Related papers: Logical limit laws for Mallows random permutations

Let $\{q_n\}_{n=0}^\infty\subset [0,1]$ satisfy $q_0=0$, $\sum_{n=0}^\infty q_n=1$, and $\gcd\{n\geq 1\mid q_n\neq 0\}=1$. We consider the following process: Let $x$ be a real number. We first set $x=0$. Then $x$ is increased by $i$ with…

The Law of Large Numbers tells us that as the sample size (N) is increased, the sample mean converges on the population mean, provided that the latter exists. In this paper, we investigate the opposite effect: keeping the sample size fixed…

We consider the problem of inferring the probability distribution associated with a language, given data consisting of an infinite sequence of elements of the languge. We do this under two assumptions on the algorithms concerned: (i) like a…

The growth dynamics of complex systems often exhibit statistical regularities involving power-law relationships. For real finite complex systems formed by countable tokens (animals, words) as instances of distinct types (species, dictionary…

We show that several classes of ordered structures (namely, convex linear orders, layered permutations, and compositions) admit first-order logical limit laws.

Using the recently developed notion of permutation limits this paper derives the limiting distribution of the number of fixed points and cycle structure for any convergent sequence of random permutations, under mild regularity conditions.…

With Zipf's law being originally and most famously observed for word frequency, it is surprisingly limited in its applicability to human language, holding over no more than three to four orders of magnitude before hitting a clear break in…

We study the convergence of the empirical spectral distribution of $\mathbf{A} \mathbf{B} \mathbf{A}$ for $N \times N$ orthogonal projection matrices $\mathbf{A}$ and $\mathbf{B}$, where $\frac{1}{N}\mathrm{Tr}(\mathbf{A})$ and…

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic…

Let $\a$ be a complex random variable with mean zero and bounded variance $\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being i.i.d. copies of $\a$. Let $\lambda_{1}, ..., \lambda_{n}$ be the eigenvalues of…

Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. In a recent paper [Random Structures Algorithms 29 (2006)…

Let $\lambda(n)$ denote the exponent of the multiplicative group modulo $n$. We show that when $q$ is odd, each coprime residue class modulo $q$ is hit equally often by $\lambda(n)$ as $n$ varies. Under the stronger assumption that…

We consider continuous relational structures with finite domain $[n] := \{1, \ldots, n\}$ and a many valued logic, $CLA$, with values in the unit interval and which uses continuous connectives and continuous aggregation functions. $CLA$…

Present human languages display slightly asymmetric log-normal (Gauss) distribution for size [1-3], whereas present cities follow power law (Pareto-Zipf law)[4]. Our model considers the competition between languages and that between cities…

It is known that, when $n$ is even, the number of permutations of $\{1,2,\dots,n\}$ all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Heged\H{u}s and Roichman recently found a…

Consider the product $G_{n}=g_{n} ... g_{1}$ of the random matrices $g_{1},...,g_{n}$ in $GL(d,\mathbb{R}) $ and the random process $ G_{n}v=g_{n}... g_{1}v$ in $\mathbb{R}^{d}$ starting at point $v\in \mathbb{R}^{d}\smallsetminus \{0\} .$…

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(\sigma)=\sum_{n=1}^\infty X_nn^{-\sigma}$. We prove that the following holds almost surely…

We derive new explicit bounds for the total variation distance between two convolution products of $n$ probability distributions, one of which having identical convolution factors. Approximations by finite signed measures of arbitrary order…

We consider random polynomials $p_n(x)=\xi_0+\xi_1+\dots+\xi_n x^n$ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded $(2+\epsilon)^{th}$ moment (for some $\epsilon>0$), also known as…

We present a bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruple of statistics considered by Behrend, Di Francesco and Zinn--Justin. This bijection involves the…