We introduce and study a model of plane random trees generalizing the famous Bienaymé-Galton-Watson model but where births and deaths are locally correlated. More precisely, given a random variable (B,H) with values in {1,2,3,…}2, given the state of the tree at some generation, the next generation is obtained (informally) by successively deleting B individuals side-by-side and replacing them with H new particles where the samplings are i.i.d. We prove that, in the critical case E[B]=E[H], and under a third moment condition on B and H, the random trees coding the genealogy of the population model converges towards the Brownian Continuum Random Tree. Interestingly, our proof does not use the classical height process or the Łukasiewicz exploration, but rather the stochastic flow point of view introduced by Bertoin and Le Gall. 

Pick a sequence of uniform points on the d-dimensional sphere. Then, link the nth point to its closest one that arrives in the past. This constructs a labelled tree called the nearest neighbour tree on the d-dimensional sphere. These trees share some properties with the random recursive tree: the height of the last arrival node, the mean degree of the root, etc. On the contrary, the number of leaves seems to depend on dimension d, but no such properties have been proved yet. In this article, we prove that the mean number of siblings depends on d.

In particular, we give explicit calculations of this number. In dimension 1, it is 1 + ln(2) and, in any dimension d, it has an explicit integral form, but unfortunately, it does not give an explicit number. Nevertheless, we show that it converges to 2 when d→∞ exponentially quick at a rate of (√3)/2.

  To prove these results, we look at the local limit of those trees and we do some fine computations about the intersection of two balls in dimension d. In particular, we obtain a non-trivial upper bound for those intersections in some precise cases.

We define a general class of random systems of horizontal and vertical broken lines on the quarter plane whose distribution is proved to be translation invariant. This invariance results from a reversibility property of the model. This class of systems generalizes many of the processes known of this kind, such as Hammersley's broken line processes involved in Last Passage Percolation theory or such as the six-vertex model for some special sets of parameters. The novelty comes from the introduction of an intensity associated with each line. Those lines are initially generated by some spatially homogeneous weighted Poisson Point Process and their evolution (turn, split, coalescence, annihilation) are ruled by a Markovian dynamic  which preserves Kirchhoff's node law for the intensities at each intersection. 

The construction of a random continuum C from independent two-sided Brownian motions as considered in arXiv:2004.01367 almost surely yields a non-degenerate indecomposable continuum. We show that C is not-hereditarily indecomposable and, in particular, it is (unfortunately) not the pseudo-arc. 

The directed last passage percolation (LPP) on the quarter-plane is a growing model. To come into the growing set, a cell needs that the cells on its bottom and on its left to be in the growing set, and then to wait a random time.

We present here a new generalisation of directed last passage percolation (GLPP). In GLPP, the waiting time of a cell depends on the difference of the coming times of its bottom and left cells. We explain in this article the physical meaning of this generalisation.

In this first work on GLPP, we study them as a growing model on the cylinders rather than on the quarter-plane, the eighth-plane or the half-plane. We focus, mainly, on the law of the front line. In particular, we prove, in some integrable cases, that this law could be given explicitly as a function of the parameters of the model. Moreover, our results applied to LPP are the first ones for LPP on the cylinders.

These new results are obtained by the use of probabilistic cellular automata (PCA) to study LPP and GLPP. 

We focus on a family of one-dimensional probabilistic cellular automata with memory two: the dynamics is such that the value of a given cell at time t+1 is drawn according to a distribution which is a function of the states of its two nearest neighbours at time t, and of its own state at time t−1. Such PCA naturally arise in the study of some models coming from statistical physics (8-vertex model, directed animals and gaz models, TASEP, etc.). 

We give conditions for which the invariant measure has a product form or a Markovian form, and we prove an ergodicity result holding in that context. The stationary space-time diagrams of these PCA present different forms of reversibility. We describe and study extensively this phenomenon, which provides families of Gibbs random fields on the square lattice having nice geometric and combinatorial properties.

This paper is devoted to the 8-vertex model and its edge correlation function. In some particular (integrable) cases, we find a closed form of the edge correlation function and we deduce also its asymptotic. In addition, we quantify influence of boundary conditions on this function.

To do this, we introduce a system of particles in interaction related to the 8-vertex model. This system, studied using various tools from analytic combinatorics, random walks and conics, permits to compute the correlation function. To study the influence of boundary conditions, we involve probabilistic cellular automata of order two.

Consider the n-th iterated Brownian motion I(n)=Bn∘⋯∘B1. Curien and Konstantopoulos proved that for any distinct numbers ti≠0, (I(n)(t1),…,I(n)(tk) converges in distribution to a limit I[k] independent of the ti's, exchangeable, and gave some elements on the limit occupation measure of I(n). 

Here, we prove under some conditions, finite dimensional distributions of n-th iterated two-sided stable processes converge, and the same holds to the reflected Brownian motions. We give a description of the law of I[k], of the finite dimensional distributions of I(n), as well as those of the iterated reflected Brownian motion iterated ad libitum. 

This paper is devoted to probabilistic cellular automata (PCA) on N, Z or Z/nZ, depending of two neighbors, with a general alphabet E (finite or infinite, discrete or not). We study the following question: under which conditions does a PCA possess a Markov chain as invariant distribution? Previous results in the literature give some conditions on the transition matrix (for positive rate PCA) when the alphabet E is finite. Here we obtain conditions on the transition kernel of PCA with a general alphabet E. In particular, we show that the existence of an invariant Markov chain is equivalent to the existence of a solution to a cubic integral equation.

One of the difficulties to pass from a finite alphabet to a general alphabet comes from some problems of measurability, and a large part of this work is devoted to clarify these issues. 

We revisit the problem of finding the conditions under which synchronous probabilistic cellular automata indexed by the line Z, or the periodic line Z/nZ, depending on 2 neighbours, admit as invariant distribution the law of a space-indexed Markov chain. Our advances concerns PCA defined on a finite alphabet, where most of existing results concern size 2 alphabet. A part of the paper is also devoted to the comparison of different structures (Z, Z/nZ, and also some structures constituted with two consecutive lines of the space time diagram) with respect to the property to possess a Markovian invariant distribution. 

We study the maximum flow on random weighted directed graphs and hypergraphs, that generalize Erdös-Rényi graphs. We show that, for a single unicast connection chosen at random, its capacity, determined by the max-flow between source and sink, converges in probability to the capacity around the source or sink. Using results from network coding, we generalize this result to different types multicast connections, whose capacity is given by the max-flow between the source(s) and sinks. Our convergence results indicate that the capacity of unicast and multicast connections using network coding are, with high probability, unaffected by network size in random networks. Our results generalize to networks with random erasures.

We investigate network coding (NC) in access point-to-multi-point (PMP) broadcast networks. Characterized by a shared unicast upstream channel and a time-shared broadcast downstream channel, PMP networks are widely deployed in optical and wireless access networks. We develop a queuing-theoretic model of NC at the medium access control (MAC) sublayer and analyze the impact of NC on packet delay. Our analysis is validated through discrete-event simulation and demonstrates significant delay advantages for NC under high loads and localized traffic.