Genealogies of Interacting Particle Systems

An ancestral recombination graph for diploid populations with skewed offspring distribution

A large offspring number diploid biparental multilocus population model of Moran type is our object of study. At each timestep, a pair of diploid individuals drawn uniformly at random contribute offspring to the population. The number of offspring can be large relative to the total population size. Similar `heavily skewed' reproduction mechanisms have been considered by various authors recently. Each diploid parental individual contributes exactly one chromosome to each diploid offspring, and hence ancestral lineages can only coalesce when in distinct individuals. A separation of timescales phenomenon is thus observed. A result of M\"{o}hle (1998) is extended to obtain convergence of the an…

Analysis of DNA sequence variation within marine species using Beta-coalescents

We apply recently developed inference methods based on general coalescent processes to DNA sequence data obtained from various marine species. Several of these species are believed to exhibit so-called shallow gene genealogies, potentially due to extreme reproductive behaviour, e.g. via Hedgecock's "reproduction sweepstakes". Besides the data analysis, in particular the inference of mutation rates and the estimation of the (real) time to the most recent common ancestor, we briefly address the question whether the genealogies might be adequately described by so-called Beta coalescents (as opposed to Kingman's coalescent), allowing multiple mergers of genealogies. The choice of the underlying…

One-dimensional random walks with self-blocking immigration

We consider a system of independent one-dimensional random walkers where new particles are added at the origin at fixed rate whenever there is no older particle present at the origin. A Poisson ansatz leads to a semi-linear lattice heat equation and predicts that starting from the empty configuration the total number of particles grows as $c \sqrt{t} \log t$. We confirm this prediction and also describe the asymptotic macroscopic profile of the particle configuration.

Non-Periodic Systems with Continuous Diffraction Measures

The present state of mathematical diffraction theory for systems with continuous spectral components is reviewed and extended. We begin with a discussion of various characteristic examples with singular or absolutely continuous diffraction, and then continue with a more general exposition of a systematic approach via stationary stochastic point processes. Here, the intensity measure of the Palm measure takes the role of the autocorrelation measure in the traditional approach. We furthermore introduce a ‘Palm-type’ measure for general complex-valued random measures that are stationary and ergodic, and relate its intensity measure to the autocorrelation measure.

Coalescing directed random walks on the backbone of a 1 +1-dimensional oriented percolation cluster converge to the Brownian web

We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 +1. A directed random walk on this backbone can be seen as an "ancestral line" of an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in [BCDG13] where a central limit theorem for a single walker was proved. Here, we consider infinitely many coalescing walkers on the same backbone starting at each space-time point. We show that, after diffusive rescaling, the collection of paths converges in distribution to the Brownian web. Hence, we prove convergence to the Brownian web for a particular system of coalescing random…

Probabilities of large values for sums of i.i.d. non-negative random variables with regular tail of index $-1$

Let $\xi_1, \xi_2, \dots$ be i.i.d. non-negative random variables whose tail varies regularly with index $-1$, let $S_n$ be the sum and $M_n$ the largest of the first $n$ values. We clarify for which sequences $x_n\to\infty$ we have $\mathbb P(S_n \ge x_n) \sim \mathbb P(M_n \ge x_n)$ as $n\to\infty$. Outside this regime, the typical size of $S_n$ conditioned on exceeding $x_n$ is not completely determined by the largest summand and we provide an appropriate correction term which involves the integrated tail of $\xi_1$.

Directed random walk on the backbone of an oriented percolation cluster

We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the ``ancestral lineage'' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e.\ for almost any realisation of the cluster) via an analysis of joint renewals of two independent walks on the same cluster.

Random walks in dynamic random environments and ancestry under local population regulation

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of the random medium on the walk is small in space-time regions where the medium is typical, we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining. Such random walks occur naturally as spatial embeddings of ancestral lineages in spatial population models with local regulation. We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high.

Importance sampling for Lambda-coalescents in the infinitely many sites model

We present and discuss new importance sampling schemes for the approximate computation of the sample probability of observed genetic types in the infinitely many sites model from population genetics. More specifically, we extend the 'classical framework', where genealogies are assumed to be governed by Kingman's coalescent, to the more general class of Lambda-coalescents and develop further Hobolth et. al.'s (2008) idea of deriving importance sampling schemes based on 'compressed genetrees'. The resulting schemes extend earlier work by Griffiths and Tavar\'e (1994), Stephens and Donnelly (2000), Birkner and Blath (2008) and Hobolth et. al. (2008). We conclude with a performance comparison o…

Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon

This article studies large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an $\alpha$-stable law, $\alpha\in (0,2]$, with emphasis on the case $\alpha=2$. There are two different scenarios: either the deviation is realised via a collective behaviour with all summands contributing to the deviation (a Gaussian scenario), or a single summand is atypically large and contributes to the deviation (a one-big-jump scenario). Such results are known when $\alpha \in (0,2)$ (large deviations always follow a one big-jump scenario) or when the random variables admit a moment of order $2+\delta$ for some $\delta>0$. We extend these results, inclu…

Statistical properties of the site-frequency spectrum associated with lambda-coalescents.

Abstract Statistical properties of the site-frequency spectrum associated with Λ-coalescents are our objects of study. In particular, we derive recursions for the expected value, variance, and covariance of the spectrum, extending earlier results of Fu (1995) for the classical Kingman coalescent. Estimating coalescent parameters introduced by certain Λ-coalescents for data sets too large for full-likelihood methods is our focus. The recursions for the expected values we obtain can be used to find the parameter values that give the best fit to the observed frequency spectrum. The expected values are also used to approximate the probability a (derived) mutation arises on a branch subtending a…

Disorder relevance for the random walk pinning model in dimension 3

We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk Y on Z^d with jump rate \rho>0, which plays the role of disorder, the law up to time t of a second independent random walk X with jump rate 1 is Gibbs transformed with weight e^{\beta L_t(X,Y)}, where L_t(X,Y) is the collision local time between X and Y up to time t. As the inverse temperature \beta varies, the model undergoes a localization-delocalization transition at some critical \beta_c>=0. A natural question is whether or not there is disorder relevance, namely whether or not \beta_c differs from the critical point \beta_c^{ann} for the annealed model. In Birkner a…

Properties of modified hepatitis B virus surface antigen particles carrying preS epitopes

The current hepatitis B virus (HBV) vaccines contain the small (S) and middle (M) viral envelope proteins in particulate form but lack the large (L) protein. Although these particles elicit protective immunity to HBV, inclusion of the immunogenic preS1 region of the L protein may enhance their efficacy. To present preS1-derived epitopes on secretable subviral particles we rearranged the HBV envelope ORF by fusing part or all of the preS1 region to either the N or C terminus of the S protein. Fusion of the first 42 residues of preS1 to either site allowed efficient secretion of the modified particles and rendered the linked sequence accessible at the surface of the particle. Conversely, fusi…