Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $ω=(ω_n)_{n\geq 1}$ uniformly distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n\geq 1}$ of positive real numbers with limit $0$. We investigate the size of the random set \[
\mathcal U (ω):=\{y\in \mathbb{T}: \ \forall N\gg 1,
\ \exists n \leq N, \ \text{s.t.} \ \| ω_n -y \| < r_N \}. \] Some sufficient conditions for $\mathcal U(ω)$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that $\mathcal U(ω)$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.
We show that, almost surely, the Hausdorff dimension $s_0$ of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension $k>s_0$. The result holds for random covering sets with a generating sequence of ball-like sets, and is obtained by investigating orthogonal projections of a class of random Cantor sets.
We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we consider covering sets generated by balls and, in the torus, we deal with general analytic generating sets.
In this paper, we study the Dvoretzky covering problem with non-uniformly distributed centers. When the probability law of the centers admits an absolutely continuous density which satisfies a regular condition on the set of essential infimum points, we give a necessary and sufficient condition for covering the circle. When the lengths of covering intervals are of the form $\ell_n = \frac{c}{n}$, we give a necessary and sufficient condition for covering the circle, without imposing any regularity on the density function.
Let $E=\limsup\limits_{n\to\infty}(g_n+ξ_n)$ be the random covering set on the torus $\mathbb{T}^d$, where $\{g_n\}$ is a sequence of ball-like sets and $ξ_n$ is a sequence of independent random variables uniformly distributed on $\T^d$. We prove that $E\cap F\neq\emptyset$ almost surely whenever $F\subset\mathbb{T}^d$ is an analytic set with Hausdorff dimension, $\dim_H(F)>d-α$, where $α$ is the almost sure Hausdorff dimension of $E$. Moreover, examples are given to show that the condition on $\dim_H(F)$ cannot be replaced by the packing dimension of $F$.
We study regular coverings of graphs and manifolds with a focus on properties of the heat equation. In particular, we look at stochastic incompleteness, the Feller property and uniform transience; and investigate the connection between the validity of these properties on the base space and its covering. For both graphs and manifolds, we prove the equivalence of stochastic incompleteness of the base and that of its cover. Along the way we also give some new conditions for the Feller property to hold on graphs.
Let ${\pmb M}$, ${\pmb N}$ and ${\pmb K}$ be $d$-dimensional Riemann manifolds. Assume that ${\bf A}:=(A_n)_{n\in{\Bbb N}}$ is a sequence of Lebesgue measurable subsets of ${\pmb M}$ satisfying a necessary density condition and ${\bf x}:=(x_n)_{n\in {\Bbb N}}$ is a sequence of independent random variables which are distributed on ${\pmb K}$ according to a measure which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets ${\bf E}({\bf x},{\bf A}):=\limsup_{n\to\infty}A_n(x_n)\subset {\pmb N}$. …
We calculate the almost sure Hausdorff dimension of the random covering set $\limsup_{n\to\infty}(g_n + ξ_n)$ in $d$-dimensional torus $\mathbb T^d$, where the sets $g_n\subset\mathbb T^d$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and $ξ_n\in\mathbb T^d$ are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.
This note provides a generalisation of a recent result by Järvenpää, Järvenpää, Koivusalo, Li, and Suomala, (to appear), on the dimension of limsup-sets of random coverings of tori. The result in this note is stronger in the sense that it provides also a large intersection property of the limsup-sets, the assumptions are weaker, and it implies the result of Järvenpää, Järvenpää, Koivusalo, Li, and Suomala as a special case. The proof is based on a recent result by Persson and Reeve from 2013.
In random sequential covering, identical objects are deposited randomly, irreversibly, and sequentially; only attempts increasing the coverage are accepted. A finite system eventually gets congested, and we study the statistics of congested configurations. For the covering of an interval by dimers, we determine the average number of deposited dimers, compute all higher cumulants, and establish the probabilities of reaching minimally and maximally congested configurations. We also investigate random covering by segments with $\ell$ sites and sticks. Covering an infinite substrate continues inde…
The classical Dvoretzky covering problem asks for conditions on the sequence of lengths $\{\ell_n\}_{n\in \mathbb{N}}$ so that the random intervals $I_n : = (ω_n -(\ell_n/2), ω_n +(\ell_n/2))$ where $ω_n$ is a sequence of i.i.d. uniformly distributed random variable, covers any point on the circle $\mathbb{T}$ infinitely often. We consider the case when $ω_n$ are absolutely continuous with a density function $f$. When $m_f=essinf_\mathbb{T}f>0$ and the set $K_f$ of its essential infimum points satisfies $\overline{\dim}_\mathrm{B} K_f<1$, where $\overline{\dim}_\mathrm{B}$ is the upper box-cou…
Let $\{B(ξ_n,r_n)\}_{n\ge1}$ be a sequence of random balls whose centers $\{ξ_n\}_{n\ge1}$ is a stationary process, and $\{r_n\}_{n\ge1}$ is a sequence of positive numbers decreasing to 0. Our object is the random covering set $E=\limsup\limits_{n\to\infty}B(ξ_n,r_n)$, that is, the points covered by $B(ξ_n,r_n)$ infinitely often. The sizes of $E$ are investigated from the viewpoint of measure, dimension and topology.
