Related papers: Crossover behaviors in branching annihilating attr…
The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from every point in space and time, while the Brownian net is an extension that also allows branching. We show here that the Brownian net is the…
In this manuscript, we are interested in the long-term behaviour of branching processes with pairwise interactions (BPI-processes). A process in this class behaves as a pure branching process with the difference that competition and…
We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as $\beta \rightarrow \beta_c-$ of the probability…
We study a particle system with hopping (random walk) dynamics on the integer lattice $\mathbb Z^d$. The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of…
Attributing a positive value \tau_x to each x in Z^d, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (\tau_x), often known as "Bouchaud's trap model". We assume that these weights are…
We study the effects of long-range hopping and long-range inter-particle interactions on quantum walk of hard-core bosons in ideal and disordered one-dimensional lattices. We find that the range of hopping has a much more significant effect…
A one-dimensional confined Nonlinear Random Walk is a tuple of $N$ diffeomorphisms of the unit interval driven by a probabilistic Markov chain. For generic such walks, we obtain a geometric characterization of their ergodic stationary…
We consider the driven dynamics of a probe particle moving through an assembly of particles with competing long-range repulsive and short-range attractive interactions, which form crystal, stripe, labyrinth, and bubble states as the ratio…
In the present paper, we consider the interacting partially-directed self-avoiding walk (IPDSAW) attracted by a vertical wall. The IPDSAW was introduced by Zwanzig and Lauritzen (J. Chem. Phys., 1968) as a manner of investigating the…
We introduce a model of interacting random walkers on a finite one dimensional chain with absorbing boundaries or targets at the ends. Walkers are of two types: informed particles that move ballistically towards a given target, and…
We treat a quantum walk (QW) on the line whose quantum coin at each vertex tends to be the identity as the distance goes to infinity. We obtain a limit theorem that this QW exhibits localization with not an exponential but a "power-law"…
The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by…
The presence of temporal correlations in random movement trajectories is a widespread phenomenon across biological, chemical and physical systems. The ubiquity of persistent and anti-persistent motion in many natural and synthetic systems…
We study crossover phenomena in a model of self-avoiding walks with medium-range jumps, that corresponds to the limit $N\to 0$ of an $N$-vector spin system with medium-range interactions. In particular, we consider the critical crossover…
We study critical branching random walks (BRWs) $U^{(n)}$ on~$\mathbb{Z}_{+}$ where for each $n$, the displacement of an offspring from its parent has drift~$2\beta/\sqrt{n}$ towards the origin and reflection at the origin. We prove that…
The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time $\R\times\R$. It was first introduced by Arratia, and later analyzed in detail by T\'{o}th and Werner. More recently, Fontes,…
We study systems of interacting Brownian particles in one dimension constructed as the diffusion scaling limits of Fisher's vicious walk models. We define two types of nonintersecting Brownian motions, in which we impose no condition (resp.…
The Levy Walk is the process with continuous sample paths which arises from consecutive linear motions of i.i.d. lengths with i.i.d. directions. Assuming speed 1 and motions in the domain of beta-stable attraction, we prove functional limit…
We study the second order of the number of excursions of a simple random walk with a bias that drives a return toward the origin along the axes introduced by P. Andreoletti and P. Debs \cite{AndDeb3}. This is a crucial step toward deriving…
We consider the branching random walk on the real line where the underlying motion is of a simple random walk and branching is at least binary and at most decaying exponentially in law. It is well known that the normalized empirical measure…