Related papers: Random polynomials: the closest roots to the unit …

Let n points be chosen randomly and independently in the unit disk. "Sylvester's question" concerns the probability p_n that they are the vertices of a convex n-sided polygon. Here we establish the link with another problem. We show that…

Define $S(n,\beta)$ to be the set of complex polynomials of degree $n \ge 2$ with all roots in the unit disk and at least one root at $\beta$. For a polynomial $P$, define $|P|_\beta$ to be the distance between $\beta$ and the closest root…

Let $\mu$ be a probability measure in $\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \dots, z_n$ be independent random variables, $z_i \sim \mu$, and consider the random polynomial $$ p_n(z) =…

We consider a uniform distribution on the set $\mathcal{M}_k$ of moments of order $k \in \mathbb{N}$ corresponding to probability measures on the interval $[0,1]$. To each (random) vector of moments in $\mathcal{M}_{2n-1}$ we consider the…

We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all…

We obtain the asymptotic variance, as the degree goes to infinity, of the normalized number of real roots of a square Kostlan-Shub-Smale random polynomial system of any size. Our main tools are the Kac-Rice formula for the second factorial…

Consider a random power series of the form $P(z) = \sum_{n\ge 1} \varepsilon_n a_n z^{n}$ where $a_n \in \mathbb{C}$ are deterministic and $\varepsilon_n$ are chosen independently and uniformly at random from $\{\pm 1\}$. Kolmogorov's…

Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem,…

The famous Posa conjecture states that every graph of minimum degree at least 2n/3 contains the square of a Hamilton cycle. This has been proved for large n by Koml\'os, Sark\"ozy and Szemer\'edi. Here we prove that if p > n^{-1/2+\eps},…

Let $\eta_t$ be a Poisson point process with intensity measure $t\mu$, $t>0$, over a Borel space $\mathbb{X}$, where $\mu$ is a fixed measure. Another point process $\xi_t$ on the real line is constructed by applying a symmetric function…

We consider random permutations on $\Sn$ with logarithmic growing cycles weights and study asymptotic behavior as the length $n$ tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables…

We prove that for large $n$, all but $o(2^{n})$ polynomials of the form $P(z) = \sum_{k=0}^{n-1}\pm z^k$ have $n/2 + o(n)$ roots inside the unit disk. This solves a problem from Hayman's book 'Research Problems in Function Theory' (1967).

Consider a polynomial of large degree n whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros…

Let K be a convex set in R d and let K $\lambda$ be the convex hull of a homogeneous Poisson point process P $\lambda$ of intensity $\lambda$ on K. When K is a simple polytope, we establish scaling limits as $\lambda$ $\rightarrow$ $\infty$…

Consider a random permutation of $\{1, \ldots, \lfloor n^{t_2}\rfloor\}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$. Next,…

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 $k\ge 3$…

Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form $\sum_kY_k\alpha_k(z_t^\epsilon(\omega))$ where $Y_k$ are vector fields, $\epsilon$ is a positive number, $z_t^\epsilon$ is a…

Let $p(n)$ denote the number of partitions of a natural number $n$. As $ n \to \infty$, the $n$th root of $p(n)$ tends to $1$, which is related to the Cauchy--Hadamard test for power series. Andrews also discovered an elementary proof. Sun…

A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $\beta$ is one of those roots, then within one unit of $\beta$ lies a root of the polynomial's derivative. If we define $r(\beta)$ to be the…

A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $\beta$ is one of those roots, then within one unit of $\beta$ lies a root of the polynomial's derivative. If we define $r(\beta)$ to be the…