In this article, we consider a configuration of weighted random balls in $\mathbb{R}^d$ generated according to a Poisson point process. The model investigated exhibits inhomogeneity, as well as dependence between the centers and the radii and heavy tails phenomena. We investigate the asymptotic behavior of the total mass of the configuration of the balls at a macroscopic level. Three different regimes appear depending on the intensity parameters and the zooming factor. Among the three limiting fields, two are stable while the third one is a Poisson integral bridging between the two stable regimes. For some particular choices of the inhomogeneity function, the limiting fields exhibit isotropy or self-similarity.
This article introduces the operator-scaling random ball model, generalizing the isotropic random ball models investigated recently in the literature to anisotropic setup. The model is introduced as a generalized random field and results on weak convergence are established in the space of tempered distributions.
We consider a collection of weighted Euclidian random balls in R^d distributed according a determinantal point process. We perform a zoom-out procedure by shrinking the radii while increasing the number of balls. We observe that the repulsion between the balls is erased and three different regimes are obtained, the same as in the weighted Poissonian case.
We consider a generalization of the weighted random ball model. The model is driven by a random Poisson measure with a product heavy tailed intensity measure. Such a model typically represents the transmission of a network of stations with a fading effect. In a previous article, the authors proved the convergence of the finite-dimensional distributions of related generalized random fields under various scalings and in the particular case when the fading function is the indicator function of the unit ball. In this paper, tightness and functional convergence are investigated. Using suitable mome…
We consider a collection of Euclidean random balls in ${\Bbb R}^d$ generated by a determinantal point process inducing interaction into the balls. We study this model at a macros\-copic level obtained by a zooming-out and three different regimes --Gaussian, Poissonian and stable-- are exhibited as in the Poissonian model without interaction. This shows that the macroscopic behaviour erases the interactions induced by the determinantal point process.
The paper investigates random fields in the ball. It studies three types of such fields: restrictions of scalar random fields in the ball to the sphere, spin, and vector random fields. The review of the existing results and new spectral theory for each of these classes of random fields are given. Examples of applications to classical and new models of these three types are presented. In particular, the Matérn model is used for illustrative examples. The derived spectral representations can be utilised to further study theoretical properties of such fields and to simulate their realisations. Th…
In this paper, we consider a cluster model of weighted Euclidean random balls generated by a shot-noise Cox process. It is an example of cluster point process. We perform a scaling on the model by shrinking the radii of the balls and compensate this effect by increasing the (mean) number of balls in each cluster, or/and increasing the (mean) number of clusters. We consider two different scenarios, say a local and a global scenarios. Heuristically, in the first scenario, we focus on the mean number of large balls in a cluster while in the second one, we focus on the global mean number of large …
We investigate the Gaussian small ball probabilities with random centers, find their deterministic a.s.-equivalents and establish a relation to infinite-dimensional high-resolution quantization.
We consider weighted random balls in $\real^d$ distributed according to a random Poisson measure with heavy-tailed intensity and study the asymptotic behaviour of the total weight of some configurations in $\real^d$. This procedure amounts to be very rich and several regimes appear in the limit, depending on the intensity of the balls, the zooming factor, the tail parameters of the radii and of the weights. Statistical properties of the limit fields are also evidenced, such as isotropy, self-similarity or dependence. One regime is of particular interest and yields $α$-stable stationary isotrop…
We present a duality relation between two systems of coalescing random walks and an analogous duality relation between two systems of coalescing Brownian motions. Our results extends previous work in the literature and we apply it to the study of a system of coalescing Brownian motions with Poisson immigration.
In this paper, we consider a multi-drawing urn model with random addition. At each discrete time step, we draw a sample of m balls. According to the composition of the drawn colors, we return the balls together with a random number of balls depending on two discrete random variables X and Y with finite means and variances. Via the stochastic approximation algorithm, we give limit theorems describing the asymptotic behavior of white balls.
