Related papers: A Correlation Inequality on Three Functions
In this note we investigate correlation inequalities for `up-sets' of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our…
We consider density-density correlations in the one-dimensional Hubbard model at half filling. On intuitive grounds one might expect them to exhibit an exponential decay. However, as has been noted recently, this is not obvious from the…
Consider critical Bernoulli percolation on $\mathbb{Z}^d$ for $d$ large; let $y_0, \dots, y_{k-1}$ be $k$ distinct points in $\mathbb{R}^d$. We prove that the probability that $\{\lfloor n y_i\rfloor\}_{i=0}^{k-1}$ all lie in the same open…
For a positive integer $n$, let $[n]$ denote $\{1, \ldots, n\}$. For a 2-dimensional integer lattice point $\mathbf{b}$ and positive integers $k\geq 2$ and $n$, a \textit{$k$-sum $\mathbf{b}$-free set} of $[n]\times [n]$ is a subset $S$ of…
We prove an analogue of the Oppenheim conjecture for a system comprising an inhomogeneous quadratic form and a linear form in $3$ variables using dynamics on the space of affine lattices.
The relative density of visible points of the integer lattice $\mathbb{Z}^d$ is known to be $1/\zeta(d)$ for $d\geq 2$, where $\zeta$ is Riemann's zeta function. In this paper we prove that the relative density of visible points in the…
We study the transverse clustering properties of high matter density peaks as traced by high column density absorption systems (either Lyman limit systems characterized by N(HI)> 2 x 10^{17} cm^{-2} or CIV systems with W_{r}> 0.5 A) at…
Feldman and Karlin conjectured that the number of isolated fixed points for deterministic models of viability selection and recombination among n possible haplotypes has an upper bound of 2^n - 1. Here a proof is provided. The upper bound…
Given $0<s<\frac d2$ with $s\leq 1$, we are interested in the large $N$-behavior of the optimal constant $\kappa_N$ in the Hardy inequality $\sum_{n=1}^N (-\Delta_n)^s \geq \kappa_N \sum_{n<m} |X_n-X_m|^{-2s}$, when restricted to…
For every positive integer $k$, we show that every graph of order $n$ at least $3k$ with more than $$\max\{{2k-1\choose 2}+(2k-1)(n-(2k-1)),{3k-1\choose 2}+(n-(3k-1))\}$$ edges has $k$ vertex disjoint cycles, which is a best possible…
The model of dense lattice polymers is studied as an example of non-unitary Conformal Field Theory (CFT) with $c=-2$. ``Antisymmetric'' correlation functions of the model are proved to be given by the generalized Kirchhoff theorem.…
We describe in this paper additively left stable sets, i.e. sets satisfying $\left((A+A)-\inf(A)\right)\cap[\inf(A),\sup(A)]=A$ (meaning that $A-\inf(A)$ is stable by addition with itself on its convex hull), when $A$ is a finite subset of…
Under explicit diophantine conditions on $(\alpha,\beta)\in\RR^2$, we prove that the local two-point correlations of the sequence given by the values $(m-\alpha)^2+\break (n-\beta)^2$, with $(m,n)\in\ZZ^2$, are those of a Poisson process.…
We present preliminary results of an investigation of the clustering properties of high matter density peaks between redshift ~2 and ~3, as traced by Lyman limit and Damped Ly-alpha systems in spectra of close QSO pairs and groups.
It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and…
We propose a generalized Bell inequality for two three-dimensional systems with three settings in each local measurement. It is shown that this inequality is maximally violated if local measurements are configured to be mutually unbiased…
We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli…
In many areas of research it is interesting how lattices can be filled with particles that have no nearest neighbors, or they are in limited quantities. Examples may be found in statistical physics, chemistry, materials science, discrete…
We construct a set of 2^(2^n) independent Bell correlation inequalities for n-partite systems with two dichotomic observables each, which is complete in the sense that the inequalities are satisfied if and only if the correlations…
For $k\geq3$, a collection of $k$ sets is said to form a \emph{weak $\Delta$-system} if the intersection of any two sets from the collection has the same size. Erd\H{o}s and Szemer\'{e}di asked about the size of the largest family…