We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned Galton–Watson tree. We consider both the distribution when both the tree and the cutting (or labels) are random, and the case when we condition on the tree. The proofs are based on Aldous’ theory of the continuum random tree.

We study a queueing system with Poisson arrivals on a bus line indexed by integers. The buses move at constant speed to the right, and the time of service per customer getting on the bus is fixed. The customers arriving at station i wait for a bus if this latter is less than $$d_i$$di stations before, where $$d_i$$di is non-decreasing. We determine the asymptotic behavior of a single bus and when two buses eventually coalesce almost surely by coupling arguments. Three regimes appear, two of which leading to a.s. coalescing of the buses. The approach relies on a connection with age-structured branching processes with immigration and varying environment. We need to prove a Kesten–Stigum-type theorem, i.e., the a.s. convergence of the successive size of the branching process normalized by its mean. The techniques developed combine a spine approach for multitype branching process in varying environment and geometric ergodicity along the spine to control the increments of the normalized process.

We show that large critical multi-type Galton–Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analogous to Kesten’s infinite monotype Galton–Watson tree. This is proven when we condition on the number of vertices of one fixed type, and with an extra technical assumption if we count at least two types. We then apply these results to study local limits of random planar maps, showing that large critical Boltzmann-distributed random maps converge in distribution to an infinite map.

We investigate a simple quantitative genetics model subjet to a gradual environmental change from the viewpoint of the phylogenies of the living individuals. We aim to understand better how the past traits of their ancestors are shaped by the adaptation to the varying environment. The individuals are characterized by a one-dimensional trait. The dynamics -births and deathsdepend on a time-changing mortality rate that shifts the optimal trait to the right at constant speed. The population size is regulated by a nonlinear non-local logistic competition term. The macroscopic behaviour can be described by a PDE that admits a unique positive stationary solution. In the stationary regime, the population can persist, but with a lag in the trait distribution due to the environmental change. For the microscopic (individual-based) stochastic process, the evolution of the lineages can be traced back using the historical process, that is, a measure-valued process on the set of continuous real functions of time. Assuming stationarity of the trait distribution, we describe the limiting distribution, in large populations, of the path of an individual drawn at random at a given time T . Freezing the non-linearity due to competition allows the use of a many-to-one identity together with Feynman-Kac’s formula. This path, in reversed time, remains close to a simple Ornstein-Uhlenbeck process. It shows how the lagged bulk of the present population stems from ancestors once optimal in trait but still in the tail of the trait distribution in which they lived.

We introduce a random finite rooted tree $\mathcal{C}$, the steady state cluster, characterized by a recursive description: $\mathcal{C}$ is a singleton with probability $1/2$ and otherwise is obtained by joining by an edge the roots of two independent trees $\mathcal{C}'$ and $\mathcal{C}''$, each having the law of $\mathcal{C}$, then re-rooting the resulting tree at a uniform random vertex. We construct a stationary regenerative stochastic process $\mathcal{C}(t)$, the steady state cluster growth process. It is characterized by a simple fixed-point property. Its stationary distribution is the law of the steady state cluster $\mathcal{C}$. We conjecture that $\mathcal{C}(t)$ is the local limit of the evolution of the cluster of a tagged vertex in the stationary state of the mean field forest fire model of Rath and Toth. We describe its explosions in terms of a Levy subordinator, using a state-dependent time change. The steady state cluster is also a multitype Galton-Watson tree with a continuum of types. The steady state cluster conditioned on its size is a random weighted spanning tree of the complete graph equipped random edge weights with a simple explicit joint distribution. The time-reversal of the steady state cluster growth process is realised as the component of a `uniform' vertex in a logging process of a critical multitype Galton-Watson tree conditioned to be infinite. We construct a stationary forest fire model on the infinite rooted tree $\mathbb{Z}^*$ with the property that the evolution of the cluster of the root is a version of the steady state cluster growth process. This model is similar in spirit to Aldous' frozen percolation model on the rooted infinite binary tree. We conjecture that it is the local weak limit of the stationary Rath-Toth model.

