Related papers: To fixate or not to fixate in two-type annihilatin…
We analyze a one-dimensional intermittent random walk on an unbounded domain in the presence of stochastic resetting. In this process, the walker alternates between local intensive search, diffusion, and rapid ballistic relocations in which…
We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for…
We consider a model of random walk in ${\mathbb Z}^2$ with (fixed or random) orientation of the horizontal lines (layers) and with non constant iid probability to stay on these lines. We prove the transience of the walk for any fixed…
Given a matching between n red points and n blue points by line segments in the plane, we consider the problem of obtaining a crossing-free matching through flip operations that replace two crossing segments by two non-crossing ones. We…
The study of first passage percolation (FPP) for the random interlacements model has been initiated in arXiv:2112.12096, where it is shown that on $\mathbb{Z}^d$, $d\geq 3$, the FPP distance is comparable to the graph distance with high…
This paper is a collection of recent results on discrete-time and continuous-time branching random walks. Some results are new and others are known. Many aspects of this theory are considered: local, global and strong local survival, the…
In this paper, we study a competitive model involving two species. When the competition is strong enough, the two species are separated by a free boundary. If the initial data has a positive bound at infinity. We prove that the solution…
We answer Problem 11.1 of Janson arXiv:1803.04207 on P\'olya urns associated with stable random walk. Our proof use neither martingales nor trees, but an approximation with a differential equation.
We consider conservative cross-diffusion systems for two species where individual motion rates depend linearly on the local density of the other species. We develop duality estimates and obtain stability and approximation results. We first…
This paper revisits the well-studied \emph{optimal stopping} problem but within the \emph{large-population} framework. In particular, two classes of optimal stopping problems are formulated by taking into account the \emph{relative…
The survival probability of immobile targets, annihilated by a population of random walkers on inhomogeneous discrete structures, such as disordered solids, glasses, fractals, polymer networks and gels, is analytically investigated. It is…
We study the problem of coexistence in a two-type competition model governed by first-passage percolation on $\Zd$ or on the infinite cluster in Bernoulli percolation. Actually, we prove for a large class of ergodic stationary passage times…
We prove a law of large numbers for a class of $\Z^d$-valued random walks in dynamic random environments, including non-elliptic examples. We assume for the random environment a mixing property called \emph{conditional cone-mixing} and that…
We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes…
Evolutionary game theory has proved to be a powerful tool to probe the self-organisation of collective behaviour by considering frequency-dependent fitness in evolutionary processes. It has shown that the stability of a strategy depends not…
We consider the branching random walks in $d$-dimensional integer lattice with time--space i.i.d. offspring distributions. Then the normalization of the total population is a nonnegative martingale and it almost surely converges to a…
Many graph coloring proofs proceed by showing that a minimal counterexample to the theorem being proved cannot contain certain configurations, and then showing that each graph under consideration contains at least one such configuration;…
We study a system of branching Brownian motions on $\mathbb R$ with annihilation: at each branching time a new particle is created and the leftmost one is deleted. In [7] it has been studied the case of strictly local creations (the new…
Random dynamical systems (RDS) evolve by a dynamical rule chosen independently with a certain probability, from a given set of deterministic rules. These dynamical systems in an interval reach a steady state with a unique well-defined…
Suppose an urn contains initially any number of balls of two colours. One ball is drawn randomly and then put back with $\alpha$ balls of the same colour and $\beta$ balls of the opposite colour. Both cases, $\beta=0$ and $\beta>0$ are well…