Anton Bovier

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• Professor
• Professor (Full) at University of Bonn

We consider a discrete-time binary branching random walk with independent standard normal increments subject to a penalty $\b$ for every pair of particles that get within distance $\e$ of each other at any time. We give a precise description of the most likely configurations of the particles under this law for $N$ large and $\b,\e$ fixed. Particles...

20 years ago, Bovier, Kurkova, and Löwe (Ann Probab 30(2):605–651, 2002) proved a central limit theorem (CLT) for the fluctuations of the free energy in the p-spin version of the Sherrington–Kirkpatrick model of spin glasses at high temperatures. In this paper we improve their results in two ways. First, we extend the range of temperatures to cover...

The drift-barrier hypothesis states that random genetic drift constrains the refinement of a phenotype under natural selection. The influence of effective population size and the genome-wide deleterious mutation rate were studied theoretically, and an inverse relationship between mutation rate and genome size has been observed for many species. How...

We study the extremes of variable speed branching Brownian motion (BBM) where the time-dependent "speed functions", which describe the time-inhomogeneous variance, converge to the identity function. We consider general speed functions lying strictly below their concave hull and piecewise linear, concave speed functions. In the first case, the log-c...

Population medical genetics aims at translating clinically relevant findings from recent studies of large cohorts into healthcare for individuals. Genetic counseling concerning reproductive risks and options is still mainly based on family history, and consanguinity is viewed to increase the risk for recessive diseases regardless of the demographic...

We derive rigorous estimates on the speed of invasion of an advantageous trait in a spatially advancing population in the context of a system of one-dimensional F-KPP equations. The model was introduced and studied heuristically and numerically in a paper by Venegas-Ortiz et al. (Genetics 196:497–507, 2014). In that paper, it was noted that the spe...

We introduce a general class of mean-field-like spin systems with random couplings that comprises both the Ising model on inhomogeneous dense random graphs and the randomly diluted Hopfield model. We are interested in quantitative estimates of metastability in large volumes at fixed temperatures when these systems evolve according to a Glauber dyna...

We consider a model of branching Brownian motion with self-repulsion. Self-repulsion is introduced via a change of measure that penalises particles spending time in an $$\epsilon $$ ϵ -neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable, and we der...

20 years ago, Bovier, Kurkova, and L\"owe [5] proved a central limit theorem (CLT) for the fluctuations of the free energy in the p-spin version of the Sherrington-Kirkpatrick model of spin glasses at high temperatures. In this paper we improve their results in two ways. First, we extend the range of temperatures to cover the entire regime where th...

Consider the complete graph on n vertices. To each vertex assign an Ising spin that can take the values $$-1$$ - 1 or $$+1$$ + 1 . Each spin $$i \in [n]=\{1,2,\dots , n\}$$ i ∈ [ n ] = { 1 , 2 , ⋯ , n } interacts with a magnetic field $$h \in [0,\infty )$$ h ∈ [ 0 , ∞ ) , while each pair of spins $$i,j \in [n]$$ i , j ∈ [ n ] interact with each oth...

We derive rigorous estimates on the speed of invasion of an advantageous trait in a spatially advancing population in the context of a system of one-dimensional F-KPP equations. The model was introduced and studied heuristically and numerically in a paper by Venegas-Ortiz et al. In that paper, it was noted that the speed of invasion by the mutant t...

Consider the complete graph on $n$ vertices. To each vertex assign an Ising spin that can take the values $-1$ or $+1$. Each spin $i \in [n]=\{1,2,\dots, n\}$ interacts with a magnetic field $h \in [0,\infty)$, while each pair of spins $i,j \in [n]$ interact with each other at coupling strength $n^{-1} J(i)J(j)$, where $J=(J(i))_{i \in [n]}$ are i....

We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an $\e$-neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable and we derive a precise d...

Despite increasing data from population-wide sequencing studies, the risk for recessive disorders in consanguineous partnerships is still heavily debated. An important aspect that has not sufficiently been investigated theoretically, is the influence of inbreeding on mutation load and incidence rates when the population sizes change. We therefore d...

The logarithmic correction for the order of the maximum for two‐speed branching Brownian motion changes discontinuously when approaching slopes , which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing and . We show that the logarithmic correction for the order of the maximum now sm...

We analyse the metastable behaviour of the dilute Curie-Weiss model subject to a Glauber dynamics. The model is a random version of a mean-field Ising model, where the coupling coefficients are Bernoulli random variables with mean $p\in (0,1)$. This model can be also viewed as an Ising model on the Erd\H{o}s-R\'enyi random graph with edge probabili...

