Related papers: Explicit universal minimal constants for polynomia…
We prove that every graph with average degree $d$ and smallest adjacency eigenvalue $|\lambda_n|\leq d^{\gamma}$ contains a clique of size $d^{1-O(\gamma)}$. A simple corollary of this yields the first polynomial bound for Chowla's cosine…
In the present paper I consider Cayley graphs of reflection groups of finite-sided Coxeter polyhedra in 3-dimensional hyperbolic space H^3, with standard sets of generators. As the main result, I prove the existence of non-trivial…
In this article we study percolation on the Cayley graph of a free product of groups. The critical probability $p_c$ of a free product $G_1*G_2*...*G_n$ of groups is found as a solution of an equation involving only the expected subcritical…
We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function $f: Z^+\rightarrow Z^+$, such that for all integers $g>0$, every graph of treewidth at least…
We prove central limit theorems (CLTs) for topological functionals of Bernoulli bond percolation on infinite graphs beyond the Euclidean lattice $\mathbb{Z}^{d}$. For quasi-transitive graphs of subexponential growth, we show that the number…
Let $P$ be a polynomial of degree $d$ in independent Bernoulli random variables which has zero mean and unit variance. The Bonami hypercontractivity bound implies that the probability that $|P| > t$ decays exponentially in $t^{2/d}$.…
Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial in the Mal'cev coordinates of…
For a general class of percolation models with long-range correlations on $\mathbb Z^d$, $d\geq 2$, introduced in arXiv:1212.2885, we establish regularity conditions of Barlow arXiv:math/0302004 that mesoscopic subballs of all large enough…
We provide a short and elementary proof that the growth constant of polyiamonds is at most $1+2z+3z^2$ for the unique real root $z$ of the equation $2z^3+z^2-1=0$. This coincidentally suffices to recover the best known upper bound $3.6108$.…
The lilypond model on a point process in $d$-space is a growth-maximal system of non-overlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a…
The celebrated result of Koml\'os, S\'ark\"ozy, and Szemer\'edi states that for any $\varepsilon>0$, there exists $0<c<1$, such that for all sufficiently large $n$, every $n$-vertex graph $G$ with $\delta(G)\geq(1/2+\varepsilon)n$ contains…
Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of…
As part of the graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because…
Consider percolation on $T\times \mathbb{Z}^d$, the product of a regular tree of degree $k\geq 3$ with the hypercubic lattice $\mathbb{Z}^d$. It is known that this graph has $0<p_c<p_u<1$, so that there are non-trivial regimes in which…
We show that sharp thresholds for Boolean functions directly imply average-case circuit lower bounds. More formally we show that any Boolean function exhibiting a sharp enough threshold at \emph{arbitrary} critical density cannot be…
It has long been known that random regular graphs are with high probability good expanders. This was first established in the 1980s by Bollob\'as by directly calculating the probability that a set of vertices has small expansion and then…
This article presents a method for finding the critical probability $p_c$ for the Bernoulli bond percolation on graphs with the so-called tree-like structure. Such a graph can be decomposed into a tree of pieces, each of which has finitely…
We give upper bounds for the number $\Phi_\ell(G)$ of matchings of size $\ell$ in (i) bipartite graphs $G=(X\cup Y, E)$ with specified degrees $d_x$ ($x\in X$), and (ii) general graphs $G=(V,E)$ with all degrees specified. In particular,…
Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral…
A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in a previous paper, is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the…