arXiv subjects

Noemi Kurt

Publications and source records attributed to Noemi Kurt.

18 recordsLinked to original sources

arXiv2025

Partially exchangeable Markov chains and characterisation of multitype Lambda-coalescents

In this paper, we study consistent and partially exchangeable sequences of Markov chains on a finite state space. We provide a characterisation of the admissible transition rates via a decomposition into individual and coordinated motion of particles. As a consequence, we find a characterisation of multitype Lambda-coalescents with multiple switches. Moreover, we provide convergence and duality results for the corresponding process of limiting relative frequencies that we call the de Finetti measure process, and discuss a number of examples from the recent literature.

math.PR↗

arXiv2024

Multitype $\Lambda$-coalescents and continuous state branching processes

We provide new connections between multitype $\Lambda$-coalescents and multitype continuous state branching processes via duality and a homeomorphism on their parameter space. The approach is based on a sequential sampling procedure for the frequency process of independent CSBPs, and provides forward and backward processes for multitype population models under $\Lambda$-type reproduction. It provides some insight on different approaches to generalise $\Lambda$-coalescents to the multitype setup.

math.PR↗

arXiv2024

Renewal contact process with dormancy

We consider the contact process with dormancy, where wake-up times follow a renewal process. Without infection between dormant individuals, we show that the process under certain conditions grows at most logarithmically. On the other hand, if infections between dormant individuals are possible, the process survives with positive probability even on finite graphs.

math.PR↗

arXiv2023

The ancestral selection graph for a $\Lambda$-asymmetric Moran model

Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanims, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure $\Lambda$ for each of the two types. Our approach consists of constructing a $\Lambda$-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Under certain conditions, that we call the "partial order of adaptation", we can couple these measures. This allows us to construct the central object of this paper, the $\Lambda-$asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time $\Lambda$-asymmetric Moran model with its ancestral process. Interestingly, the construction also provides a connection to the theory of optimal transport. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.

math.PR↗

arXiv2021

Particle systems with coordination

We consider a generalization of spatial branching coalescing processes in which the behaviour of individuals is not (necessarily) independent, on the contrary, individuals tend to take simultaneous actions. We show that these processes have moment duals, which happen to be multidimensional diffusions with jumps. Moment duality provides a general framework to study structural properties of the processes in this class. We present some conditions under which the expectation of the process is not affected by coordination and comment on the effect of coordination on the variance. We analyse several examples in more detail, including the nested coalescent, the peripatric coalescent with selection and coordinated migration, and the Parabolic Anderson Model.

math.PR↗

arXiv2020

Population genetic models of dormancy

In the present article, we investigate the effects of dormancy on an abstract population genetic level. We first provide a short review of seed bank models in population genetics, and the role of dormancy for the interplay of evolutionary forces in general, before we discuss two recent paradigmatic models, referring to spontaneous resp. simultaneous switching of individuals between the active and the dormant state. We show that both mechanisms give rise to non-trivial mathematical objects, namely the (continuous) seed bank diffusion and the seed bank diffusion with jumps, as well as their dual processes, the seed bank coalescent and the seed bank coalescent with simultaneous switching.

math.PR↗

arXiv2018

The seed bank coalescent with simultaneous switching

We introduce a new Wright-Fisher type model for seed banks incorporating "simultaneous switching", which is motivated by recent work on microbial dormancy. We show that the simultaneous switching mechanism leads to a new jump-diffusion limit for the scaled frequency processes, extending the classical Wright-Fisher and seed bank diffusion limits. We further establish a new dual coalescent structure with multiple activation and deactivation events of lineages. While this seems reminiscent of multiple merger events in general exchangeable coalescents, it actually leads to an entirely new class of coalescent processes with unique qualitative and quantitative behaviour. To illustrate this, we provide a novel kind of condition for coming down from infinity for these coalescents using recent results of Griffiths.

q-bio.PE↗

arXiv2018

Probability to be positive for the membrane model in dimensions 2 and 3

We consider the membrane model on a box $V_N\subset \mathbb{Z}^n$ of size $(2N+1)^n$ with zero boundary condition in the subcritical dimensions $n=2$ and $n=3$. We show optimal estimates for the probability that the field is positive in a subset $D_N$ of $V_N$. In particular we obtain for $D_N=V_N$ that the probability to be positive on the entire domain is exponentially small and the rate is of the order of the surface area $N^{n-1}$.

math.PR↗

arXiv2017

Exponential decay of covariances for the supercritical membrane model

We consider the membrane model, that is the centered Gaussian field on $\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a $δ-$pinning condition, giving a reward of strength $\varepsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\geq 5$ covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result.

math.PR↗

arXiv2016

A new coalescent for seed-bank models

We identify a new natural coalescent structure, which we call the seed-bank coalescent, that describes the gene genealogy of populations under the influence of a strong seed-bank effect, where "dormant forms" of individuals (such as seeds or spores) may jump a significant number of generations before joining the "active" population. Mathematically, our seed-bank coalescent appears as scaling limit in a Wright-Fisher model with geometric seed-bank age structure if the average time of seed dormancy scales with the order of the total population size $N$. This extends earlier results of Kaj, Krone and Lascoux [J. Appl. Probab. 38 (2011) 285-300] who show that the genealogy of a Wright-Fisher model in the presence of a "weak" seed-bank effect is given by a suitably time-changed Kingman coalescent. The qualitatively new feature of the seed-bank coalescent is that ancestral lineages are independently blocked at a certain rate from taking part in coalescence events, thus strongly altering the predictions of classical coalescent models. In particular, the seed-bank coalescent "does not come down from infinity," and the time to the most recent common ancestor of a sample of size $n$ grows like $\log\log n$. This is in line with the empirical observation that seed-banks drastically increase genetic variability in a population and indicates how they may serve as a buffer against other evolutionary forces such as genetic drift and selection.