Antigen loss is a key mechanism how tumor cells escape from T-cell immunotherapy. Using a mouse model of melanoma we directly compared antigen downregulation by phenotypic adaptation with genetically hardwired antigen loss. Unexpectedly, genetic ablation of Pmel, the melanocyte differentiation antigen targeted by adoptively transferred CD8+ T-cells...

We consider an asexually reproducing population on a finite type space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event. On the individual-based level this population can be modelled as a measure-valued Markov process. Multiple variations of this system have been st...

I discuss the so-called stochastic individual based model of adaptive dynamics and in particular how different scaling limits can be obtained by taking limits of large populations, small mutation rate, and small effect of single mutations together with appropriate time rescaling. In particular, one derives the trait substitution sequence, polymorph...

It is well-known that many diffusion equations can be recast as Wasserstein gradient flows. Moreover, in recent years, by modifying the Wasserstein distance appropriately, this technique has been transferred to further evolution equations and systems; see e.g. [15], [11], [7]. In this paper we establish such a gradient flow representation for evolu...

We consider an asexually reproducing population on a finite trait space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event. On the individual-based level this population can be modeled as a measure-valued Markov process. Multiple variations of this system have been st...

We study the large population limit of a stochastic individual-based model which describes the time evolution of a diploid hermaphroditic population reproducing according to Mendelian rules. In [Neukirch, Bovier, 2016] it is proved that sexual reproduction allows unfit alleles to survive in individuals with mixed genotype much longer than they woul...

The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $\sigma_1^2=\sigma_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $\sigma_1^2=1\pm t^{-\alpha}$ and $\sigma_2^2=1\pm t^{-\a...

In this paper we study a class of stochastic individual-based models that describe the evolution of haploid populations where each individual is characterised by a phenotype and a genotype. The phenotype of an individual determines its natural birth- and death rates as well as the competition kernel, $c(x,y)$ which describes the induced death rate...

We consider the dynamics of a class of spin systems with unbounded spins interacting with local mean-field interactions. We prove convergence of the empirical measure to the solution of a McKean–Vlasov equation in the hydrodynamic limit and propagation of chaos. This extends earlier results of Gärtner, Comets and others for bounded spins or strict...

It is well-known that many diffusion equations can be recast as Wasserstein gradient flows. Moreover, in recent years, by modifying the Wasserstein distance appropriately, this technique has been transferred to further evolution equations and systems. In this paper we establish such a gradient flow representation for evolution equations that depend...

We consider the dynamics of a class of spin systems with unbounded spins interacting with local mean field interactions. We proof convergence of the empirical measure to the solution of a McKean-Vlasov equation in the hydrodynamic limit and propagation of chaos. This extends earlier results of G\"artner, Comets and others for bounded spins or stric...

We consider a stochastic model of population dynamics where each individual is characterised by a trait in {0,1,...,L} and has a natural reproduction rate, a logistic death rate due to age or competition and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape...

We consider a stochastic model of population dynamics where each individual is characterised by a trait in {0,1,...,L} and has a natural reproduction rate, a logistic death rate due to age or competition and a probability of mutation towards neighbouring traits at each reproduction event. We choose parameters such that the induced fitness landscape...

In this paper we study a class of stochastic individual-based models that describe the evolution of haploid populations where each individual is characterised by a phenotype and a genotype. The phenotype of an individual determines its natural birth- and death rates as well as the competition kernel, $c(x,y)$ which describes the induced death rate...

We extend the results of Arguin et al and A\"\i{}d\'ekon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson tree. We show that the limit is a cluster point process on $\mathbb{R}_+\times \mathbb{R}$ where each cluster...

Adaptive dynamics are well studied for the haploid reproduction model. In this paper we are interested in the genetic evolution of a diploid hermaphroditic population, which is modelled by a three-type nonlinear birth-and-death process with competition and Mendelian reproduction. In a recent paper, Collet et al. (2013) have shown that on the mutati...

We study a model for Darwinian evolution in an asexual population with a large but non-constant populations size. The model incorporates the basic evolutionary mechanisms, namely natural birth, density-depending logistic death due to age and competition and a probability $u$ of mutation at each birth event. The difference between mothers and mutant...

We study the large population limit of a stochastic individual-based model which describes the time evolution of a diploid hermaphroditic population reproducing according to Mendelian rules. In [Neukirch, Bovier, 2016] it is proved that sexual reproduction allows unfit alleles to survive in individuals with mixed genotype much longer than they woul...