Place an obstacle with probability $1-p$ independently at each vertex of $\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\geq 2$ and $p$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that for environments with probability tending to one as $n\to \infty$ there exists a unique discrete Euclidean ball of volume $d \log_{1/p} n$ asymptotically such that the following holds: conditioned on survival up to time $n$ we have …
We prove a limit theorem for the the maximal interpoint distance (also called the diameter) for a sample of n i.i.d. points in the unit ball of dimension 2 or more. The exact form of the limit distribution and the required normalisation are derived using assumptions on the tail of the interpoint distance for two i.i.d. points. The results are specialised for the cases when the points have spherical symmetric distributions, in particular, are uniformly distributed in the whole ball and on its boundary.
We show that for finite-range, symmetric random walks on general transient Cayley graphs, the expected occupation time of any given ball of radius $r$ is $O(r^{5/2})$.. We also study the volume-growth property of the wired spanning forests on general Cayley graphs, showing that the expected number of vertices in the component of the identity inside any given ball of radius $r$ is $O(r^{11/2})$.
We study biased random walk on the infinite connected component of supercritical percolation on the integer lattice $\mathbb{Z}^d$ for $d\geq 2$. For this model, Fribergh and Hammond showed the existence of an exponent $γ$ such that: for $γ<1$, the random walk is sub-ballistic (i.e. has zero velocity asymptotically), with polynomial escape rate described by $γ$; whereas for $γ>1$, the random walk is ballistic, with non-zero speed in the direction of the bias. They moreover established, under the usual diffusive scaling about the mean distance travelled by the random walk in the direction of th…
Let $\{Z_n\}_{n\geq 0 }$ be a critical or subcritical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on $\mathbb{R}^d$. Denote by $R_n:=\sup\{u>0:Z_n(\{x\in\mathbb{R}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to\infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random wal…
We consider a random walk in random environment in the low disorder regime on $\mathbb Z^d$. That is, the probability that the random walk jumps from a site $x$ to a nearest neighboring site $x+e$ is given by $p(e)+εξ(x,e)$, where $p(e)$ is deterministic, $\{\{ξ(x,e):|e|_1=1\}:x\in\mathbb Z^d\}$ are i.i.d. and $ε>0$ is a parameter which is eventually chosen small enough. We establish an asymptotic expansion in $ε$ for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to firs…
We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this set: a union of disjoint disks, a union of disjoint balls, a union of disjoint line segments, and the boundary of a polyhedron. In the first three cases, we provide closed-form expressions of the cumulative distribution function and the density. In the last case, we propose an algorithm with complexity O(n ln n), n being the number of edges of the polyhedron…
Sharp large deviation results of Bahadur-Ranga Rao type are provided for the $q$-norm of random vectors distributed on the $\ell _{p}^{n}$-ball ${\mathbb{B}}^{n}_{p}$ according to the cone probability measure or the uniform distribution for $1 \le q<p < \infty $, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different $\ell _{p}^{n}$-balls in the spirit of Schechtman and Schmuckenschläger, and for the length of the projection of an $\ell _{p}^{n}$-ba…
We consider a random walk in $\mathbb Z^d$ which jumps from a site $x$ to a nearest neighboring site $x+e$ (where $e\in V:=\{x\in\mathbb Z^d: |x|_1=1\}$) with probability $p_0(e)+εξ(x,e)$. Here $\sum_e p_0(e)=1$, $p_0(e)> 0$, $ε$ is a small parameter while $\{\{ξ(x,e):e\in V\}: x\in\mathbb Z^d\}$ are i.i.d. random variables with an absolute value bounded by $1$. We review recent progress in the non-vanishing velocity case, giving an asymptotic expansion in $ε$ of the invariant measure of the environmental process, and bounds for the velocity.
