Related papers: Fluctuations of Biggins' martingales at complex pa…
Let $W_t(\theta)$ be the Biggins martingale of a supercritical branching L\'evy process with non-local branching mechanism, and denote by $W_\infty(\theta)$ its limit. In this paper, we first study moment properties of $W_t(\theta)$ and…
In many-particle diffusions, particles that move the furthest and fastest can play an outsized role in physical phenomena. A theoretical understanding of the behavior of such extreme particles is nascent. A classical model, in the spirit of…
In this paper, we study the scaling limit of a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. If the probability of…
Let $\mm_n, n=0,1,...$ be the supercritical branching random walk, in which the number of direct descendants of one individual may be infinite with positive probability. Assume that the standard martingale $W_n$ related to $\mm_n$ is…
Based on a martingale theory approach, we present a complete characterization of the asymptotic behaviour of a lazy reinforced random walk (LRRW) which shows three different regimes (diffusive, critical and superdiffusive). This allows us…
We consider branching random walks on the Euclidean lattice in dimensions five and higher. In this non-Markovian setting, we first obtain a relationship between the equilibrium measure and Green's function, in the form of an approximate…
We study the fluctuations of the area $A=\int_0^T x(t) dt$ under a one-dimensional Brownian motion $x(t)$ in a trapping potential $\sim |x|$, at long times $T\to\infty$. We find that typical fluctuations of $A$ follow a Gaussian…
We consider $n$ independent, identically distributed one-dimensional Brownian motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth density function $f$. The empirical quantiles, or pointwise order statistics, are denoted by…
A famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet non identically distributed. This phenomenon is at the heart of the asymptotic analysis…
We prove the emergence of stable fluctuations for reaction-diffusion in random environment with Weibull tails. This completes our work around the quenched to annealed transition phenomenon in this context of reaction diffusion. In [9], we…
The aim of this paper is first the detection of multiple abrupt changes of the long-range dependence (respectively self-similarity, local fractality) parameters from a sample of a Gaussian stationary times series (respectively time series,…
We give a simple non-analytic proof of Biggins' theorem on martingale convergence for branching random walks.
Branching random flights are key to describing the evolution of many physical and biological systems, ranging from neutron multiplication to gene mutations. When their paths evolve in bounded regions, we establish a relation between the…
Statistical fluctuations of the light emitted from amplifying random media are studied theoretically and numerically. The characteristic scales of the diffusive motion of light lead to Gaussian or power-law (Levy) distributed fluctuations…
We study occupation time statistics in ergodic continuous-time random walks. Under thermal detailed balance conditions, the average occupation time is given by the Boltzmann-Gibbs canonical law. But close to the non-ergodic phase, the…
We analyze the fluctuations of incomplete $U$-statistics over a triangular array of independent random variables. We give criteria for a Central Limit Theorem (CLT, for short) to hold in the sense that we prove that an appropriately scaled…
We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial…
When the memory parameter of the elephant random walk is above a critical threshold, the process becomes superdiffusive and, once suitably normalised, converges to a non-Gaussian random variable. In a recent paper by the three first…
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…
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…