math.PR↗

arXiv2016

An individual-based model for the Lenski experiment, and the deceleration of the relative fitness

The Lenski experiment investigates the long-term evolution of bacterial populations. Its design allows the direct comparison of the reproductive fitness of an evolved strain with its founder ancestor. It was observed by Wiser et al. (2013) that the relative fitness over time increases sublinearly, a behaviour which is commonly attributed to effects like clonal interference or epistasis. In this paper we present an individual-based probabilistic model that captures essential features of the design of the Lenski experiment. We assume that each beneficial mutation increases the individual reproduction rate by a fixed amount, which corresponds to the absence of epistasis in the continuous-time (intraday) part of the model, but leads to an epistatic effect in the discrete-time (interday) part of the model. Using an approximation by near-critical Galton-Watson processes, we prove that under some assumptions on the model parameters which exclude clonal interference, the relative fitness process converges, after suitable rescaling, in the large population limit to a power law function.

math.PR↗

arXiv2016

Fast decay of covariances under $δ-$pinning in the critical and supercritical membrane model

We consider the membrane model, that is the centered Gaussian field on $\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a $δ-$pinning condition, giving a reward of strength $\varepsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\geq 4$ covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in $d\geq 5$ and logarithmic in $d=4.$ The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.

math.PR↗

arXiv2014

Genealogy of a Wright Fisher model with strong seed bank component

We investigate the behaviour of the genealogy of a Wright-Fisher population model under the influence of a strong seed-bank effect. More precisely, we consider a simple seed-bank age distribution with two atoms, leading to either classical or long genealogical jumps (the latter modeling the effect of seed-dormancy). We assume that the length of these long jumps scales like a power $N^β$ of the original population size $N$, thus giving rise to a `strong' seed-bank effect. For a certain range of $β$, we prove that the ancestral process of a sample of $n$ individuals converges under a non-classical time-scaling to Kingman's $n-$coalescent. Further, for a wider range of parameters, we analyze the time to the most recent common ancestor of two individuals analytically and by simulation.

math.PR↗

arXiv2014

On the notion(s) of duality for Markov processes

We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory and give functional analytic results including existence and uniqueness criteria and a comparison of the spectra of dual semi-groups. The analytic framework builds on the notion of dual pairs, convex geometry, and Hilbert spaces. In addition, we formalize the notion of pathwise duality as it appears in population genetics and interacting particle systems. We discuss the relation of duality with rescalings, stochastic monotonicity, intertwining, symmetries, and quantum many-body theory, reviewing known results and establishing some new connections.

math.PR↗

arXiv2013

The ancestral process of long term seed bank models

We present a new model for seed banks, where direct ancestors of individuals may have lived in the near as well as the very far past. The classical Wright-Fisher model, as well as a seed bank model with bounded age distribution considered by Kaj, Krone and Lascoux (2001) are special cases of our model. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. We prove that for age distributions with finite mean, the ancestral process converges to a time-changed Kingman coalescent, while in the case of infinite mean, ancestral lineages might not merge at all with positive probability. Further, we present a construction of the forward in time process in equilibrium. The mathematical methods are based on renewal theory, the urn process introduced by Kaj et al., as well as on a paper by Hammond and Sheffield (2011).

math.PR↗

arXiv2013

Graphical representation of certain moment dualities and application to population models with balancing selection

We investigate dual mechanisms for interacting particle systems. Generalizing an approach of Alkemper and Hutzenthaler in the case of coalescing duals, we show that a simple linear transformation leads to a moment duality of suitably rescaled processes. More precisely, we show how dualities of interacting particle systems of the form $H(A,B)=q^{|A\cap B|}, A,B\subset\{0,1\}^N, q\in[-1,1),$ are rescaled to yield moment dualities of rescaled processes. We discuss in particular the case $q=-1,$ which explains why certain population models with balancing selection have an annihilating dual process. We also consider different values of $q,$ and answer a question by Alkemper and Hutzenthaler.

math.PR↗

arXiv2009

Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension

We consider the real-valued centered Gaussian field on the four-dimensional integer lattice, whose covariance matrix is given by the Green's function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. $d=4$ is the critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field.

math.PR↗

arXiv2006

Entropic repulsion for a class of Gaussian interface models in high dimensions

Consider the centered Gaussian field on the lattice $\mathbb{Z}^d,$ $d$ large enough, with covariances given by the inverse of $\sum_{j=k}^K q_j(-Δ)^j,$ where $Δ$ is the discrete Laplacian and $q_j \in \mathbb{R},k\leq j\leq K,$ the $q_j$ satisfying certain additional conditions. We extend a previously known result to show that the probability that all spins are nonnegative on a box of side-length $N$ has an exponential decay at rate of order $N^{d-2k}\log{N}.$ The constant is given in terms of a higher-order capacity of the unit cube, analogous to the known case of the lattice free field. This result then allows us to show that, if we condition the field to stay positive in the $N-$box, the local sample mean of the field is pushed to a height of order $\sqrt{\log N}.$

math.PR↗