Related papers: On simple connectivity of random 2-complexes

We consider the random interlacements process with intensity $u$ on ${\mathbb Z}^d$, $d\ge 5$ (call it $I^u$), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on ${\mathbb Z}^d$. For $k\ge 3$…

We identify the asymptotic probability of a configuration model $\mathrm{CM}_n(\boldsymbol{d})$ to produce a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well…

We prove that for $c>0$ a sufficiently small universal constant that a random set of $c d^2/\log^4(d)$ independent Gaussian random points in $\mathbb{R}^d$ lie on a common ellipsoid with high probability. This nearly establishes a…

In this paper, firstly, for some $4n$-dimensional almost complex manifolds $M_{i}, ~1\le i \le \alpha$, we prove that $\left(\sharp_{i=1}^{\alpha} M_{i}\right) \sharp (\alpha{-}1) \mathbb{C} P^{2n}$ must admits an almost complex structure,…

An $n$-tuple $D=(d(1),\dots,d(n))$ is a \emph{feasible degree sequence} if there is a graph on $\{1,\dots,n\}$ such that $i$ has degree $d(i)$. Any such graph will have $m=\sum_{i=1}^n d(i)/2$ edges. Letting $G(D)$ be a graph chosen…

Simplicial complexes are gaining increasing scientific attention as they are generalized network structures that can represent the many-body interactions existing in complex systems raging from the brain to high-order social networks.…

Consider a random multigraph G* with given vertex degrees d_1,...,d_n, contructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges (d_1+...+d_n)/2 tending to infinity, the…

We consider the interlacement Poisson point process on the space of doubly-infinite Z^d-valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times. The set of vertices and edges visited by at least…

Consider two graphs $X$ and $Y$, each with $n$ vertices. The friends-and-strangers graph $\mathsf{FS}(X,Y)$ of $X$ and $Y$ is a graph with vertex set consisting of all bijections $\sigma :V(X) \mapsto V(Y)$, where two bijections $\sigma$,…

We deduce a structurally inductive description of the determinantal probability measure associated with Kalai's celebrated enumeration result for higher--dimensional spanning trees of the $n-1$--simplex. As a consequence, we derive the…

Let $P$ be a set of $n\geq 3$ points in general position in the plane. The edge disjointness graph $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in…

We present high statistics data on the distribution of shortest path lengths between two near-by points on the same cluster at the percolation threshold. Our data are based on a new and very efficient algorithm. For $d=2$ they clearly…

Within crystallization theory, (Matveev's) complexity of a 3-manifold can be estimated by means of the combinatorial notion of GM-complexity. In this paper, we prove that the GM-complexity of any lens space L(p,q), with p greater than 2, is…

We consider the problem $(\mathrm{P})$ of fitting $n$ standard Gaussian random vectors in $\mathbb{R}^d$ to the boundary of a centered ellipsoid, as $n, d \to \infty$. This problem is conjectured to have a sharp feasibility transition: for…

Hoang, Levit, Mandrescu and Pham asked for structural conditions ensuring that the independence polynomial of a $\W_p$ graph is log-concave, or at least unimodal, and conjectured that a connected $\W_2$ graph is $2$-quasi-regularizable if…

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worst-case hardness lies at the core of computational complexity theory, for example in the form of NP-hardness and the (Strong) Exponential…

Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the cavity method to put forward precise conjectures on phase transitions in random problems [Mezard, Parisi, Zecchina: Science 2002]. The cavity…

We consider bootstrap percolation on the binomial random graph $G(n,p)$ with infection threshold $r\in \mathbb{N}$, an infection process which starts from a set of initially infected vertices and in each step every vertex with at least $r$…

Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its…

We study graph bootstrap percolation on the Erd\H{o}s-R\'enyi random graph ${\mathcal G}_{n,p}$. For all $r \ge 5$, we locate the sharp $K_r$-percolation threshold $p_c \sim (\gamma n)^{-1/\lambda}$, solving a problem of Balogh, Bollob\'as…