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There has been much recent interest in the satisfiability of random Boolean formulas. A random k-SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known…
Models based on assumptions of multivariate regular variation and hidden regular variation provide ways to describe a broad range of extremal dependence structures when marginal distributions are heavy tailed. Multivariate regular variation…
The collective dynamics of neural populations are often characterized in terms of correlations in the spike activity of different neurons. Open questions surround the basic nature of these correlations. In particular, what leads to…
The Erd\H os Matching Conjecture states that the maximum size $f(n,k,s)$ of a family $\mathcal{F}\subseteq \binom{[n]}{k}$ that does not contain $s$ pairwise disjoint sets is $\max\{|\mathcal{A}_{k,s}|,|\mathcal{B}_{n,k,s}|\}$, where…
Schools with the highest average student performance are often the smallest schools; localities with the highest rates of some cancers are frequently small and the effects observed in clinical trials are likely to be largest for the…
We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the…
We prove a Large Deviation Principle for the random spec- tral measure associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N) and e is a fixed unit vector (and more generally in the $\beta$- extension of this model). The…
Let $Z(F)$ be the number of solutions of a random $k$-satisfiability formula $F$ with $n$ variables and clause density $\alpha$. Assume that the probability that $F$ is unsatisfiable is $O(1/\log(n)^{1+\e})$ for $\e>0$. We show that…
An equivalent condition for the product of elements of an independent random sample on a compact algebraic group converging in distribution to some random variable as the sample size increases is obtained. Namely, a limit distribution…
Neglecting many motivating details for the Park-Pham theorem (previously known as the Kahn-Kalai conjecture), the result starts with a finite set $X$, a non-trivial upper set $\mathcal{F} \subseteq 2^X$, and a particular parameterized…
In the panoply of pattern classification techniques, few enjoy the intuitive appeal and simplicity of the nearest neighbor rule: given a set of samples in some metric domain space whose value under some function is known, we estimate the…
By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special…
Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We…
Quantum Mechanics (QM) predicts the correlation between measurements performed in remote regions of a spatially spread entangled state to be higher than allowed by the intuitive concepts of Locality and Realism (LR). This high correlation…
A strong direct product theorem states that if we want to compute $k$ independent instances of a function, using less than $k$ times the resources needed for one instance, then the overall success probability will be exponentially small in…
Let (S_0,S_1,...) be a supermartingale relative to a nondecreasing sequence of \sigma-algebras (H_{\le0},H_{\le1},...), with S_0\le0 almost surely (a.s.) and differences X_i:=S_i-S_{i-1}. Suppose that for every i=1,2,... there exist…
In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to $(1+\sqrt{\gamma})^2$…
We extend the classical preferential attachment random graph model to random simplicial complexes. At each stage of the model, we choose one of the existing $k$-simplices with probability proportional to its $k$-degree. The chosen…
We give an asymptotic formula for correlations \[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are bounded "pretentious" multiplicative functions, under certain natural hypotheses. We then deduce…
Let $\mu$ be a measure that samples a subset of a finite ground set, and let $\mathcal{A}_e$ be the event that element $e$ is sampled. The measure $\mu$ is negatively correlated if for any pair of elements $e, f$ one has $\mu(\mathcal{A}_e…