2008, Birkner, Matthias, Sun, Rongfeng

We study a random walk pinning model, where conditioned on a simple random walk $Y$ on $Z^d$ acting as a random medium, the path measure of a second independent simple random walk $X$ up to time $t$ is Gibbs transformed with Hamiltonian $-L_t(X,Y)$, where $L_t(X,Y)$ is the collision local time between $X$ and $Y$ up to time $t$. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with Brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits a localization-delocalization transition as the inverse temperature $beta$ varies. We show that in dimensions $d=1,2$, the annealed and quenched critical values of $beta$ are both 0, while in dimensions $dgeq 4$, the quenched critical value of $beta$ is strictly larger than the annealed critical value (which is positive). This implies the existence of certain intermediate regimes for the parabolic Anderson model with Brownian noise and the directed polymer model. For $dgeq 5$, the same result has recently been established by Birkner, Greven and den Hollander via a quenched large deviation principle. Our proof is based on a fractional moment method used recently by Derrida, Giacomin, Lacoin and Toninelli to establish the non-coincidence of annealed and quenched critical points for the pinning model in the disorder-relevant regime. The critical case $d=3$ remains open.

2007, Birkner, Matthias, Blath, Jochen

We review recent progress in the understanding of the interplay between population models, measure-valued diffusions, general coalescent processes and inference methods for evolutionary parameters in population genetics. Along the way, we will discuss the powerful and intuitive (modified) lookdown construction of Donnelly and Kurtz, Pitman's and Sagitov's $Lambda$-coalescents as well as recursions and Monte Carlo schemes for likelihood-based inference of evolutionary parameters based on observed genetic types.

2006, Birkner, Matthias, Depperschmidt, Andrej

We study a discrete time spatial branching system on Zd with logistic-type local regulation at each deme depending on a weighted average of the population in neighbouring demes. We show that the system survives for all time with positive probability if the competition term is small enough. For a restricted set of parameter values, we also obtain uniqueness of the non-trivial equilibrium and complete convergence, as well as long-term coexistence in a related two-type model.

2008, Birkner, Matthias, Greven, Andreas, den Hollander, Frank

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere ...

2007, Birkner, Matthias, Blath, Jochen

We show how Donnelly and Kurtz' (modified) lookdown construction for measure-valued processes can be used to analyse the longterm- and scaling properties of spatially stable generalised $Lambda$-Fleming Viot processes, exhibiting a rare ``natural'' example of a scaling family converging in f.d.d. sense, but not in any of Skorohod's topologies on path space. This completes results of Fleischmann and Wachtel (2004) about the spatial Neveu process and complements results of Dawson and Hochberg (1982) about the classical Fleming Viot process. The lookdown construction provides an elegant machinery and clear intuition to describe the path properties of the family in terms of a ``flicker effect'', clarifying ``what can go wrong.''

2007, Birkner, Matthias, Blath, Jochen

One of the main problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of a classical Fleming-Viot process, then the standard means to describe the corresponding genealogy is Kingman's coalescent. For this process, powerful inference methods are well-established. An important feature of this class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised Lambda Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called Lambda-coalescents. Moreover, Eldon & Wakeley (2006) have provided evidence that such more general coalescents, which allow m ultiple collisions, might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier & Griffiths (1987) and Griffiths & Tavaré (1994) to obtain a likelihood based inference method for general Lambda-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of Lambda-coalescents, the parametrisable sub-family of Beta$(2-alpha,alpha)$-coalescents, where $alpha in (1,2]$, are of particular biological relevance. We apply our method in this case to simulated and real data (taken from Árnason (2004)). We conclude that for populations with extreme reproductive behaviour, the Kingman-coalescent as standard model might have to be replaced by more general coalescents, in particular by Beta$(2-alpha,alpha)$-coalescents.

2007, Birkner, Matthias, Blath, Jochen

One of the central problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of the classical Fleming-Viot process, such as the Wright-Fisher- or the Moran model, then the standard means to describe its genealogy is Kingman's coalescent. For this coalescent process, powerful inference methods are well-established. An important feature of the above class of models is, roughly speaking, that the number of offspring of each individual is small when compared to the total population size, and hence all ancestral collisions are binary only. Recently, more general population models have been studied, in particular in the domain of attraction of so-called generalised $Lambda$-Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called $Lambda$-coalescents, which allow multiple collisions. Moreover, Eldon and Wakeley (2006) provide evidence that such more general coalescents might actually be more adequate to describe real populations with extreme reproductive behaviour, in particular many marine species. In this paper, we extend methods of Ethier and Griffiths (1987) and Griffiths and Tavaré (1994, 1995) to obtain a likelihood based inference method for general $Lambda$-coalescents. In particular, we obtain a method to compute (approximate) likelihood surfaces for the observed type probabilities of a given sample. We argue that within the (vast) family of $Lambda$-coalescents, the parametrisable sub-family of Beta$(2-alpha, alpha)$-coalescents, where $alpha in (1,2]$, are of particular relevance. We illustrate our method using simulated datasets, thus obtaining maximum-likelihood estimators of mutation and demographic parameters.

2007, Birkner, Matthias

We consider branching random walks in $d ge 3$ with a Lipschitz branching rate function and show that the system, starting either in a Poisson field or in equilibrium, decorrelates over long time horizons, and employ this to obtain an ergodic theorem. We use coupling and a stochastic representation of the Palm distribution.

2008, Baake, Michael, Birkner, Matthias, Moody, Robert V.

Stochastic point sets are considered that display a diffraction spectrum of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. Several pairs of autocorrelation and diffraction measures are discussed that show a duality structure that may be viewed as analogues of the Poisson summation formula for lattice Dirac combs.

2008, Birkner, Matthias, Blath, Jochen, Möhle, Martin, Steinrücken, Matthias, Tams, Johanna

Let $Lambda$ be a finite measure on the unit interval. A $Lambda$-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions ($Lambda$-coalescent) in analogy to the duality known for the classical Fleming Viot process and Kingman's coalescent, where $Lambda$ is the Dirac measure in $0$. We explicitly construct a dual process of the coalescent with simultaneous multiple collisions ($Xi$-coalescent) with mutation, the $Xi$-Fleming-Viot process with mutation, and provide a representation based on the empirical measure of an exchangeable particle system along the lines of Donnelly and Kurtz (1999). We establish pathwise convergence of the approximating systems to the limiting $Xi$-Fleming-Viot process with mutation. An alternative construction of the semigroup based on the Hille-Yosida theorem is provided and various types of duality of the processes are discussed. In the last part of the paper a populations is considered which undergoes recurrent bottlenecks. In this scenario, non-trivial $Xi$-Fleming-Viot processes naturally arise as limiting models.