelling-wave equation of speed − ρ with solutions f : [ 0 , ∞ ) → [ 0 , 1 ] satisfying f ( 0 ) = 1 and f ( ∞ ) = 0 has a unique solution with a particular asymptotic when ρ 2 β , and no solutions otherwise. Our analysis is in the spirit of the standard BBM studies of [S.C. Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proc. Roy. Soc. Edinburgh Sect. A 129 (3) (1999) 503–517] and [A.E. Kyprianou, Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis, Ann. Inst. H. Poincare Probab. Statist. 40 (1) (2004) 53–72] and includes an intuitive application of a change of measure inducing a spine decomposition that, as a by product, gives the new result that the asymptotic speed of the right-most particle in the killed BBM is 2 β − ρ on the survival set. Secondly, we introduce and discuss the convergence of an additive martingale for the killed BBM, W λ , that appears of fundamental importance as well as facilitating some new results on the almost-sure exponential growth rate of the number of particles of speed λ ∈ ( 0 , 2 β − ρ ) . Finally, we prove a new result for the asymptotic behaviour of the probability of finding the right-most particle with speed λ > 2 β − ρ . This result combined with Chauvin and Rouault's [B. Chauvin, A. Rouault, KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees, Probab. Theory Related Fields 80 (2) (1988) 299–314] arguments for standard BBM readily yields an analogous Yaglom-type conditional limit theorem for the killed BBM and reveals W λ as the limiting Radon–Nikodým derivative when conditioning the right-most particle to travel at speed λ into the distant future. We consider exclusion processes on a rooted @math -regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For @math , we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process "seen from the tagged particle" has an ergodic invariant measure.