In this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C}\mathbb{P}^1,\textrm{conj})$. We prove that the space of degree $d$ real branched coverings having "many" real branched points (for example more than $\sqrt{d}^{1+α}$, for any $α>0$) has exponentially small measure. In particular, maximal real branched coverings, that is real branched coverings such that all the branched points are real, are exponentially rare.
This article presents an algebraic topology perspective on the problem of finding a complete coverage probability of a one dimensional domain $X$ by a random covering, and develops techniques applicable to the problem beyond the one dimensional case. In particular we obtain a general formula for the chance that a collection of finitely many compact connected random sets placed on $X$ has a union equal to $X$. The result is derived under certain topological assumptions on the shape of the covering sets (the covering ought to be {\em good}, which holds if the diameter of the covering elements do…
In this paper we study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when $d\ge 4$.
We analyze a deterministic form of the random walk on the integer line called the {\em liar machine}, similar to the rotor-router model, finding asymptotically tight pointwise and interval discrepancy bounds versus random walk. This provides an improvement in the best-known winning strategies in the binary symmetric pathological liar game with a linear fraction of responses allowed to be lies. Equivalently, this proves the existence of adaptive binary block covering codes with block length $n$, covering radius $\leq fn$ for $f\in(0,1/2)$, and cardinality $O(\sqrt{\log \log n}/(1-2f))$ times th…
We study random subcube intersection graphs, that is, graphs obtained by selecting a random collection of subcubes of a fixed hypercube $Q_d$ to serve as the vertices of the graph, and setting an edge between a pair of subcubes if their intersection is non-empty. Our motivation for considering such graphs is to model `random compatibility' between vertices in a large network. For both of the models considered in this paper, we determine the thresholds for covering the underlying hypercube $Q_d$ and for the appearance of s-cliques. In addition we pose some open problems.
Let p(w) denote the probability that four random circular caps of angular radius 70deg<w<90deg cover the unit sphere S^2. An exact expression for p(w) is unknown. We give nontrivial lower bounds for p(w) when w>84deg; no improvement on the inequality p(w)>=0 for w<84deg is yet feasible. A dual problem involving randomly inscribed well-centered tetrahedra is also examined.
In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the n…
Moderate deviation principles (MDPs) for random walks on covering graphs with groups of polynomial volume growth are discussed in a geometric point of view. They deal with any intermediate spatial scalings between those of laws of large numbers and those of central limit theorems. The corresponding rate functions are given by quadratic forms determined by the Albanese metric associated with the given random walks. We apply MDPs to establish laws of the iterated logarithm on the covering graphs by characterizing the set of all limit points of the normalized random walks.
The aim of this article is to study some asymptotics of a natural model of random ramified coverings on the disk of degree $N$. We prove that the monodromy field, called also the holonomy field, converges in probability to a non-random field as $N$ goes to infinity. In order to do so, we use the fact that the monodromy field of random uniform labelled simple ramified coverings on the disk of degree $N$ has the same law as the $\mathfrak{S}(N)$-Yang-Mills measure associated with the random walk by transposition on $\mathfrak{S}(N)$. This allows us to restrict our study to random walks on $\math…
In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We first prove a semigroup CLT for the family of random walks by realizing the nilpotent covering graph into a nilpotent Lie group via discrete harmonic maps. The limiting diffusion semigroup i…
Given a closed Riemann surface $Σ$ equipped with a volume form $ω$, we construct a natural probability measure on the space $\mathcal{M}_d(Σ)$ of degree $d$ branched coverings from $Σ$ to the Riemann sphere $\mathbb{C}\mathbb{P}^1.$ We prove a large deviations principle for the number of critical points in a given open set $U\subset Σ$: given any sequence $ε_d$ of positive numbers, the probability that the number of critical points of a branched covering deviates from $2d\cdot\textrm{Vol}(U)$ more than $ε_d$ is smaller than $\exp(-C_Uε^3_d d)$, for some positive constant $C_U$. In particular, …
In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$. For $d\ge 4$, it has been shown in [5] that such probability decays exponentially with respect to $N$. For $d=3$, however, the same technique does not apply, and in this paper we obtain a slightly weaker upper bound: $\forall \varepsilon>0,\exists c_\varepsilon>0,$ $$P\left({\rm Trace}(\mathcal{P})\subseteq {\rm Trace}\big(\{X_n\}_{n=0}^\infty\big) \right)\le \exp\left(-c_\varepsilon N\log^{…
We investigate the relations between the Poissonnian loop ensembles , their occupation fields, non ramified Galois coverings of a graph, the associated gauge fields, and random Eulerian networks.
