We consider branching Brownian motion in which initially there is one particle at x, particles produce a random number of offspring with mean m+1 at the time of branching events, and each particle branches at rate β= 1/2m. Particles independently move according to Brownian motion with drift -1 and are killed at the origin. It is well-known that this process eventually dies out with positive probability. We condition this process to survive for an unusually large time t and study the behavior of the process at small times s ≪ t using a spine decomposition. We show, in particular, that the time when a particle gets furthest from the origin is of the order t^5/6.

We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a nonpositive power of the arrival time of $v$, continuing until the graph has $n$ vertices. This class includes uniformly grown random graphs and inhomogeneous random graphs of preferential-attachment type. Whenever the critical percolation threshold is positive, we show that the critical window has width of order $(\log n)^{-2}$ and a secondary phase transition at its finite upper boundary. Inside this window the largest component has size of order $\sqrt{n}/\log n$, and the susceptibility remains finite and independent of the position in the window. The proofs couple component explorations to branching random walks killed outside an interval of length $\log n$, allowing sharp control of the barely subcritical and critical regimes.

We study necessary and sufficient criteria for global survival of discrete or continuous-time branching Markov processes. We relate these to several definitions of generalised principle eigenvalues for elliptic operators due to Berestycki and Rossi. In doing so, we extend these notions to fairly general semigroups of linear positive operators. We use this relation to prove new results about the generalised principle eigenvalues, as well as about uniqueness and non-uniqueness of stationary solutions of a generalised FKPP equation. The probabilistic approach through branching processes gives rise to relatively simple and transparent proofs under much more general assumptions, as well as constructions of (counter-)examples to certain conjectures.

For a branching random walk that drifts to infinity, consider its Malthusian martingale, i.e. the additive martingale with parameter theta being the smallest root of the characteristic equation. When particles are killed below the origin, we show that the limit of this martingale admits an exponential tail, contrary to the case without killing, where the tail is polynomial. In the critical case, where the characteristic equation has a single root, the same holds for the (truncated) derivative martingale, as we show. This study is motivated by recent work on first passage percolation on Erdos-R & eacute;nyi graphs.

The continuous random energy model (CREM) is a Gaussian process indexed by a binary tree of depth $T$, introduced by Derrida and Spohn, then Bovier and Kurkova, as a toy model of a spin glass. Addario-Berry and the second author have established the existence of an algorithmic hardness threshold $x_*$ in the CREM: finding a state of energy lower than $-x T$ is possible in polynomial time if $x < x_*$, and takes exponential time if $x > x_*$, with high probability. In this article, we are interested in the nature of the transition from polynomial to exponential complexity near the algorithmic hardness threshold. We do this by studying in detail the performance of a certain beam-search algorithm of beam width $N = N(T)$ depending on $T$ -- we believe this algorithm to be natural and asymptotically optimal. The algorithm turns out to be essentially equivalent to the time-inhomogeneous version of the so-called $N$-particle branching Brownian motion ($N$-BBM), which has seen a lot of interest in the last two decades. Studying the performance of the algorithm then amounts to investigating the maximal displacement at time $T$ of the time-inhomogeneous $N$-BBM. In doing so, we are able to quantify precisely the nature of the transition from polynomial to exponential complexity, proving that the transition happens when the log-complexity is of the order of $T^{1/3}$. This result appears to be the first of its kind and we believe that this phenomenon should extend to other models.

We study the Bolker-Pacala-Dieckmann-Law (BPDL) model of population dynamics in the regime of large population density. The BPDL model is a particle system in which particles reproduce, move randomly in space, and compete with each other locally. We rigorously prove global survival as well as a shape theorem describing the asymptotic spread of the population, when the population density is sufficiently large. In contrast to most previous studies, we allow the competition kernel to have an arbitrary, even infinite range, whence the term non-local competition. This makes the particle system non-monotone and of infinite-range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands-on approach. Some ideas in the proof are inspired by works on the non-local Fisher-KPP equation, but the stochasticity of the model creates new difficulties.

The N-particle branching Brownian motion (N-BBM) is a branching Markov process which describes the evolution of a population of particles undergoing reproduction and selection. It has attracted a lot of interest due to its relations to the study of front propagation phenomena on the one hand, and to (hierarchical) physical p-spin models on the other hand, among which the continuous random energy model (CREM). This paper investigates the asymptotic displacement of the N-BBM in a time-inhomogeneous setting, and when the time horizon T and the number of particles N jointly tend to infinity. We estimate the maximal displacement of the process up to the second order, and show that the latter undergoes a transition at the scale log N≈ T^1/3. In particular when log N≪ T^1/3 we recover the Brunet-Derrida behavior which was proven in a time-homogeneous setting and for T→+∞ then N→+∞. Furthermore, our results can also be interpreted from the perspective of algorithmic optimisation on some spin glass models, since the time-inhomogeneous N-BBM can be seen as the realization of an optimization procedure called beam search on the aforementioned CREM. The CREM has been proven by L. Addario-Berry and the second author to undergo an algorithm hardness threshold phenomenon, and the results of the present paper describe precisely the efficiency of the beam search algorithm around that threshold.

