36 problems

Let $X^{(n)}_3$ denote the number of distinct adjacent pairs of distinct letters in a geometrically distributed word of length $n$. Gaussian-limit conjecture. The asymptotic distri…

Let $(\mathcal{M},\rho)$ be a quantum probability space, and let $A_1,\ldots,A_k$ be self-adjoint elements with mean $0$ and symmetric covariance matrix. For each $t$, write … and…

Critical logarithmic CLT conjecture. Let $d\in\mathbb{N}$. There exists a constant $σ_{d,d/2}^2\in(0,\infty)$ such that

Penrose–Ong conjecture. Suppose $d\in\mathbb{N}$. The limit theorems for $\d835α\in(0,d/4)$, namely the variance limits and the corresponding Gaussian convergence for the binomial…

GOE nodal-count central limit conjecture. For every $k\in[n]$,

Let $H$) be a fixed graph containing $t$ triangles and $s$ two stars. Under the same conditions as in the two-star conditional central limit theorem, let $F_n$ denote the number of…

Janson's asymptotic Gaussianity conjecture. Under this assumption, the number of vertices $|V_n|$ is asymptotically Gaussian.

Let $T>0$, let $\nu_p$ be the product Bernoulli measure of density $p$, and let $\hat{\mathcal{Y}}^N$ and $\hat{\mathcal{Y}}$ be the discrete and limiting random fields defined in…

Two-dimensional occupation-time central limit conjecture. If $d=2$ and $\eta_0$ is distributed according to $\nu_p$, then

Let $t\geq 0$ and let $\psi_1,\psi_2\in\mathcal{D}(\mathbb{T}^d)$. Write $\eta_N$ for the scaling factor, $P_t^N(\psi_1,\psi_2)$ for the isolation-by-distance quantity, $Z_t$ for t…

Let $X_n$ be an $n\times n$ generalized random Toeplitz, circulant, or symmetric circulant matrix with symmetric standard sub-Gaussian entries, meaning mean-zero, variance-one sub-…

Let $X_n$) be an $n\times n$ generalized random Toeplitz, circulant, or symmetric circulant matrix with standard Gaussian entries. Let $p\geq 2$ be a positive integer and set…

Let $m$, $n$, and $A(n)$ be as in Theorem, and for each $n$ let $c=c(n)$ be a uniformly random unit cost vector independent of $A(n)$. Write $z^*$ for the optimal objective value.…

Let $(X_n)_{n\geq 0}$ be the Ulam–Kac adder sequence, and let convergence in law be denoted by $\to$. Central-limit conjecture. There are positive constants $\mu$ and $\sigma$ such…

For fixed $d$, let ${\mathcal{F}}_d(n)$ be the set of permutations of $\{1,\dots,n\}$ satisfying $i-d\leq\sigma(i)\leq i+1$ for every $i$, and let the number of $k$-cycles in such…

Let $(\phi_j)_{j=1}^r$ be a finite orthogonal family of Hecke–Maaß cusp forms for $\operatorname{SL}(2,\mathbb{Z})$. Let $\mathcal{V}(x)$ be the associated vector of additive-twist…

Let $X$ be a random vector in $\mathbb{R}^n$ with independent and identically distributed entries, … Assume that $\mathbb{E}x_1=0$, $\mathbb{E}x_1^2=1$, and…

Configuration-count conjecture. For every vincular pattern $\sigma$ with $j$ blocks and every $1\leq l\leq M_{\underline{\sigma}}$,

Let $S_n$ be the number of non-zero elements of the random adjacency matrix under $\mathbb{P}_{n;\alpha,h}$, let $(\alpha_c,h_c)$ be the critical point, and let $u^*(\alpha_c,h_c)$…

Euler Sudler-product central limit conjecture. For every $t\in\mathbb R$,

Let $f\in S_2(\Gamma_0(q))$, let $c$ tend to infinity through positive integers satisfying $(c,q)=d$, where $d$ divides $q$, and let $a$ range over $(\mathbb Z/c\mathbb Z)^*$. Let…

Let $n\geqslant 1$, let $f\in L^{\infty}(\mathbb{R}^n)\cap L^{1}(\mathbb{R}^n)\cap L^{2}(\mathbb{R}^n)$, and let $B_{\lambda}$ be the block map. Quantum central limit conjecture. U…

Neumann surplus central limit conjecture. Then

Let $m$ and $n$ be the dimensions of the two subsystems, let $c=m/n$, and let $X$ be the standardized von Neumann entropy. Consider the fixed-trace ensemble of reduced density matr…

For each $i\in\mathbb N$ and strip size $K$, let $D_{i,K}$ be the random variable defined in the paper, with expectation $E(D_{i,K})$ and standard deviation $\sigma(D_{i,K})$. For…