We give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton–Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton–Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton–Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree. The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding theorems for this model. This survey paper contains many known results from many different sources, together with some new results.

We extend the infinite-allele simple branching process of Griffiths and Pakes (1988) allowing the offspring to change types and labels. The model is developed and limit theorems are given for the growth of the number of labels of a specific type. We also discuss the asymptotics of the frequency spectrum. Finally, we present an application of the model’s use in tumorigenesis.

We describe and study the delayed multi-type branching process, a finite-time delayed multi-type branching process in which individuals are active (can reproduce offspring) during a finite time interval of random length bounded by D. We show that the criterion for extinction is similar to that for the non-delayed case but is based on the sum of the mean matrices rather than a single mean matrix. In the supercritical case we impose the condition that the mean matrices at each delay offset share the right and left Perron-Frobenius eigenvectors. In this case we are able to give explicit analytic expressions for various quantities derived from the limit of the geometrically weighted mean evolution of the process. When the mean of the offspring distribution is 1 at every delay offset in the period of time an individual is actively reproducing, the mean evolution of the process is described by the D-Fibonacci sequences.

We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, Sly '11], under an appropriate "continuity" property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with $+$ and $-$ boundary conditions on that tree. Under the extra assumptions that $G_n$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization, is the Ising measure with $+$ boundary condition on the limiting tree. The "continuity" property holds except possibly for countably many choices of $\beta$, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton Watson trees.

We consider the evolution of populations under the joint action of mutation and differential reproduction, or selection. The population is modelled as a finite-type Markov branching process in continuous time, and the associated genealogical tree is viewed both in the forward and the backward direction of time. The stationary type distribution of the reversed process, the so-called ancestral distribution, turns out as a key for the study of mutation–selection balance. This balance can be expressed in the form of a variational principle that quantifies the respective roles of reproduction and mutation for any possible type distribution. It shows that the mean growth rate of the population results from a competition for a maximal long-term growth rate, as given by the difference between the current mean reproduction rate, and an asymptotic decay rate related to the mutation process; this tradeoff is won by the ancestral distribution. We then focus on the case when the type is determined by a sequence of letters (like nucleotides or matches/mismatches relative to a reference sequence), and we ask how much of the above competition can still be seen by observing only the letter composition (as given by the frequencies of the various letters within the sequence). If mutation and reproduction rates can be approximated in a smooth way, the fitness of letter compositions resulting from the interplay of reproduction and mutation is determined in the limit as the number of sequence sites tends to infinity. Our main application is the quasispecies model of sequence evolution with mutation coupled to reproduction but independent across sites, and a fitness function that is invariant under permutation of sites. In this model, the fitness of letter compositions is worked out explicitly. In certain cases, their competition leads to a phase transition.

We consider distributional fixed point equations of the form X D = ∨iTiXi in a systematic way. The distribution of T = (T1, T2, . . .) is given in advance. The positive rvs T,Xi, i ∈ IN are independent and the Xi have the same distribution as X. We present a systematic approach in order to find solutions using the monotonicity of the corresponding operator. These equations in the limit come up in the natural setting of trees with finite or countable many branches. Examples are in branching processes and the analysis of algorithms (for parallel computing).

We consider branching processes in discrete time for structured population in varying environment. Each individual has a trait which belongs to some general state space and both its reproduction law and the trait inherited by its offsprings may depend on its trait and the environment. We study the long-time behavior of the population and the ancestral lineage of typical individuals under general assumptions. We focus on the mean growth rate and the trait distribution among the population. The approach relies on many-to-one formulae and the analysis of the genealogy, and a key role is played by well-chosen (possibly non-homogeneous) Markov chains. The applications use large deviations principles and the Harris ergodicity for these auxiliary Markov chains.

We consider Markov jump processes describing structured populations with interactions via density dependance. We propose a Markov construction with a distinguished individual which allows to describe the random tree and random sample at a given time via a change of probability. This spine construction involves the extension of type space of individuals to include the state of the population. The jump rates outside the spine are also modified. We apply this approach to some issues concerning evolution of populations and competition. For single type populations, we derive the diagram phase of a growth fragmentation model with competition and the growth of the size of birth and death processes with multiple births. We also describe the ancestral lineages of a uniform sample in multitype populations.