We study exit laws from large balls in $\mathbb{Z}^d$, $d\geq3$, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary segments. As an application, we prove transience of the random walks in random environment. …
The convex hull peeling of a point set is obtained by taking the convex hull of the set and repeating iteratively the operation on the interior points until no point remains. The boundary of each hull is called a layer. We study the number of k-dimensional faces and the outer defect intrinsic volumes of the first layers of the convex hull peeling of a homogeneous Poisson point process in the unit ball whose intensity goes to infinity. More precisely we provide asymptotic limits for their expectation and variance as well as a central limit theorem. In particular, the growth rates do not depend …
We give a new criterion for ballistic behavior of random walks in random environments which are low disorder perturbations of the simple symmetric random walk on $\mathbb{Z}^d$, for $d\geq 2$. This extends the results established by Sznitman in 2003 and, in particular, allow us to give new examples of ballistic RWREs in dimension $d=3$ which do not satisfy Kalikow's condition. Essentially, this new criterion states that ballisticity occurs whenever the average local drift of the walk is not too small when compared to the standard deviation of the environment. Its proof relies on applying coars…
We study biased random walk on subcritical and supercritical Galton-Watson trees conditioned to survive in the transient, sub-ballistic regime. By considering offspring laws with infinite variance, we extend previously known results for the walk on the supercritical tree and observe new trapping phenomena for the walk on the subcritical tree which, in this case, always yield sub-ballisticity. This is contrary to the walk on the supercritical tree which always has some ballistic phase.
We study properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is of order $n$, most of the harmonic measure is supported on a boundary set of size approximately equal to $n^β$, where $β\approx0.78$ is a universal constant. To derive such results, we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree, and we first deal with the case of critical Galton-Watson trees conditioned to have height greater than $n$. An important ingredien…
We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.
This note contains two types of small ball estimates for random vectors in finite dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain Littlewood-Offord type estimates for quasi-norms. This generalizes a result which was previously obtained by Friedland and Sodin and by Rudelson and Vershynin.
We consider products of uniform random variables from the Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^n$, $k \le n$, and random vectors from the $n$-dimensional $\ell_p^n$-ball $\mathbb{B}_p^n$ with certain $p$-radial distributions, $p\in[1,\infty)$. The distribution of this product geometrically corresponds to the projection of the $p$-radial distribution on $\mathbb{B}^n_p$ onto a random $k$-dimensional subspace. We derive large deviation principles (LDPs) on the space of probability measures on $\mathbb{R}^k$ for sequences of such projections.
Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\ell^p$ ball in $\mathbb{R}^n$ onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension $n$ goes to $\infty$, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for $p\in(1,\infty]$ (but not for $p=1$), the corresponding rate function is "universal", in the sense that …
We study approximations of smooth convex bodies by random ball-polytopes. We examine the following probability model: let $K\subset{\bf R}^d$ be a convex body such that $K$ slides freely in a ball of radius $R>0$ and has $C^2$ smooth boundary. Let $x_1,\ldots, x_n$ be i.i.d. uniform random points in $K$. For $r\geq R$, let $K^r_{(n)}$ denote the intersection of all radius $r$ closed balls that contain $x_1,\ldots, x_n$. Then $K^r_{(n)}$ is a (uniform) random ball-polytope (of radius $r$) in $K$. We study the asymptotic properties of the expectation of the number of facets of $K_{(n)}^r$ as $n\…
In order to study how well a finite group might be generated by repeated random multiplications, P. Diaconis suggested the following urn model. An urn contains some balls labeled by elements which generate a group G. Two are drawn at random with replacement and a ball labeled with the group product (in the order they were picked) is added to the urn. We give a proof of his conjecture that the limiting fraction of balls labeled by each group element almost surely approaches 1/|G|.
