In this paper, we study the size of the giant component $C_G$ in the random geometric graph $G=G(n,r_n,f)$ of $n$ nodes independently distributed each according to a certain density $f(·)$ in ${[0,1]}^2$ satisfying ${inf}_{x\in {[0,1]}^2}f\left(x\right)gt;$$0$. If $\frac{c_1}{n}\le r_n^2\le c_2\frac{logn}{n}$ for some positive constants $c_1$, $c_2$ and $n{r}_n^2⟶\infty$ as $n\to \infty$, we show that the giant component of $G$ contains at least $n-\mathrm{o}\left(n\right)$ nodes with probability at least $1-{\mathrm{e}}^{-\beta n{r}_n^2}$ for all $n$ and for some positive constant $\beta$....

The aim of this paper is to study the threshold behavior for the satisfiability property of a random $k$-XOR-CNF formula or equivalently for the consistency of a random Boolean linear system with $k$ variables per equation. For $k\ge 3$ we show the existence of a sharp threshold for the satisfiability of a random $k$-XOR-CNF formula, whereas there are smooth thresholds for $k=1$ and $k=2$.

The aim of this paper is to study the threshold behavior for the satisfiability property of a random k-XOR-CNF formula or equivalently for the consistency of a random Boolean linear system with k variables per equation. For k ≥ 3 we show the existence of a sharp threshold for the satisfiability of a random k-XOR-CNF formula, whereas there are smooth thresholds for k=1 and k=2.

L'article passe en revue quelques Solutions de Tournois (correspondances de choix définies sur les tournois). On compare ces solutions entre elles, et on mentionne certaines de leurs propriétés.

Suppose that in a ballot candidate $A$ scores $a$ votes and candidate $B$ scores $b$ votes and that all possible $\left(a+ba\right)$ voting sequences are equally probable. Denote by ${\alpha }_r$ and by ${\beta }_r$ the number of votes registered for $A$ and for $B$, respectively, among the first $r$ votes recorded, $r=1,\cdots ,a+b$. The purpose of this paper is to derive, for $a\ge b-c$, the probability distributions of the random variables defined as the number of subscripts $r=1,\cdots ,a+b$ for which (i) ${\alpha }_r={\beta }_r-c$, (ii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$, (iii) ${\alpha }_r={\beta }_r-c$ but ${\alpha }_{r-1}={\beta }_{r-1}-c\pm 1$ and ${\alpha }_{r+1}={\beta }_{r+1}-c\pm 1$, where $c=0,\pm 1,\pm 2,\cdots$.

We consider a planar Poisson process and its associated Voronoi map. We show that there is a proper coloring with 6 colors of the map which is a deterministic isometry-equivariant function of the Poisson process. As part of the proof we show that the 6-core of the corresponding Delaunay triangulation is empty. Generalizations, extensions and some open questions are discussed.

In this paper we study in detail the associativity property of the discrete copulas. We observe the connection between discrete copulas and the empirical copulas, and then we propose a statistic that indicates when an empirical copula is associative and obtain its main statistical properties under independence. We also obtained asymptotic results of the proposed statistic. Finally, we study the associativity statistic under different copulas and we include some final remarks about associativity...

Si considera, sul gruppo degli interi, una passeggiata aleatoria uscente dall’origine, i cui passi ammettano due soli possibili valori: uno strettamente negativo, l’altro strettamente positivo. Nel caso particolare in cui il primo di questi valori sia $-1$, si dà un’espressione esplicita per la legge del primo istante di ritorno nell’origine.

In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory $\gamma$ between two points on the boundary of a finite subdomain of ${\mathbb{Z}}^d$ is proportional to ${\mu }^{-\text{length}\left(\gamma \right)}$. When $\mu$ is supercritical (i.e. $\mu lt;{\mu }_c$ where ${\mu }_c$ is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.