Related papers: Independent sets in random subgraphs of the hyperc…

Random interlacements at level u is a one parameter family of connected random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the vacant set at level u, exhibits a non-trivial percolation phase transition in u, as shown…

Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a (random) graph on $n$ vertices obtained by drawing uniformly at random a $d$-regular (simple)…

In this paper, we study the asymptotic behavior of the volume of spheres in metric measure spaces. We first introduce a general setting adapted to the study of asymptotic isoperimetry in a general class of metric measure spaces. We then…

Given $\alpha \in (0, \infty)$ and $r \in (0, \infty)$, let ${\cal D}_{r, \alpha}$ be the disc of radius $r$ in the hyperbolic plane having curvature $-\alpha^2$. Consider the Poisson point process having uniform intensity density on ${\cal…

This article aims to develop the uniformization and boundary theory of random infinite ideal hyperbolic polyhedra (abbr. IHP) and their dual 1-skeleton, i.e., ideal angled graphs (abbr. IAG) from multiple perspectives, including…

In image detection, one problem is to test whether the set, though mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve, for example, a curve with…

Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[\gamma]$ of a closed curve $\gamma \in \pi_{1}(S_{g,n})$ contains a unique closed…

We study average-case complexity of branch-and-bound for maximum independent set in random graphs under the $\mathcal{G}(n,p)$ distribution. In this model every pair $(u,v)$ of vertices belongs to $E$ with probability $p$ independently on…

In this note we analyze two algorithms, one for producing a matching and one for an independent set, on $k$-uniform $d$-regular hypergraphs of large girth. As a result we obtain new lower bounds on the size of a maximum matching or…

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]=\{1,2,\ldots,n\}$ with $m$ edges, whenever $n\to\infty$ and $n-1\le m=m(n)\le \binom{n}{2}$. We give an asymptotic formula for the…

We consider the stochastic behavior of a class of local $U$-statistics of Poisson processes$-$which include subgraph and simplex counts as special cases, and amounts to quantifying clustering behavior$-$for point clouds lying in diverging…

Inspired by the Erd\H{o}s R\'enyi model, we propose a new model for freesquare random monomial ideals generated by edges and covers of a graph. This permit us to investigate the conditions of normality for which we obtain asymptotic…

We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {\mathbb K}[x_1, \dots x_n]$, utilizing methods from the Erd\"{o}s-R\'{e}nyi model of random graphs. Here for a graph $G \sim G(n, p)$ we consider…

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $\lambda(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let…

Counting independent sets in graphs and hypergraphs under a variety of restrictions is a classical question with a long history. It is the subject of the celebrated container method which found numerous spectacular applications over the…

Let $G$ be a triangle-free graph with $n$ vertices and average degree $t$. We show that $G$ contains at least \[ e^{(1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t (\frac{1}{2}\ln t-1)} \] independent sets. This improves a recent result of the…

We prove an asymptotic formula for the number of $k$-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that is asymptotically equal to the number of $d$-regular $k$-uniform…

We consider the vacant set of random interlacements on Z^d, with d bigger or equal to 3, in the percolative regime. Motivated by the large deviation principles obtained in our recent work arXiv:1304.7477, we investigate the asymptotic…

This work revisits the PCP Verifiers used in the works of Hastad [Has01], Guruswami et al.[GHS02], Holmerin[Hol02] and Guruswami[Gur00] for satisfiable Max-E3-SAT and Max-Ek-Set-Splitting, and independent set in 2-colorable 4-uniform…

The critical 2-dimensional Ising model is studied with four types boundary conditions: free, fixed ferromagnetic, fixed antiferromagnetic, fixed double antiferromagnetic. Using Bond Propagation algorithms with surface fields, we obtained…