Let $A$ be an $n\times n$ random matrix with i.i.d. entries of zero mean, unit variance and a bounded subgaussian moment. We show that the condition number $s_{\max}(A)/s_{\min}(A)$ satisfies the small ball probability estimate $${\mathbb P}\big\{s_{\max}(A)/s_{\min}(A)\leq n/t\big\}\leq 2\exp(-c t^2),\quad t\geq 1,$$ where $c>0$ may only depend on the subgaussian moment. Although the estimate can be obtained as a combination of known results and techniques, it was not noticed in the literature before. As a key step of the proof, we apply estimates for the singular values of $A$, ${\mathbb P}\…
In this paper, we prove a multivariate central limit theorem for $\ell_q$-norms of high-dimensional random vectors that are chosen uniformly at random in an $\ell_p^n$-ball. As a consequence, we provide several applications on the intersections of $\ell_p^n$-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an $\ell_p^n$-ball onto a line spanned by a random direction $θ\in\mathbb S^{n-1}$. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreov…
We find the asymptotic total variation distance between two distributions on configurations of m balls in n labeled bins: in the first, each ball is placed in a bin uniformly at random; in the second, k balls are planted in an arbitrary but fixed arrangement and the remaining m-k balls placed uniformly at random.
In this work we study the rate of convergence in the central limit theorem for the Euclidean norm of random orthogonal projections of vectors chosen at random from an $\ell_p^n$-ball which has been obtained in [Alonso-Gutiérrez, Prochno, Thäle: Gaussian fluctuations for high-dimensional random projections of $\ell_p^n$-balls, Bernoulli 25(4A), 2019, 3139--3174]. More precisely, for any $n\in\mathbb N$ let $E_n$ be a random subspace of dimension $k_n\in\{1,\ldots,n\}$, $P_{E_n}$ the orthogonal projection onto $E_n$, and $X_n$ be a random point in the unit ball of $\ell_p^n$. We prove a Berry-Es…
Random variables equidistributed on convex bodies have received quite a lot of attention in the last few years. In this paper we prove the negative association property (which generalizes the subindependence of coordinate slabs) for generalized Orlicz balls. This allows us to give a strong concentration property, along with a few moment comparison inequalities. Also, the theory of negatively associated variables is being developed in its own right, which allows us to hope more results will be available. Moreover, a simpler proof of a more general result for $\ell_p^n$ balls is given.
Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite Markov chains, one expects a "weak concentration" bound for the distribution of the cover time to hold under minimal assumptions. We give two such results, one for random fixed-radius balls and the other for sequentially arriving randomly-centered and deterministically growing balls. Each is in fact a simple application of a different more general bound, the f…
In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $\ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guédon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of $\ell_p^n$-balls to lo…
In this paper, we study high-dimensional random projections of $\ell_p^n$-balls. More precisely, for any $n\in\mathbb N$ let $E_n$ be a random subspace of dimension $k_n\in\{1,\ldots,n\}$ and $X_n$ be a random point in the unit ball of $\ell_p^n$. Our work provides a description of the Gaussian fluctuations of the Euclidean norm $\|P_{E_n}X_n\|_2$ of random orthogonal projections of $X_n$ onto $E_n$. In particular, under the condition that $k_n\to\infty$ it is shown that these random variables satisfy a central limit theorem, as the space dimension $n$ tends to infinity. Moreover, if $k_n\to\i…
In this paper, we investigate the small scale equidistribution properties of randomised sums of Laplacian eigenfunctions (i.e. random waves) on a compact manifold. We prove small scale expectation and variance results for random waves on all compact manifolds. Here, "small scale" refers to balls of radius $r(λ)\to 0$ such that $r/r_{\text{Planck}}\to\infty$, where $r_{\text{Planck}}$ is the Planck scale. For balls at a larger scale (although still $r(λ)\to 0$) we also obtain estimates showing that the probability that a random wave fails to equidistribute decays exponentially with the eigenval…
We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of "valleys" of height $\log t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.