In this paper, we study a discrete-time analogue of a Hawkes process, modelled as a Poisson autoregressive process whose parameters depend on the past of the trajectory. The model is characterized to allow these parameters to take negative values, modelling inhibitory dynamics. More precisely, the model is the stochastic process ( X_n)_n≥0 with parameters a_1,…,a_p ∈, p∈ and λ > 0, such that for all n≥ p, conditioned on X_0,…, X_n-1, X_n is Poisson distributed with parameter (a_1 X_n-1 + ⋯ + a_p X_n-p + λ)_+. This process can be seen as a discrete time Hawkes process with inhibition with a memory of length p. extension of a prior work where we studied the specific case p = 2, for which we were able to classify the asymptotic behaviour of the process for the whole range of parameters, except for boundary cases. We first provide a sufficient condition for stability in the general case which is the analog of a condition for continuous time Hawkes processes from . We then focus on the case p=3, extending the results derived for the p=2 case in a previous work . In particular, we show that the process may be stable even if one of the coefficients a_i is much greater than one.

We consider random coefficient autoregressive models of infinite order (AR($\infty$)) under the assumption of non-negativity of the coefficients. We develop novel methods yielding sufficient or necessary conditions for finiteness of moments, based on combinatorial expressions of first and second moments. The methods based on first moments recover previous sufficient conditions by Doukhan and Wintenberger in our setting. The second moment method provides in particular a necessary and sufficient condition for finiteness of second moments which is different, but shown to be equivalent to the classical criterion of Nicholls and Quinn in the case of finite order. We further illustrate our results through two examples.

We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n)_{n\ge0}$ with parameters $a_1,\ldots,a_p \in \mathbb{R}$ , $p\in\mathbb{N}$ , and $\lambda \ge 0$ , such that, for all $n\ge p$ , conditioned on $X_0,\ldots,X_{n-1}$ , $X_n$ is Poisson distributed with parameter $(a_1 X_{n-1} + \cdots + a_p X_{n-p} + \lambda)_+$ . This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p . In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case $p = 2$ , for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-2} + \lambda$ remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$ , in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.

We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin.We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one.We establish precise asymptotics for the probability that the process survives for a large time t, building on previous results by Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014).We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, providing an essentially complete answer to a question first addressed by Kesten (1978).An important tool in the proofs of these results is the convergence of a certain observable to a continuous state branching process.Our proofs incorporate new ideas which might be of use in other branching models.

We consider a Markovian evolution on point processes, the $\Psi$--process, on the unit interval in which points are added according to a rule that depends only on the spacings of the existing point configuration. Having chosen a spacing, a new point is added uniformly within it. Building on previous work of the authors and of Junge, we show that the empirical distribution of points in such a process is always equidistributed under mild assumptions on the rule, generalizing work of Junge. A major portion of this article is devoted to the study of a particular growth--fragmentation process, or cell process, which is a type of piecewise--deterministic Markov process (PDMP). This process represents a linearized version of a size--biased sampling from the $\Psi$--process. We show that this PDMP is ergodic and develop the semigroup theory of it, to show that it describes a linearized version of the $\Psi$--process. This PDMP has appeared in other contexts, and in some sense we develop its theory under minimal assumptions.

Disordered systems such as spin glasses have been used extensively as models for high-dimensional random landscapes and studied from the perspective of optimization algorithms. In a recent paper by L. Addario-Berry and the second author, the continuous random energy model (CREM) was proposed as a simple toy model to study the efficiency of such algorithms. The following question was raised in that paper: what is the threshold $\beta_G$, at which sampling (approximately) from the Gibbs measure at inverse temperature $\beta$ becomes algorithmically hard? This paper is a first step towards answering this question. We consider the branching random walk, a time-homogeneous version of the continuous random energy model. We show that a simple greedy search on a renormalized tree yields a linear-time algorithm which approximately samples from the Gibbs measure, for every $\beta<\beta_c$, the (static) critical point. More precisely, we show that for every $\varepsilon>0$, there exists such an algorithm such that the specific relative entropy between the law sampled by the algorithm and the Gibbs measure of inverse temperature $\beta$ is less than $\varepsilon$ with high probability. In the supercritical regime $\beta>\beta_c$, we provide the following hardness result. Under a mild regularity condition, for every $\delta>0$, there exists $z>0$ such that the running time of any given algorithm approximating the Gibbs measure stochastically dominates a geometric random variable with parameter $e^{-z\sqrt{N}}$ on an event with probability at least $1-\delta$.

