We prove that given any finite set of Z d , with ≥ 3, there is a subset whose capacity and volume are both the same order as initial set.As an application, we obtain estimates on probability transient random walk covers uniformly set.Finally, characterize some folding events, under optimal hypotheses.For instance, knowing folds to produce atypically high occupation density somewhere, show region most likely ball-like, asymptotically length goes infinity.

We consider a cluster growth model on the d-dimensional lattice, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at origin, one time, and stop moving when reaching site not occupied by previous walks. It is known that asymptotic shape of spherical. When dimension 2 or more, we prove fluctuations with respect to sphere are most power logarithm its radius in d larger than equal 2. so doing, introduce closely related call flashing process, whose...

Consider a continuous-time Markov process with transition rates matrix Q in the state space Λ ⋃ {0}. In associated Fleming-Viot N particles evolve independently until one of them attempts to jump 0. At this moment particle jumps positions other particles, chosen uniformly at random. When is finite, we show that empirical distribution fixed time converges as → ∞ single same conditioned on not touching Furthermore, profile unique invariant measure for quasistationary one-particle motion. A key...

We study the capacity of range a transient simple random walk on $\mathbb {Z}^d$. Our main result is central limit theorem for $d\ge 6$. present few open questions in lower dimensions.

Nous considérons $N$ particules indépendantes. Chaque particule suit l'évolution d'un processus de Galton–Watson sous-critique jusqu'au moment où elle touche $0$. À cet instant, cette choisit uniformément au hasard la position d'une des autres et y saute. Ce est appelé Fleming–Viot. montrons que pour chaque entier $N$, il existe une unique mesure invariante le Fleming–Viot, empirique stationnaire converge vers loi quasi-stationnaire minimale conditionné à ne pas mourir.

We consider internal diffusion limited aggregation in dimension larger than or equal to two. This is a random cluster growth model, where walks start at the origin of $d$-dimensional lattice, one time, and stop moving when reaching site that not occupied by previous walks. It known asymptotic shape sphere. When two more, we have shown paper inner (resp., outer) fluctuations its radius most order $\log(\mathrm{radius})$ [resp., $\log^{2}(\mathrm{radius})$]. Using same approach, improve upper...

Activated Random Walks, on Z d for any 1, is an interacting particle system, where particles can be in either of two states: active or frozen.Each performs a continuous-time simple random walk during exponential time parameter λ, after which it stays still the frozen state, until another shares its location, and turns instantaneously back into activity.This model known to have phase transition, we show that critical density, controlling less than one dimension value sleep rate λ.We provide...

We study the scaling limit of capacity range a random walk on integer lattice in dimension four. establish strong law large numbers and central theorem with non-Gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall ’86 [Comm. Math. Phys. 104 (1986) 471–507] for volume two.

We consider symmetric activated random walks on Z, and show that the critical densitywhere λ denotes sleep rate.

We consider a random walk in scenery {Xn=η(S0)+⋯+η(Sn),n∈N}, where centered {Sn,n∈N} is independent of the {η(x),x∈Zd}, consisting symmetric i.i.d. with tail distribution P(η(x)>t)∼exp(−cαtα), 1⩽α<d/2. study probability, when averaged over both randomness, that {Xn>ny} for y>0, and n large. In this note, we show large deviation estimate order exp(−c(ny)a), a=α/(α+1). Soit une marche aléatoire en paysage Xn=η(S0)+⋯+η(Sn). La {Sn} est centrée, et évolue indépendamment d'un formé d'une suite...

Consider a continuous-time Markov process with transition rates matrix Q in the state space Λ ⋃ {0}. In associated Fleming-Viot N particles evolve independently until one of them attempts to jump 0. At this moment particle jumps positions other particles, chosen uniformly at random. When is finite, we show that empirical distribution fixed time converges as → ∞ single same conditioned on not touching Furthermore, profile unique invariant measure for quasistationary one-particle motion. A key...

We obtain estimates for large and moderate deviations the capacity of range a random walk on $\mathbb {Z}^{d}$, in dimension $d\ge 5$, both upward downward directions. The results are analogous to those we obtained volume two companion papers [AS17a, AS19]. Interestingly, main steps strategy developed latter apply this seemingly different setting, yet details analysis different.

Nous étudions un modèle d’agrégation limitée par diffusion interne (DLA), sur le peigne bidimensionnel. Le est arbre couvrant du réseau cubique, et DLA de croissance aléatoire : des marches simples, lancées une après l’autre à l’origine peigne, s’arrêtent lorsqu’elles atteignent premier sommet inexploré. Une forme asymptotique suggérée borne inférieure Huss Sava (Electron. J. Probab. 17 (2012) 30). les fluctuations rapport cette asymptotique.

We study quasi-stationary measures for conservative particle systems in the finite lattice. Existence of is established a fairly general class reversible systems. For special cases system independent random walks and symmetric simple exclusion process, it shown that qualitative features change drastically with dimension.

We study downward deviations of the boundary range a transient walk on Euclidean lattice. describe optimal strategy adopted by in order to shrink its range. The technics we develop apply equally well range, and provide pathwise statements for {\it Swiss cheese} picture Bolthausen, van den Berg Hollander \cite{BBH}.

We consider N nearest neighbor random walks on the positive integers with a drift towards origin. When one walk reaches origin, it jumps to position of other N-1 walks, chosen uniformly at random. show that this particle system is ergodic, and establish some exponential moments rightmost position, under stationary measure.

We consider a cluster growth model on Z^d, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one time, and stop moving when reaching site not occupied by previous walks. It is known that asymptotic shape of spherical. Also, dimension 2 or more, has volume $n^d$, it fluctuations radius are most order $n^{1/3}$. improve estimate to $n^{1/(d+1)}$, in 3 more. so doing, we introduce closely related call flashing process, whose...