Description Contents Resources Courses About the Authors Branching Brownian motion (BBM) is a classical object in probability theory with deep connections to partial differential equations. This book highlights the connection to classical extreme value theory and to the theory of mean-field spin glasses in statistical mechanics. Starting with a con...

We propose an extension of a standard stochastic individual-based model in population dynamics which broadens the range of biological applications. Our primary motivation is modeling of immunotherapy of malignant tumors. The main expansions of the model are distinguishing phenotype and genotype, including environment-dependent phenotypic plastici...

We describe the behaviour of a particle system with long-range interactions, in which the range of interactions is allowed to depend on the size of the system. We give conditions on the interaction strength under which the scaling limit of the particle system is a well-posed stochastic PDE. As a corollary we obtain that the metastable behaviour of...

We study the large deviation behaviour of $S_n=\sum_{j=1}^n W_jZ_j$, where $(W_j)_{j \in \mathbb N}$ and $(Z_j)_{j \in \mathbb N}$ are sequences of real-valued, independent and identically distributed random variables satisfying certain moment conditions, independent of each other. More precisely, we prove a conditional strong large deviation resul...

I review recent work on the construction of the extremal process of branching Brownian motion. I place this in the context of spin glass theory and in particular the Generalised Random Energy models of Derrida and Gardner. The main emphasis is on a review of the construction of the extremal process of branching Brownian motion, done in collaboratio...

In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is extended here to the entire system of particles that are extremal, i.e. close to the maximum. Namely, it is pro...

We prove the convergence of the extremal processes for variable speed branching Brownian motions with speed functions that lie below their convex hull and satisfy a certain weak regularity condition. These limiting objects are universal in the sense that they only depend on the slope of the speed function at $0$ and the final time $t$. The proof is...

We investigate a special part of the human immune system, namely the activation of T-Cells, using stochastic tools, especially sharp large deviation results. T-Cells have to distinguish reliably between foreign and self peptides which are both presented to them by antigen presenting cells. Our work is based on a model studied by Zint, Baake, and de...

In this chapter we look at Glauber dynamics in infinite volume in two metastable regimes. In Sect. 23.1 we consider the limit of small temperature at positive magnetic field, while in Sect. 23.2 we turn to the limit of small magnetic field at positive temperature. We restrict ourselves to presenting the main ideas only, omitting proofs.

Chapter 9 collects techniques that are basic for the study of metastability and that are used throughout the rest of the book. Section 9.1 focusses on capacity estimates, and derives upper and lower bounds on capacities with the help of variational principles, namely, the Dirichlet principle and the Berman-Konsowa principle. We outline the strategi...

In this chapter we review some important aspects of Markov processes in continuous time. Their study involves much more analytic work than in the discrete-time setting. However, there is also a lot of common structure. The basic definition of a Markov process was already given in Chap. 4. Clearly, one-step transition kernels no longer make sense an...

Sections 1.1 and 1.2 provide a brief description of the phenomenon of metastability, Sect. 1.3 offers a brief historical perspective of the mathematical theories that were developed to obtain a quantitative understanding of this phenomenon, starting with the Arrhenius law in reaction-rate theory. In particular, we discuss the pathwise approach, the...

There are several challenges within metastability that as yet remain unsolved, but are potentially within reach of the conceptual and technical machinery described in the book. This chapter is devoted to some models representing some of these challenges. We state a few theorems—without proofs—and point to a few open problems. Section 22.1 looks at...

Adding a random magnetic field to the Curie-Weiss model adds a considerable amount of complexity to the model. The case when the random fields may take only a finite number of values is still tractable with lumping techniques. This case is treated in this chapter. In Sect. 14.1 we introduce the model. In Sect. 14.2 we define the associated Gibbs me...

The first steps towards describing metastability for models with a non-discrete state space lead to finite-dimensional diffusions. These are the processes originally studied by Freidlin and Wentzell. In the case of gradient drifts, we are able to recover the heuristic predictions by Eyring and Kramers explained in Chap. 2. The presentation contains...

In Chap. 3 we recall some basic notions from probability theory in order to set notation and to have easy references for later use. Proofs are omitted. Section 3.1 defines key ingredients such as probability spaces, random variables, integrals and Radon-Nikodým derivative. Section 3.2 defines stochastic processes and states the Kolmogorov extension...

