Genealogies of Interacting Particle Systems

The following sections are included:

• Contents
• Foreword
• Preface

We discuss some stochastic spatial generalizations of the Lotka–Volterra model for competing species. The generalizations take the forms of spin systems on general discrete sets and interacting diffusions on integer lattices. Methods for proving coexistence in these generalizations and some related open questions are discussed. We use duality as the central point of view. It relates coexistence of the models to survival of their dual processes.

We survey results on the description of stochastically evolving genealogies of populations and marked genealogies of multitype populations or spatial populations via tree-valued Markov processes on (marked) ultrametric measure spaces. In particular we explain the choice of state spaces and their topologies, describe the dynamics of genealogical Fleming-Viot and branching models by well-posed martingale problems, and formulate the typical results on the longtime behavior. Furthermore we discuss the basic techniques of proofs and sketch as two key tools of analysis the different forms of duality and the Girsanov transformation.

This survey article gives an elementary introduction to the algebraic approach to Markov process duality, as opposed to the pathwise approach. In the algebraic approach, a Markov generator is written as the sum of products of simpler operators, which each have a dual with respect to some duality function. We discuss at length the recent suggestion by Giardinà, Redig, and others, that it may be a good idea to choose these simpler operators in such a way that they form an irreducible representation of some known Lie algebra. In particular, we collect the necessary background on representations of Lie algebras that is crucial for this approach. We also discuss older work by Lloyd and Sudbury on duality functions of product form and the relation between intertwining and duality.

Branching Brownian motion (BBM) is used as a population model where the location of a particle on the real line represents his or her fitness which evolves according to a Brownian motion due to mutations. In this context, the genealogical tree of the particles is important to understand the evolution of the population over time.

The main goal of this article is to give an overview and motivations for some of the results on the genealogy of extremal particles of BBM.

In the first part of this article, we discuss the results of L.-P. Arguin, A. Bovier and N. Kistler from [Genealogy of Extremal Particles of BBM, 2011] and [Poissonian Statistics in the Extremal Process of BBM, 2012]. Their articles give a partial answer to what the distribution of the limiting extremal process of BBM looks like. These two articles highlight, how genetical information can successfully be used to obtain rather extensive information for the limiting particle picture.

In the second part, we discuss the results of J. Berestycki, N. Berestycki and J. Schweinsberg from [The Genealogy of BBM with Absorption, 2013]. BBM with absorption is related to a population model with selection. This article shows that the genealogy of the surviving particles (with a selective advantage), which are ‘local’ extremal particles, is then governed by a Bolthausen-Sznitman coalescent process. This result is in sharp contrast to the standard population models without selection.

We study Markov processes with values in the space of general two-dimensional arrays whose distribution is exchangeable. The results of this paper are inspired by the theory of exchangeable dynamical random graphs developed by H. Crane in [4, 5].

A radix sort tree arises when storing distinct infinite binary words in the leaves of a binary tree such that for any two words their common prefixes coincide with the common prefixes of the corresponding two leaves. If one deletes the out-degree 1 vertices in the radix sort tree and “closes up the gaps”, then the resulting PATRICIA tree maintains all the information that is necessary for sorting the infinite words into lexicographic order.

We investigate the PATRICIA chains — the tree-valued Markov chains that arise when successively building the PATRICIA trees for the collection of infinite binary words Z₁, … ,Z_n, n = 1, 2, …, where the source words Z₁, Z₂, … are independent and have a common diffuse distribution on {0, 1}^∞. It turns out that the PATRICIA chains share a common collection of backward transition probabilities and that these are the same as those of a chain introduced by Rémy for successively generating uniform random binary trees with larger and larger numbers of leaves. This means that the infinite bridges of any PATRICIA chain (that is, the chains obtained by conditioning a PATRICIA chain on its remote future) coincide with the infinite bridges of the Rémy chain. The infinite bridges of the Rémy chain are characterized concretely in Evans, Grübel, and Wakolbinger (2017) and we recall that characterization here while adding some details and clarifications.

Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi, Newman and Ravishankar (2004). One of their convergence condition, called (B2), states that the probability of the event that there exist three distinct paths for a time interval of length t (> 0), all starting within a segment of length ε, is of small order of ε. This condition is often verified by applying an FKG type correlation inequality together with a coalescing time tail estimate for two paths. For many models where paths have complex interactions, it is hard to establish FKG type inequalities. In this article, we show that for a non-crossing path model, with certain assumptions, a suitable upper bound on expected first collision time among three paths can be obtained directly using Lyapunov functions. This, in turn, provides an alternate verification of Condition (B2). We further show that in case of independent simple symmetric one-dimensional random walks or in case of independent Brownian motions, the expected value can be computed explicitly. We apply this alternate method of verification of (B2) to several models in the basin of attraction of the Brownian web studied earlier in the literature ([22], [14], [11]).

We record two natural tightness criteria for random sets of càdlàg paths, within the state space introduced by Etheridge et al. in [4]. This state space is the equivalent, for càdlàg paths, of the (continuous path) state space introduced for the Brownian web by Fontes et al. in [6].

We discuss some implications of these criteria, in particular for cases of coalescing random walkers with heavy tailed jumps, in which tightness fails.

We consider the contact process on a countable-infinite and connected graph of bounded degree. Our focus is on its mixing properties in the supercritical regime. In particular, we prove that the projection of the stationary contact process onto a finite subset forms a process which is ϕ-mixing whenever the infection parameter is sufficiently large. The proof of this is based on large deviation estimates for the spread of an infection and general correlation inequalities. In the special case of the contact process on ℤ^d, d ≥ 1, we furthermore show the cutoff phenomenon, valid in the entire supercritical regime.

We review the results in Chen and Yao [5], [6] which concerns the contact process in a static random environment on the half space ℤ^d × ℤ⁺ and make some additions to them. Furthermore, we explain why our methods cannot apply to the whole space case and compare our results with some related works.