Geometric properties of $N$ random points distributed independently and uniformly on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$ with respect to surface area measure are obtained and several related conjectures are posed. In particular, we derive asymptotics (as $N \to \infty$) for the expected moments of the radii of spherical caps associated with the facets of the convex hull of $N$ random points on $\mathbb{S}^{d}$. We provide conjectures for the asymptotic distribution of the scaled radii of these spherical caps and the expected value of the largest of these radii (the covering…
We prove bounds for the covering numbers of classes of convex functions and convex sets in Euclidean space. Previous results require the underlying convex functions or sets to be uniformly bounded. We relax this assumption and replace it with weaker integral constraints. Existing results can be recovered as special cases of our results.
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…
Let $A=(a_{ij})$ be an $n\times n$ random matrix with i.i.d. entries such that $\mathbb{E} a_{11} = 0$ and $\mathbb{E} {a_{11}}^2 = 1$. We prove that for any $δ>0$ there is $L>0$ depending only on $δ$, and a subset $\mathcal{N}$ of $B_2^n$ of cardinality at most $\exp(δn)$ such that with probability very close to one we have $$A(B_2^n)\subset \bigcup_{y\in A(\mathcal{N})}\bigl(y+L\sqrt{n}B_2^n\bigr).$$ As an application, we show that for some $L'>0$ and $u\in[0,1)$ depending only on the distribution law of $a_{11}$, the smallest singular value $s_n$ of the matrix $A$ satisfies $\mathbb{P}\{s_n…
Let $V$ be an $n$-set, and let $X$ be a random variable taking values in the powerset of $V$. Suppose we are given a sequence of random coupons $X_1, X_2, \ldots $, where the $X_i$ are independent random variables with distribution given by $X$. The covering time $T$ is the smallest integer $t\geq 0$ such that $\bigcup_{i=1}^tX_i=V$. The distribution of $T$ is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focussed almost exclusively on the case where $X$ is assumed to be symmetric and/or uniform in some way. In this pa…
A sufficient condition for the uniform modulus of continuity of a random field $X = \{X(t), t \in \R^N\}$ is provided. The result is applicable to random fields with heavy-tailed distribution such as stable random fields.
Elephant Random Walks on Coverings of Dipole Graphs
Faculty of Advanced Science and Technology, Kumamoto University（熊本大学先进科学技术学部）；Graduate School of Science and Technology, Kumamoto University（熊本大学理工学研究科）
We study logical limit laws for uniform attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time $n+1$, the vertex $n+1$ is introduced together with $m$ edges joining the new vertex with $m$ different vertices chosen uniformly at random from $1,\ldots,n$. We prove that this random graph obeys convergence law for first-order sentences with at most $m-2$ variables.
The uniform law for sojourn times of processes with cyclically exchangeable increments is extended to the case of random fields, with general parameter sets, that possess a suitable invariance property.
In this note we discuss uniform integrability of random variables. In a probability space, we introduce two new notions on uniform integrability of random variables, and prove that they are equivalent to the classic one. In a sublinear expectation space, we give de La Vallée Poussin criterion for the uniform integrability of random variables and do some other discussions.
We study a one-dimensional random walk among random conductances, with unbounded jumps. Assuming the ergodicity of the collection of conductances and a few other technical conditions (uniform ellipticity and polynomial bounds on the tails of the jumps) we prove a quenched \textit{uniform} invariance principle for the random walk. This means that the rescaled trajectory of length $n$ is (in a certain sense) close enough to the Brownian motion, uniformly with respect to the choice of the starting location in an interval of length $O(\sqrt{n})$ around the origin.
We investigate the asymptotic number of induced subgraphs in power-law uniform random graphs. We show that these induced subgraphs appear typically on vertices with specific degrees, which are found by solving an optimization problem. Furthermore, we show that this optimization problem allows to design a linear-time, randomized algorithm that distinguishes between uniform random graphs and random graph models that create graphs with approximately a desired degree sequence: the power-law rank-1 inhomogeneous random graph. This algorithm uses the fact that some specific induced subgraphs appear …
Unbiased random vectors i.e. distributed uniformly in n-dimensional space, are widely applied and the computational cost of generating a vector increases only linearly with n. On the other hand, generating uniformly distributed random vectors in its subspaces typically involves the inefficiency of rejecting vectors falling outside, or re-weighting a non-uniformly distributed set of samples. Both approaches become severely ineffective as n increases. We present an efficient algorithm to generate uniformly distributed random directions in n-dimensional cones, to aid searching and sampling tasks …