We study almost sure separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0-1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov-Sobolev spaces $B_{2}^σ\left(\mathbb{B}_{d}\right)$, in the range $0 < σ\leq1 / 2$. For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $\mathrm{H}^2(\mathbb{D}^d)$ and its multiplier algebra $\mathrm{H}^\infty(\mathbb{D}^d)$: in th…
In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector $X_n$ being in a thin shell of radius $\sqrt{n}$ times the asymptotic value of $n^{-1/2}\left(\mathbb E\left[\| X_n\|_2^2\right]\right)^{1/2}$ (as $n\to\infty$), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter $t=t_n$ goes down to ze…
The unit ball random geometric graph $G=G^d_p(λ,n)$ has as its vertices $n$ points distributed independently and uniformly in the $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most $λ$. Like its cousin the Erdos-Renyi random graph, $G$ has a connectivity threshold: an asymptotic value for $λ$ in terms of $n$, above which $G$ is connected and below which $G$ is disconnected (and in fact has isolated vertices in most cases). In the connected zone, we determine upper and lower bounds for the graph diameter of $G$. Specifically, almost always, $\di…
We prove the large-dimensional Gaussian approximation of a sum of $n$ independent random vectors in $\mathbb{R}^d$ together with fourth-moment error bounds on convex sets and Euclidean balls. We show that compared with classical third-moment bounds, our bounds have near-optimal dependence on $n$ and can achieve improved dependence on the dimension $d$. For centered balls, we obtain an additional error bound that has a sub-optimal dependence on $n$, but recovers the known result of the validity of the Gaussian approximation if and only if $d=o(n)$. We discuss an application to the bootstrap. We…
The paper provides a description of the large deviation behavior for the Euclidean norm of projections of $\ell_p^n$-balls to high-dimensional random subspaces. More precisely, for each integer $n\geq 1$, let $k_n\in\{1,\ldots,n-1\}$, $E^{(n)}$ be a uniform random $k_n$-dimensional subspace of $\mathbb R^n$ and $X^{(n)}$ be a random point that is uniformly distributed in the $\ell_p^n$-ball of $\mathbb R^n$ for some $p\in[1,\infty]$. Then the Euclidean norms $\|P_{E^{(n)}}X^{(n)}\|_2$ of the orthogonal projections are shown to satisfy a large deviation principle as the space dimension $n$ tend…
We establish Central Limit Theorems for the volumes of intersections of $B_{p}^n$ (the unit ball of $\ell_p^n$) with uniform random subspaces of codimension $d$ for fixed $d$ and $n\to \infty$. As a corollary we obtain higher order approximations for expected volumes, refining previous results by Koldobsky and Lifschitz and approximations obtained from the Eldan--Klartag version of CLT for convex bodies. We also obtain a Central Limit Theorem for the Minkowski functional of the intersection body of $B_p^n$, evaluated on a random vector distributed uniformly on the unit sphere.
Let $X$ be an isotropic random vector in $R^d$ that satisfies that for every $v \in S^{d-1}$, $\|<X,v>\|_{L_q} \leq L \|<X,v>\|_{L_p}$ for some $q \geq 2p$. We show that for $0<\varepsilon<1$, a set of $N = c(p,q,\varepsilon) d$ random points, selected independently according to $X$, can be used to construct a $1 \pm \varepsilon$ approximation of the $L_p$ unit ball endowed on $R^d$ by $X$. Moreover, $c(p,q,\varepsilon) \leq c^p \varepsilon^{-2}\log(2/\varepsilon)$; when $q=2p$ the approximation is achieved with probability at least $1-2\exp(-cN \varepsilon^2/\log^2(2/\varepsilon))$ and if $q$…
Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into $d$-dimensional Euclidean spaces, $\R^{d}$, have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggest that the problem is quite challenging. This paper provides extension theorem for multiradial characteristic functions that are defined in balls embedded in $\R^d$ cross, either $…
Balls are sequentially allocated into $n$ bins as follows: for each ball, an independent, uniformly random bin is generated. An overseer may then choose to either allocate the ball to this bin, or else the ball is allocated to a new independent uniformly random bin. The goal of the overseer is to reduce the load of the most heavily loaded bin after $Θ(n)$ balls have been allocated. We provide an asymptotically optimal strategy yielding a maximum load of $(1+o(1))\sqrt{\frac{8\log n}{\log\log n}}$ balls.