Chapter 2 discusses in an informal way some basic principles underlying the mathematical analysis of metastability. In Sect. 2.1 we describe two paradigmatic models that serve as a red thread throughout the book: the Kramers model of a Brownian motion in a double-well potential; finite-state Markov processes with exponentially small transition rate...

The goal of this chapter is to extend the analysis in Chap. 19 to Kawasaki dynamics. Again, the average time until the appearance of a critical droplet somewhere is inversely proportional to the volume, and is driven by the same quantities as for small volumes. However, in the proof we encounter several difficult issues, all coming from the fact th...

The goal of this chapter is to extend the analysis of Chap. 17 to volumes that grow moderately fast as the temperature decreases. We run the Glauber dynamics on a large torus starting from a random initial configuration where all the droplets (= clusters of plus-spins) are small. For low temperature, and in the parameter range corresponding to the...

In this chapter we apply the results obtained in Chap. 16 to the lattice gas in two and three dimensions subject to Kawasaki dynamics. Particles live in a finite box, hop between nearest-neighbour sites, feel an attractive interaction when they sit next to each other, and are created, respectively, annihilated at the boundary of the box in a way th...

In this chapter we apply the results obtained in Chapter 16 to Ising spins in two and three dimensions subject to Glauber dynamics. Spins live in a finite box, flip up and down, want to align when they sit next to each other, and want to align with an external magnetic field. We are interested in how the system magnetises, i.e., how the dynamics al...

Chapter 16 describes the metastable behaviour of lattice systems in small volumes at low temperatures subject to a Metropolis dynamics. These theorems are derived under two hypotheses on the energy landscape, i.e., on the interaction Hamiltonian. The theorems themselves are model-independent, and therefore amplify the universal nature of metastabil...

The random-field Curie-Weiss model with general distributions of the magnetic fields is a key example where non-exact coarse-graining methods can be shown to work efficiently in the context of the potential-theoretic approach. The main results are described in Sect. 15.1. The coarse-graining in carried out Sect. 15.2. Bounds on capacities are deriv...

Most systems of interest in statistical physics are extremely high-dimensional, and become infinite-dimensional in the thermodynamic limit. Hence, their metastable behaviour cannot be read off from the energy of paths alone, because a true interplay between energy and entropy of paths takes place. This makes the analysis of such systems hard. A pro...

A natural generalisation of the finite-dimensional diffusions are stochastic partial differential equations. In this chapter we focus on the Allen-Cahn equation introduced in Section 5. 7 in one spatial dimension. Section 5. 7 gives the main theorem and a rough outline of its proof. Section 12.2 lists some approximation properties for the potential...

One of the simplest settings in which the general theory of metastability can be applied is that of discrete diffusions. By this we understand discrete-time or continuous-time (nearest-neighbour) random walks on \(d\)-dimensional lattices, subject to a drift field derived from a potential that may have several local minima, and subject to a small n...

This chapter gives a summary introduction to large deviations. Although large deviation theory is not our main interest in this book, it is an essential element in our conceptual understanding of metastability. Moreover, it provides tools to obtain estimates, which often serve as preliminary steps towards more refined estimates. Section 6.1 recalls...

Markov processes are the basic class of stochastic processes that we use to model metastable systems. There is a substantial difference in the mathematical difficulties involved in dealing with discrete time and continuous time. In this chapter we give an outline of the theory of discrete-time Markov processes (also called Markov chains). Section 4...

The zero-range process offers yet another example of a system for which potential-theoretic methods can be used to describe metastable behaviour. The free energy landscape is of a different nature than what we encountered in the models treated so far. In particular, there is no temperature parameter, and the key quantity to control is entropy. This...

In this chapter we introduce the basic setup for our approach to metastability. The guiding principle is to provide a definition of metastable sets, representing metastable states in model systems, that is verifiable in concrete models and implies the type of behaviour that is associated with metastability. The intuitive picture we have in mind com...

The martingale problem and the stopping times that were described in Chaps. 4 and 5 provide the key link between reversible Markov processes and Dirichlet problems. This chapter gives a detailed account of their connection. Although the basic principles are the same in discrete time and in continuous time, we split the presentation: discrete time a...

Focusing on the mathematics that lies at the intersection of probability theory, statistical physics, combinatorics and computer science, this volume collects together lecture notes on recent developments in the area. The common ground of these subjects is perhaps best described by the three terms in the title: Random Walks, Random Fields and Disor...

Genome-wide assessment of protein–DNA interaction by chromatin immunoprecipitation followed by massive parallel sequencing (ChIP-seq) is a key technology for studying transcription factor (TF) localization and regulation of gene expression. Signal-to-noise-ratio and signal specificity in ChIP-seq studies depend on many variables, including antibody...

