23 problems

Hitczenko–Kwapień conjecture. $C=7/32$, with equality attained by the example $a_1=\dots=a_6$.

Let $W$ be an $n\times n$ matrix with independent entries distributed as $N(0,1)_{\mathbb C}$, and let $p$ be a polynomial in its arguments. Permanent anti-concentration conjecture…

Let $X\sim\mu$, where $\mu$ is a log-concave measure on $\mathbb{R}^n$, and let $f:\mathbb{R}^n\to\mathbb{R}$ be a polynomial of degree $d$. For a Borel measure $\nu$ on…

Let $d\in\mathbb{N}$. Let $H_n$ be the hyperoctahedral group acting on $\mathbb{R}^n$ by coordinate permutations and sign changes. A measure $\mu$ on $\mathbb{R}^n$ is isotropic if…

Let $U$ be a Haar-random unitary matrix, and let $\operatorname{perm}(U)$ denote its permanent. The quantity $n!$ is the relevant Gaussian-scale second moment, and…

Lowther's conjecture. The function $f$ is piecewise constant on $\mathbb{R}_{\geq 0}$, takes seven different values, and has discontinuities at

Let $\mathbb{T}$ denote the unit circle, let $X$ be sampled from the complex Gaussian ensemble $\mathcal{G}=\mathcal{N}_{\mathbb{C}}(0,1)^{n\times n}$, and let…

Let $G_{n,p}$ be the binomial random graph with $n$ vertices, where $p=n^{\gamma}$ and $-1<\gamma<-2/3$, and let $\alpha(G_{n,p})$ denote its independence number. For an arbitrary…

Let $G_{n,m}$ be the random graph with $n$ vertices and $m$ edges, and let $falpha(G_{n,m})$ denote its independence number. Suppose … where $falpha$ is a constant such that…

Convex-position Littlewood–Offord conjecture. If $n$ is sufficiently large in terms of $\varepsilon$ and $d$, then

Optimal point-concentration conjecture. In the setting of Theorem baby-forwards, one has $\Pr(X\in S)\le C_S/\sqrt n$ for some constant $C_S$ depending only on $S$.

Let $X=\sum_{i=1}^n a_i\varepsilon_i$ be a Rademacher sum, where the $\varepsilon_i$ are independent uniformly random signs taking values in $\{-1,1\}$, and suppose that…

Let $X\in\{0,\ldots,n\}$ be a random variable with $p_0p_n>0$ and probability generating function $f_X$. Let $\delta>0$ and $R\geqslant1$. If every zero $\zeta$ of $f_X$ satisfies…

Pemantle's conjecture. If $\sigma(X)\gg1$, then $X$ is approximately normally distributed. This conjecture was the original motivation for studying anti-concentration from zero-fre…

Let $\mathcal{D}$ be a distribution over $\mathbb{R}$ and set $p=\|\mathcal{D}\|_{\infty}$, where $\|\mathcal{D}\|_{\infty}=\max_x \mathcal{D}(x)$. Let $M$ be a random $n \times n$…

Variance-based anti-concentration conjecture. For every $x\in\mathbb{Z}$,

Fix $p\in(0,1)$ and a graph $H$ with $h$ non-isolated vertices. Let $G\in\mathbb{G}(n,p)$ be a binomial random graph, and let $X_H$ denote the number of copies of $H$ in $G$. For a…

Alon–Hefetz–Krivelevich–Tyomkyn's conjecture. One has

Relative anti-concentration conjecture. The probability of this event is at most a constant multiple of the ratio of the Euclidean norms of the two given vectors, up to an additive…

Anti-concentration norm-sum conjecture. There exist consecutive indices $i_0,\dots,i_0+n^{1-\varepsilon}$ such that

Let $P$ be a degree-$d$ multilinear polynomial in independent Rademacher variables $\xi_1,\dots,\xi_n$, let $r=\operatorname{rank}(P)\geq 2$, and let $I$ be an interval of length…

Let $n\geq 1$ and $t\geq 0$, let $\mu_t=\frac{1-e^{-t}}{2}\delta_{-1}+\frac{1+e^{-t}}{2}\delta_1$, and let $\mu=\mu_\infty$ be the uniform probability measure on $\{-1,1\}$. Write…

Let $n\geq 1$ and let $\gamma_n$ be the standard Gaussian probability measure on $\mathbb R^n$. For $t>0$, let $U_t$ denote the Gaussian diffusion operator, and for a measurable no…