We study mixing times of the symmetric and asymmetric simple exclusion process on segment where particles are allowed to enter exit at endpoints. consider different regimes depending entering exiting rates as well in bulk, show that exhibits pre-cutoff some cases cutoff. Our main contribution is for with open boundaries. order time can be linear or exponential size choice boundary parameters, proving a strikingly (and richer) behavior boundaries than closed segment. arguments combine...

We consider a biased random walk Xn on Galton–Watson tree with leaves in the sub-ballistic regime. prove that there exists an explicit constant γ = γ(β) ∈ (0, 1), depending bias β, such |Xn| is of order nγ. Denoting Δn hitting time level n, we Δn/n1/γ tight. Moreover, show does not converge law (at least for large values β). along sequences nλ(k) ⌊λβγk⌋, converges to certain infinitely divisible laws. Key tools proof are classical Harris decomposition trees, new variant regeneration times...

We consider the probability that a weighted sum of n i.i.d. random variables $X_j, j = 1,\ldots,n$, with stretched exponential tails is larger than its expectation and determine rate decay, under suitable conditions on weights. show decay subexponential, identify function in terms $X_j$ Our result generalizes large deviation principle given by Kiesel Stadtmüller as well tail asymptotics for sums provided Nagaev. As an application our result, motivated projections high-dimensional vectors, we...

We study $\lambda$-biased branching random walks on Bienaym\'e--Galton--Watson trees in discrete time. consider the maximal displacement at time $n$, $\max_{\vert u \vert =n} X(u) \vert$, and show that it almost surely grows a deterministic, linear speed. characterize this speed with help of large deviation rate function walk single particle. A similar result is given for minimal $\min_{\vert \vert$.

Let (Zn)n∈N be a d-dimensional random walk in scenery, i.e., Zn=∑k=0n−1Y(Sk) with (Sk)k∈N0 Zd and (Y(z))z∈Zd an i.i.d. independent of the walk. The walker's steps have mean zero some finite exponential moments. We identify speed rate logarithmic decay P(1nZn>bn) for various choices sequences (bn)n [1,∞). Depending on upper tails we different regimes variational formulas functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations Brownian motion Probab. Theory Related...

Considérons une marche aléatoire branchante surcritique à temps discret. Nous nous intéressons la probabilité qu'il existe un rayon infini du support de branchante, le long duquel elle croît plus vite qu'une fonction linéaire pente γ − ε, où désigne vitesse asymptotique position particule droite dans branchante. Sous des hypothèses générales peu restrictives, prouvons que, lorsque ε → 0, cette décroît comme exp{−(β+o(1)) / ε1/2}, β est constante strictement positive dont valeur dépend loi...

We consider the two-dimensional simple random walk conditioned on never hitting origin. This process is a Markov chain, namely it Doob h-transform of with respect to potential kernel. It known be transient and we show that “almost recurrent” in sense each infinite set visited infinitely often, almost surely. prove that, for “large” set, proportion its sites by approximately Uniform[0,1] variable. Also, given G⊂ℝ 2 does not “surround” origin, a.s. there an number k’s such kG∩ℤ unvisited....

We investigate recurrence and transience of Branching Markov Chains (BMC) in discrete time. are clouds particles which move (according to an irreducible underlying Chain) produce offspring independently. The distribution can depend on the location particle. If is constant for all locations, these Tree-Indexed chains sense \cite{benjamini94}. Starting with one particle at $x$, we denote by $α(x)$ probability that $x$ visited infinitely often cloud. Due irreducibility Chain, there three...

Abstract We consider reversible diffusions in a random environment and prove the Einstein relation for this model. It says that derivative at 0 of effective velocity under an additional local drift equals diffusivity model without drift. The is conjectured to hold variety models but so far it has only been proved particular cases. Our proof makes use homogenization arguments, Girsanov transform, refinement regeneration times introduced by Shen. © 2011 Wiley Periodicals, Inc.

Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from suitably normalized $\ell^{p}$ ball in $\mathbb{R}^{n}$ onto an independent unit sphere. We show that sequences such projections, when normalized, satisfy large deviation principle (LDP) as dimension $n$ goes to $\infty$, which can be viewed annealed LDP. also establish quenched LDP (conditioned on fixed sequence directions) and for $p\in(1,\infty]$ (but not $p=1$), corresponding rate function is "universal," sense...

We consider random walk among i.i.d., uniformly elliptic conductances on $\mathbb{Z}^{d}$, and prove the Einstein relation (see Theorem 1). It says that derivative of velocity a biased as function bias equals diffusivity in equilibrium. For fixed bias, we show there is an invariant measure for environment seen from particle. These measures are often called steady states. The follows at least $d\geq3$, expansion states 2), which can be considered our main result. This proved $d\geq3$. In...

We consider real-valued branching random walks and prove a large deviation result for the position of rightmost particle. The particle is maximum collection number dependent walks. characterise rate function as solution variational problem. same independent walks, show that walk dominated by For we derive principle well. It turns out functions upper deviations coincide, but in general lower do not.

We consider an in .nite Galton–Watson tree $\Gamma$ and label the vertices $v$ with a collection of i.i.d.random variables $(Y_v)_{v \in \Gamma}$. In case where upper tail distribution $Y_v$ is semiexponential, we then determine speed corresponding tree-indexed random walk. contrast to classical have finite exponential moments, normalization definition depends on $Y_v$. Interpreting as displacements offspring from parent, \Gamma}$ describes branching The result gives limit theorem for...

We consider a directed random walk on the backbone of infinite cluster generated by supercritical oriented percolation, or equivalently space-time embedding "ancestral lineage'' an individual in stationary discrete-time contact process. prove law large numbers and annealed central limit theorem (i.e., averaged over realisations cluster) using regeneration approach. Furthermore, we obtain quenched (i.e. for almost any realisation via analysis joint renewals two independent walks same cluster.

Nous étudions une marche aléatoire branchante uni-dimensionelle quand les déplacements n’ont pas des moments exponentiels. Plus précisement, la queue d’un déplacement X se comporte comme suit : P[X>t]∼aexp{−λtr}, pour constantes a,λ>0 et r∈(0,1). donnons description détaillée du comportement asymptotique maximum, en montrant lois limites presque sûres, theorèmes de convergence loi dichotomie basée sur condition croissance. Ces diverses font apparaître différences interéssantes entre deux...