Using an associated branching process as the basis of our approximation, we show that typical inter-point distances in a multi-type random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 179–209, 2011 (Supported by Schweizer Nationalfonds Projekt (No. 20-117625/1); Institut Mittag–Leffler, Djursholm, Sweden; EPSRC and BBSRC through OCISB.)

This paper is centered on covariant dynamics on unimodular random graphs and random networks, namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here. The first result of the paper is a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils. The latter are discrete analogues the stable manifold of the dynamics. The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three classes of foliations in a connected component: F/F (with finitely many finite foils), I/F (infinitely many finite foils), and I/I (infinitely many infinite foils). An infinite connected component of the graph of a vertex-shift on a random network forms an infinite tree with one selected end which is referred to as an Eternal Family Tree. Such trees can be seen as stochastic extensions of branching processes. Unimodular Eternal Family Trees can be seen as extensions of critical branching processes. The class of offspring-invariant Eternal Family Trees, which is introduced in the paper, allows one to analyze dynamics on networks which are not necessarily unimodular. These can be seen as extensions of non-necessarily critical branching processes. Several construction techniques of Eternal Family Trees are proposed, like the joining of trees or moving the root to a far descendant. Eternal Galton-Watson Trees and Eternal Multi-type Galton-Watson Trees are also introduced as special cases of Eternal Family Trees satisfying additional independence properties. These examples allow one to show that the results on Eternal Family Trees unify and extend to the dependent case several well known theorems of the literature on branching processes.

The notion of information pervades informal descriptions of biological systems, but formal treatments face the problem of defining a quantitative measure of information rooted in a concept of fitness, which is itself an elusive notion. Here, we present a model of population dynamics where this problem is amenable to a mathematical analysis. In the limit where any information about future environmental variations is common to the members of the population, our model is equivalent to known models of financial investment. In this case, the population can be interpreted as a portfolio of financial assets and previous analyses have shown that a key quantity of Shannon’s communication theory, the mutual information, sets a fundamental limit on the value of information. We show that this bound can be violated when accounting for features that are irrelevant in finance but inherent to biological systems, such as the stochasticity present at the individual level. This leads us to generalize the measures of uncertainty and information usually encountered in information theory.

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like the X log X condition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.

It is conjectured in the Physics literature that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing & Schaeffer, have shown that the radius of a random quadrangulation with $n$ faces converges in distribution once rescaled by $n^{1/4}$ to the diameter of the Brownian snake, up to a scaling constant. Using a bijection due to Bouttier, di Francesco \&\ Guitter between bipartite planar maps and a family of labeled trees, we show the corresponding invariance principle for a class of random maps that follow a Boltzmann distribution: the radius of such maps, conditioned to have $n$ faces (or $n$ vertices) and under a criticality assumption, converges in distribution once rescaled by $n^{1/4}$ to a scaled version of the diameter of the Brownian snake. Convergence results for the so-called profile of maps are also provided, as well as convergence of rescaled bipartite maps to the Brownian map, as introduced by Marckert & Mokkadem. The proofs of these results rely on a new invariance principle for two-type spatial Galton-Watson trees.

In this paper, we provide a pathwise spine decomposition for superprocesses with both local and non-local branching mechanisms under a martingale change of measure. This result complements the related results obtained in Evans (1993), Kyprianou et al. (2012) and Liu, Ren and Song (2009) for superprocesses with purely local branching mechanisms and in Chen, Ren and Song (2016) and Kyprianou and Palau (2016) for multitype superprocesses. As an application of this decomposition, we obtain necessary/sufficient conditions for the limit of the fundamental martingale to be non-degenerate. In particular, we obtain extinction properties of superprocesses with non-local branching mechanisms as well as a Kesten-Stigum $L\log L$ theorem for the fundamental martingale.

In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. In addition, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.