Zero-temperature Glauber dynamics on {\mathbb{Z}^d}

We study zero-temperature Glauber dynamics on ${\mathbb{Z}^d}$ , which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define ${p_c(\mathbb{Z}^d)}$ to be the infimum over p such that the system fixates at ???+??? with probability 1. It is a folklore conjecture that ${p_c(\mathbb{Z}^d) = 1/2}$ for every ${2 \le d \in \mathbb{N}}$ . We prove that ${p_c(\mathbb{Z}^d) \to 1/2}$ as d ?? ??.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

Discrete fractal dimensions of the ranges of random walks in {{\mathbb Z}^d} associate with random conductances

Let ${X= \{X_t, t \ge 0\}}$ be a continuous time random walk in an environment of i.i.d. random conductances ${\{\mu_e \in [1,\infty), e \in E_d\}}$ , where E _d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice ${\mathbb{Z}^d}$ and d ≥ 3. Let ${{\rm R} = \{x \in \mathbb{Z}^d: X_t = x {\rm \,for\, some}\,t \ge 0\}}$ be the range of X. It is proved that, for almost every realization of the environment, dim_H R = dim_P R = 2 almost surely, where dim_H and dim_P denote, respectively, the discrete Hausdorff and packing dimension. Furthermore, given any set ${A \subseteq \mathbb{Z}^d}$ , a criterion for A to be hit by X _t for arbitrarily large t > 0 is given in terms of dim_H A. Similar results for Bouchoud’s trap model in ${\mathbb{Z}^d}$ (d ≥ 3) are also proven.     

Invariance principle for the random conductance model

We study a continuous time random walk X in an environment of i.i.d. random conductances ${\mu_{e} \in [0,\infty)}$ in ${\mathbb{Z}^d}$ . We assume that ${\mathbb{P}(\mu_{e} > 0) > p_c}$ , so that the bonds with strictly positive conductances percolate, but make no other assumptions on the law of the μ _e . We prove a quenched invariance principle for X, and obtain Green’s functions bounds and an elliptic Harnack inequality.     

Martin boundary of a reflected random walk on a half-space

The complete representation of the Martin compactification for reflected random walks on a half-space ${\mathbb{Z}^d\times\mathbb{N}}$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the “radial” compactification obtained by Ney and Spitzer for the homogeneous random walks in ${\mathbb{Z}^d}$ : convergence of a sequence of points ${z_n\in\mathbb{Z}^{d-1}\times\mathbb{N}}$ to a point of on the Martin boundary does not imply convergence of the sequence z _n /|z _n | on the unit sphere S ^d . Our approach relies on the large deviation properties of the scaled processes and uses Pascal’s method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.     

Convergence of symmetric Markov chains on {\mathbb{Z}^d}

For each n let ${Y^{(n)}_t}$ be a continuous time symmetric Markov chain with state space ${n^{-1} \mathbb{Z}^d}$ . Conditions in terms of the conductances are given for the convergence of the ${Y^{(n)}_t}$ to a symmetric Markov process Y _t on ${\mathbb{R}^d}$ . We have weak convergence of $\{{Y^{(n)}_t: t \leq t_0\}}$ for every t ₀ and every starting point. The limit process Y has a continuous part and may also have jumps.     

Strong Central Limit Theorem for isotropic random walks in {\mathbb{R}^d}

We prove an optimal Gaussian upper bound for the densities of isotropic random walks on ${\mathbb{R}^d}$ in spherical case (d ?? 2) and ball case (d ?? 1). We deduce the strongest possible version of the Central Limit Theorem for the isotropic random walks: if ${\tilde S_n}$ denotes the normalized random walk and Y the limiting Gaussian vector, then ${\mathbb{E} f(\tilde S_{n}) \rightarrow \mathbb{E} f(Y)}$ for all functions f integrable with respect to the law of Y. We call such result a ??Strong CLT??. We apply our results to get strong hypercontractivity inequalities and strong Log-Sobolev inequalities.     

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

Lower semicontinuity for functionals with (p, q)-growth in Carnot groups

Let ?? be a bounded open subset of ${\mathbb{G}}$ , where ${\mathbb{G}}$ is a Carnot group, and let ${u: \Omega \rightarrow \mathbb{R}^d}$ be a vector valued function. We prove a lower semicontinuity result in the weak topology of the horizontal Sobolev space ${W^{1,p}_X(\Omega,\mathbb{R}^d)}$ , with p?>?1, of the integral functional of the calculus of variations of the type $$F(u)=\int\limits_\Omega f(Xu)\,dx$$ where f is a X-quasiconvex function satisfying a non-standard growth conditions and Xu is the horizontal gradient of u.     

Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family H(k) of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system k ∈ $\mathbb{T}^d $ , where $\mathbb{T}^d $ is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ?^d, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ $\mathbb{T}^d $ if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator H₊(k), k ∈ $\mathbb{T}^d $ , corresponding to a two-particle system with repulsive interaction.     

On fixed points of a generalized multidimensional affine recursion

Let G be a multiplicative subsemigroup of the general linear group Gl ${(\mathbb{R}^d)}$ which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a G-valued random matrix A, we consider the following generalized multidimensional affine equation $$R\stackrel{\mathcal{D}}{=} \sum_{i=1}^N A_iR_i+B,$$ where N ≥ 2 is a fixed natural number, A ₁, . . . , A _N are independent copies of ${A, B \in \mathbb{R}^d}$ is a random vector with positive entries, and R ₁, . . . , R _N are independent copies of ${R \in \mathbb{R}^d}$ , which have also positive entries. Moreover, all of them are mutually independent and ${\stackrel{\mathcal{D}}{=}}$ stands for the equality in distribution. We will show with the aid of spectral theory developed by Guivarc’h and Le Page (Simplicité de spectres de Lyapounov et propriété d’isolation spectrale pour une famille d’opérateurs de transfert sur l’espace projectif. Random Walks and Geometry, Walter de Gruyter GmbH & Co. KG, Berlin, 2004; On matricial renewal theorems and tails of stationary measures for affine stochastic recursions, Preprint, 2011) and Kesten’s renewal theorem (Kesten in Ann Probab 2:355–386, 1974), that under appropriate conditions, there exists χ >  0 such that ${{\mathbb{P}(\{\langle R, u \rangle > t\})\asymp t^{-\chi}}}$ , as t → ∞, for every unit vector ${u \in \mathbb{S}^{d-1}}$ with positive entries.     

Almost periodic pseudodifferential operators and Gevrey classes

We study almost periodic pseudodifferential operators acting on almost periodic functions ${G_{\rm ap}^s(\mathbb {R}^d)}$ of Gevrey regularity index s ≥ 1. We prove that almost periodic operators with symbols of H?rmander type ${S_{\rho,\delta}^m}$ satisfying an s-Gevrey condition are continuous on ${G_{\rm ap}^s(\mathbb {R}^d)}$ provided 0 < ρ ≤ 1, δ?=?0 and s ρ ≥ 1. A calculus is developed for symbols and operators using a notion of regularizing operator adapted to almost periodic Gevrey functions and its duality. We apply the results to show a regularity result in this context for a class of hypoelliptic operators.     

A lower bound on the critical parameter of interlacement percolation in high dimension

We investigate the percolative properties of the vacant set left by random interlacements on ${\mathbb{Z}^d}$ , when d is large. A non-negative parameter u controls the density of random interlacements on ${\mathbb{Z}^d}$ . It is known from Sznitman (Ann Math, 2010), and Sidoravicius and Sznitman (Comm Pure Appl Math 62(6):831?C858, 2009), that there is a non-degenerate critical value u _*, such that the vacant set at level u percolates when u < u _*, and does not percolate when u > u _*. Little is known about u _*, however, random interlacements on ${\mathbb{Z}^d}$ , for large d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, where the corresponding critical parameter can be explicitly computed, see Teixeira (Electron J Probab 14:1604?C1627, 2009). We show in this article that lim inf _d ?u _*/ log d ?? 1. This lower bound is in agreement with the above mentioned heuristics.     

Lower large deviations for the maximal flow through a domain of {\mathbb{R}^d} in first passage percolation

We consider the standard first passage percolation model in the rescaled graph ${\mathbb{Z}^d/n}$ for d??? 2, and a domain ?? of boundary ?? in ${\mathbb{R}^d}$ . Let ??¹ and ??² be two disjoint open subsets of ??, representing the parts of ?? through which some water can enter and escape from ??. We investigate the asymptotic behaviour of the flow ${\phi_n}$ through a discrete version ?? _n of ?? between the corresponding discrete sets ${\Gamma^{1}_{n}}$ and ${\Gamma^{2}_{n}}$ . We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of ${\phi_n/ n^{d-1}}$ below a certain constant are of surface order.     