We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni \cite{FZ_BM}, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is $\s_1$ for $s\leq bt$ and $\s_2$ when $bt\leq s\leq t$. In the case $\s_1>\s_2$, the process i...

In this paper we consider two continuous-mass population models as analogues of logistic branching random walks, one is supported on a finite trait space and the other one is supported on an infinite trait space. For the first model with nearest-neighbor competition and migration, we justify a well-described evolutionary path to the short-term equi...

We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with exponential density. The law of the individual clusters is characterized as bra...

The zero-range process in the high density phase is known to show condensation behaviour, i.e, a macroscopic fraction of particles is localised on a single site under the canonical equilibrium measure. Recently, Beltran and Landim analysed some aspects of the metastable behaviour of this process in one dimension for finite systems in the limit of i...

This paper extends recent results on ageing in mean field spin glasses on short time scales, obtained by Ben Arous and Gün (Commun Pure Appl Math 65:77-127, 2012) in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward in (Gayrar...

We derive a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible, extending recent results by Gayrard [15,16], based on general criterion for convergence of sums of dependent random variables due to Durrett and Resnick [13]. We demonstrate...

Immunotherapies, signal transduction inhibitors and chemotherapies can successfully achieve remissions in advanced stage cancer patients, but durable responses are rare. Using malignant melanoma as a paradigm, we propose that therapy-induced injury to tumour tissue and the resultant inflammation can activate protective and regenerative responses th...

We prove a conjecture of Lalley and Sellke [Ann. Probab. 15 (1987)] asserting that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a double exponential, or Gumbel, distribution with a random shift. The method of proof is based on the decorrelation of the maximal displacement...

As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time $t$, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, o...

Kramer's equation of a diffusion in a double well potential has been the pardigm for a metastable system since 1940. The theme of this note is to partially explain, why and in what sense this is a good model for metastable systems. In the process, I review recent progress in a variety of models, ranging from mean field spin systems to stochastic pa...

We show how coupling techniques can be used in some metastable systems to prove that mean metastable exit times are almost constant as functions of the starting microscopic configuration within a "meta-stable set." In the example of the Random Field Curie Weiss model, we show that these ideas can also be used to prove asymptotic exponentiallity of...

We consider a fitness-structured population model with competition and migration between nearest neighbors. Under a combination of large population and rare migration limits we are particularly interested in the asymptotic behavior of the total population partition on supporting trait sites. For the population without mutation on a finite trait spa...

Branching Brownian Motion describes a system of particles which diffuse in space and split into offsprings according to a certain random mechanism. In virtue of the groundbreaking work by M. Bramson on the convergence of solutions of the Fisher-KPP equation to traveling waves, the law of the rightmost particle in the limit of large times is rather...

A method to describe unresolved processes in meteorological models by physically based stochastic processes (SP) is proposed by the example of an energy budget model (EBM). Contrary to the common approach using additive white noise, a suitable variable within the model is chosen to be represented by a SP. Spectral analysis of ice core time series s...

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process (K(t), Y(t)), where K(t) is a autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance, while Y(t) is an additive functional of K(t). We prove that under a suitable re...

In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let $\b$ denote the inverse temperature and let $\L_\b \subset \Z^2$ be a square box with periodic boundary conditions such that $\lim_{\b\to\in...

We consider a coupled bistable N-particle system driven by a Brownian noise, with a strong coupling corresponding to the synchronised regime. Our aim is to obtain sharp estimates on the metastable transition times between the two stable states, both for fixed N and in the limit when N tends to infinity, with error estimates uniform in N. These esti...

We give a brief introduction to the theory of mean field models of spin glasses. This includes a concise presentation of the Random Energy model and the Generalized Random Energy model and the connection to the corresponding asymptotic models based on Poisson cascades. We also explain the nature of the Parisi solution of the Sherrington-Kirkpatrick...

In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was r...

We prove upper and lower bounds on the free energy in the Sherrington-Kirkpatrick model with multidimensional (e.g., Heisenberg) spins in terms of the variational inequalities based on the corresponding Parisi functional. We employ the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the generalised random energy model-...

In these lectures we will discuss Markov processes with a particular interest for a phenomenon called metastability. Basically this refers to the existence of two or more time-scales over which the system shows very different behaviour: on the short time scale, the systems reaches quickly a “pseudo-equilibrium” and remains effectively in a restrict...