Let μ_t denote the critical derivative Gibbs measure of branching Brownian motion at time t. It has been proved by Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470–502) and Maillard and Zeitouni (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 3, 1144–1160) that μ_t converges weakly to the random measure Z_∞√(2/π) x^2 e^-x^2/2 1_x >0 d x, where Z_∞ is the limit of the derivative martingale. In this paper, we are interested in the fluctuations that occur in this convergence and prove for a large class of functions F that √(t)( ∫_ℝ F d μ_t - Z_∞∫_0^∞ F(x) √(2/π) x^2 e^-x^2/2 d x - c(F) log t/√(t) Z_∞) → S(F), in law, as t→∞, where c(F) is a constant depending on F and, given Z_∞, S(F) has an explicit 1-stable distribution. Moreover, we extend this result to a functional convergence, and we identify precisely the particles responsible for the fluctuations. In particular, this proves the following result for the critical additive martingale (W_t)_t≥ 0: √(t)( √(t) W_t - √(2/π) Z_∞) C Z_∞, in law, where here C is a Cauchy variable independent of Z_∞, confirming a conjecture by Mueller and Munier (Phys. Rev. E 90 (2014), 042143) in the physics literature.

We characterize all random point measures which are in a certain sense stable under the action of branching. Denoting by ⊛ the branching convolution operation introduced by Bertoin and Mallein (2019), and by 𝒵 the law of a random point measure on the real line, we are interested in solutions to the fixed point equation ℰ = 𝒵⊛ℰ, with ℰ a random point measure distribution. Under suitable assumptions, we characterize all solutions of this equation as shifted decorated Poisson point processes with a uniquely defined shift.

We characterize all random point measures which are in a certain sense stable under the action of branching. Denoting by ⊛ the branching convolution operation introduced by Bertoin and Mallein, and by Z the law of a random point measure on the real line, we are interested in solutions to the fixed-point equation E=Z⊛E, with E a random point measure distribution. Under suitable assumptions, we characterize all solutions of this equation as shifted decorated Poisson point processes with a uniquely defined shift.

We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale $1/\varepsilon$ in space and time. This can be seen as a model for the spatial dynamics of a biological population in a habitat which is heterogeneous at a large scale (mountains, temperature or precipitation gradient...). The model incorporates another parameter, $K$, which is a measure of the local population density. We study the model in the limit when first $\varepsilon\to 0$ and then $K\to\infty$. In this asymptotic regime, we show that the rescaled position of the front as a function of time converges to the solution of an explicit ODE. We further discuss the relation with another popular model of population dynamics, the Fisher-KPP equation, which arises in the limit $K\to\infty$. Combined with known results on the Fisher-KPP equation, our results show in particular that the limits $\varepsilon\to0$ and $K\to\infty$ do not commute in general. We conjecture that an interpolating regime appears when $\log K$ and $1/\varepsilon$ are of the same order.

We consider branching random walks with a spine in the domain of attraction of an $α$-stable Lévy process. For this process, the classical derivative martingale in general degenerates in the limit. We first determine the quantity replacing the derivative martingale and show that it converges to a non-degenerate limit under a certain LlogL-type condition which we assume to be optimal. We go on to give the Seneta-Heyde norming for the critical additive martingale under the same assumptions. The proofs are based on the methods introduced in our previous paper which considered the finite variance case [Boutaud and Maillard (2019), EJP, vol. 24, paper no. 99].

We study the branching tree of the perimeters of the nested loops in the non-generic critical O(n) model on random quadrangulations. We prove that after renormalization it converges towards an explicit continuous multiplicative cascade whose offspring distribution (x(i))(i >= 1) is related to the jumps of a spectrally positive alpha-stable Levy process with alpha = 3/2 +/- 1/pi arccos(n/2) and for which we have the surprisingly simple and explicit transform E[Sigma(i >= 1)(x(i))(theta)] = sin(pi(2 - alpha))/sin(pi(theta - alpha)), for theta is an element of (alpha, alpha + 1). An important ingredient in the proof is a new formula of independent interest on first moments of additive functionals of the jumps of a left-continuous random walk stopped at a hitting time. We also identify the scaling limit of the volume of the critical O(n)-decorated quadrangulation using the Malthusian martingale associated to the continuous multiplicative cascade.

Let $(Z_t)_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, i.e.\@ the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke [\textit{Ann. Probab.}, 15(3):1052--1061, 1987] says that this martingale converges almost surely to a limit $Z_\infty$, positive on the event of survival. In this paper, our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture by Mueller and Munier [\textit{Phys. Rev. E}, 90:042143, 2014]: \[ \sqrt{t} \left( Z_\infty - Z_t + \frac{\log t}{\sqrt{2\pi t}} Z_\infty \right) \xrightarrow[t\to\infty]{} S_{Z_\infty}, \quad \text{in law}, \] where $S$ is a spectrally positive 1-stable Levy process independent of $Z_\infty$. In a first part of the paper, a relatively short proof of (a slightly stronger form of) this convergence is given based on the functional equation satisfied by the characteristic function of $Z_\infty$ together with tail asymptotics of this random variable. We then set up more elaborate arguments which yield a more thorough understanding of the trajectories of the particles contributing to the fluctuations. In this way, we can upgrade our convergence result to functional convergence. This approach also sets the ground for a follow-up paper, where we study the fluctuations of more general functionals including the renormalized critical additive martingale. All proofs in this paper are given under the hypothesis $E[L(\log L)^3] < \infty$, where the random variable $L$ follows the offspring distribution of the branching Brownian motion. We believe this hypothesis to be optimal.