14 problems

A high-dimensional random walk on a discrete point process is a random walk on such a point process in sufficiently large dimension. Failure conjecture. There are random walks on d…

Let $N_{\mathbb{R},\boldsymbol{\xi}}$ be the number of real zeros of a random polynomial from any model considered in Section. Assume that the coefficient distribution is subgaussi…

Let $\mu\in\mathcal{P}(\mathbb{R}^d)$ be an isotropic distribution, meaning that … Let $\nu_n$ denote the normalized $n$-fold convolution sequence from the central limit theorem, s…

Weighted fifth-order central limit conjecture. There exists such a weighting function $\bar w$ and a universal constant $C_3>0$ for which

Let $\alpha(n)$ be the random completely multiplicative function appearing in the paper, let $\mu$ be the Möbius function, and let $m_{x,\infty}$ be the random measure defined in t…

Asymptotic normality conjecture. There are functions $f(n)$ and $g(n)$ such that, at least for good $n$,

Let $M$ be a real algebraic manifold of dimension $n$, let $s_d$ be a random section in the complex Fubini--Study model in dimension $n$ and codimension $r\in\{1,\dots,n\}$, and le…

Let $\mathcal N_n$ and $\mathcal L_n$ denote the relevant families of Newman and Littlewood polynomials of degree $n$, and let $N$ and $\widetilde N$ count zeros in the open unit d…

Fluctuation-regime conjecture. (i) If $E(N,2)=\mathcal{O}(1/N)$ as $N\to\infty$, then the proportion of heads obeys a central limit theorem. (ii) If $E(N,2)=\mathcal{O}(1/N)$ fails…

Infinite-second-moment examples conjecture. There exist random walks on discrete point processes that satisfy a Central Limit Theorem but do not have all of their second moments fi…

Finite-second-moment conjecture. Theorem $$ remains true under the weaker assumption that only the second moments are finite.

A random walk on a discrete point process is the model studied in the paper, and a central limit theorem means convergence of its appropriately normalized position to a Gaussian li…

Let $a_n(h)$ be the first-passage percolation time across a cylinder of length $n$ and half-height $h$, and define … For distributions satisfying the stated moment and admissibilit…

Let $u{\kappa,\beta,mu}$ be the distribution of the random point measure $4xi{\kappa}4$ on $4\mathbb{R}^d4$ for $4\kappa04$, and let $4\mathfrak{L}4$ denote Lebesgue measure on…