Local percolative properties of the vacant set of random interlacements with small intensity

Alexander Drewitz; Balázs Ráth; Artëm Sapozhnikov

Displaying similar documents to “Local percolative properties of the vacant set of random interlacements with small intensity”

Localization and delocalization for heavy tailed band matrices

Florent Benaych-Georges, Sandrine Péché (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider some random band matrices with band-width $N^{μ}$ whose entries are independent random variables with distribution tail in $x^{- α}$ . We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when $α l t; 2 (1 + μ^{- 1})$ , the largest eigenvalues have order $N^{(1 + μ) / α}$ , are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices...

Ergodic behaviour of “signed voter models”

G. Maillard, T. S. Mountford (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We answer some questions raised by Gantert, Löwe and Steif ( (2005) 767–780) concerning “signed” voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site $x$ and a site $y$ is negative (respectively positive) the site $y$ will contribute towards the flip rate of $x$ if and only if the two current spin values are equal (respectively opposed). ...

Convergence rates for the full gaussian rough paths

Peter Friz, Sebastian Riedel (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Under the key assumption of finite $ρ$ -variation, $ρ \in [1, 2)$ , of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), $ρ = 1$ resp. $ρ = 1 / (2 H)$ , we recover and extend the respective results of ( (2009) 2689–2718) and ( (2012) 518–550). In particular, we establish an a.s. rate $k^{- (1 / ρ - 1 / 2 - ε)}$ , any $ε g t; 0$ , for Wong–Zakai and Milstein-type approximations...

Limit theorems for one and two-dimensional random walks in random scenery

Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pène (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Random walks in random scenery are processes defined by $Z_{n} : = \sum_{k = 1}^{n} ξ_{X_{1} + \dots + X_{k}}$ , where $(X_{k}, k \geq 1)$ and $(ξ_{y}, y \in ℤ^{d})$ are two independent sequences of i.i.d. random variables with values in $ℤ^{d}$ and $ℝ$ respectively. We suppose that the distributions of $X_{1}$ and $ξ_{0}$ belong to the normal basin of attraction of stable distribution of index $α \in (0, 2]$ and $β \in (0, 2]$ . When $d = 1$ and $α \neq 1$ , a functional limit theorem has been established in ( (1979) 5–25) and a local limit theorem in (To appear). In this paper, we establish the convergence in distribution...

The critical barrier for the survival of branching random walk with absorption

Bruno Jaffuel (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We study a branching random walk on $ℝ$ with an absorbing barrier. The position of the barrier depends on the generation. In each generation, only the individuals born below the barrier survive and reproduce. Given a reproduction law, Biggins et al. [ (1991) 573–581] determined whether a linear barrier allows the process to survive. In this paper, we refine their result: in the boundary case in which the speed of the barrier matches the speed of the minimal position of a...

Spatially adaptive density estimation by localised Haar projections

Florian Gach, Richard Nickl, Vladimir Spokoiny (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Given a random sample from some unknown density $f_{0} : ℝ \to [0, \infty)$ we devise Haar wavelet estimators for $f_{0}$ with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny ( (1997) 927–947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of $f_{0}$ , simultaneously for every point $x$ in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in...

On the limiting velocity of random walks in mixing random environment

Xiaoqin Guo (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider random walks in strong-mixing random Gibbsian environments in $ℤ^{d}$ , $d \geq 2$ . Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment ( (2005) 36–44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ( $d \geq 5$ ).

An integral transform and its application in the propagation of Lorentz-Gaussian beams

A. Belafhal, E.M. El Halba, T. Usman (2021)

Communications in Mathematics

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The aim of the present note is to derive an integral transform $I = \int_{0}^{\infty} x^{s + 1} e^{- β x^{2} + γ x} M_{k, ν} (2 ζ x^{2}) J_{μ} (χ x) d x,$ involving the product of the Whittaker function $M_{k, ν}$ and the Bessel function of the first kind $J_{μ}$ of order $μ$ . As a by-product, we also derive certain new integral transforms as particular cases for some special values of the parameters $k$ and $ν$ of the Whittaker function. Eventually, we show the application of the integral in the propagation of hollow higher-order circular Lorentz-cosh-Gaussian beams through an ABCD optical system...

The fundamental theorem of prehomogeneous vector spaces modulo $p^{m}$ (With an appendix by F. Sato)

Raf Cluckers, Adriaan Herremans (2007)

Bulletin de la Société Mathématique de France

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For a number field $K$ with ring of integers $𝒪_{K}$ , we prove an analogue over finite rings of the form $𝒪_{K} / 𝒫^{m}$ of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where $𝒫$ is a big enough prime ideal of $𝒪_{K}$ and $m > 1$ . In the appendix, F.Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J.Denef and A.Gyoja [, Compos. Math., (1998), 237–346] to the functional equation of $L$ -functions of Dirichlet type associated with prehomogeneous...

Large scale behaviour of the spatial $𝛬$ -Fleming–Viot process

N. Berestycki, A. M. Etheridge, A. Véber (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the spatial $𝛬$ -Fleming–Viot process model ( (2010) 162–216) for frequencies of genetic types in a population living in $ℝ^{d}$ , in the special case in which there are just two types of individuals, labelled $0$ and $1$ . At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type $0$ . We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the...

Actions of the additive group $G_{a}$ on certain noncommutative deformations of the plane

Ivan Kaygorodov, Samuel A. Lopes, Farukh Mashurov (2021)

Communications in Mathematics

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We connect the theorems of Rentschler [] and Dixmier [] on locally nilpotent derivations and automorphisms of the polynomial ring $A_{0}$ and of the Weyl algebra $A_{1}$ , both over a field of characteristic zero, by establishing the same type of results for the family of algebras $A_{h} = 〈 x, y ∣ y x - x y = h (x) 〉,$ where $h$ is an arbitrary polynomial in $x$ . In the second part of the paper we consider a field $𝔽$ of prime characteristic and study $𝔽 [t]$ comodule algebra structures on $A_{h}$ . We also compute the Makar-Limanov invariant of absolute...