Crystallographic number systems

We introduce the notion of crystallographic number systems, generalizing matrix number systems. Let Γ be a group of isometries of ${\mathbb{R}^d,g}$ an expanding affine mapping of ${\mathbb{R}^d}$ with ${g\circ\Gamma\circ g^{-1}\subset\Gamma}$ and ${\mathcal{D}\subset\Gamma}$ . We say that ${(\Gamma,g,\mathcal{D})}$ is a Γ-number system if every isometry ${\gamma\in \Gamma}$ has a unique expansion $$\gamma=g^n\delta_n g^{-n}\,g^{n-1}\delta_{n-1} g^{-(n-1)}\dots g\delta_{1} g^{-1}\,\delta_0,$$ for some ${n\in \mathbb{N}}$ and ${\delta_0,\ldots,\delta_n\in \mathcal{D}}$ . A tile can be attached to a Γ-number system. We show fundamental topological properties of this tile: they admit the fixed point of g as interior point and tesselate the space by the whole group Γ. Moreover, we give several examples, among them a class of p2-number systems, where p2 is the crystallographic group generated by the π-rotation and two independent translations.     

Percolation in the vacant set of Poisson cylinders

We consider a Poisson point process on the space of lines in ${{\mathbb R}^d}$ , where a multiplicative factor u?>?0 of the intensity measure determines the density of lines. Each line in the process is taken as the axis of a bi-infinite cylinder of radius 1. We investigate percolative properties of the vacant set, defined as the subset of ${{\mathbb R}^d}$ that is not covered by any such cylinder. We show that in dimensions d ≥ 4, there is a critical value ${u_*(d) \in (0,\infty)}$ , such that with probability 1, the vacant set has an unbounded component if u?<?u _*(d), and only bounded components if u?>?u _*(d). For d?=?3, we prove that the vacant set does not percolate for large u and that the vacant set intersected with a two-dimensional subspace of ${{\mathbb R}^d}$ does not even percolate for small u?>?0.     

Weighted Function Spaces and Dunkl Transform

We introduce first weighted function spaces on ${\mathbb{R}^d}$ using the Dunkl convolution that we call Besov-Dunkl spaces. We provide characterizations of these spaces by decomposition of functions. Next we obtain in the real line and in radial case on ${\mathbb{R}^d}$ weighted L ^p -estimates of the Dunkl transform of a function in terms of an integral modulus of continuity which gives a quantitative form of the Riemann-Lebesgue lemma. Finally, we show in both cases that the Dunkl transform of a function is in L ¹ when this function belongs to a suitable Besov-Dunkl space.     

On Isomorphisms of Hardy Spaces Associated with Schrödinger Operators

Let L=?Δ+V is a Schrödinger operator on $\mathbb{R}^{d}$ , d≥3, V≥0. Let $H^{1}_{L}$ denote the Hardy space associated with L. We shall prove that there is an L-harmonic function w, 0<δ≤w(x)≤C, such that the mapping $$H_L^1 \ni f\mapsto wf\in H^1\bigl(\mathbb{R}^d\bigr) $$ is an isomorphism from the Hardy space $H_{L}^{1}$ onto the classical Hardy space $H^{1}(\mathbb{R}^{d})$ if and only if $\Delta^{-1}V(x)=-c_{d}\int_{\mathbb{R}^{d}} |x-y|^{2-d} V(y) dy$ belongs to $L^{\infty}(\mathbb{R}^{d})$ .     

Proper holomorphic mappings, Bell’s formula, and the Lu Qi-Keng problem on the tetrablock

We consider proper holomorphic maps ${\pi : D\rightarrow G}$ , where D and G are domains in ${\mathbb{C}^{n}}$ . Let ${\alpha\in \mathcal{C}(G,\mathbb{R}_{ > 0})}$ . We show that every π induces some subspace H of ${\mathbb{A}^{2}_{\alpha\circ\pi}(D)}$ such that ${\mathbb{A}^{2}_{\alpha}(G)}$ is isometrically isomorphic to H via some unitary operator Γ. Using this isomorphism we construct the orthogonal projection onto H, and we derive Bell’s transformation formula for the weighted Bergman kernel function under proper holomorphic mappings. As a consequence of the formula, we get that the tetrablock is not a Lu Qi-Keng domain.