142 problems

Central limit conjecture. There exists a sequence of positive numbers $(\sigma_n^2)_{n\ge1}$ such that

Consider a branching random walk in which the offspring variables are independent and identically distributed, independent of the parent's position, and the displacements are indep…

Let $\mathcal D_n$ be the disordered random full binary tree with $n$ leaves, and let $N_t(\mathcal D_n)$ denote the number of occurrences of a fixed full binary tree $t$ as a frin…

Spohn's central-limit conjecture. For every fixed $\alpha>0$, as $T\to\infty$ this rescaled path increment should converge in distribution to a centered three-dimensional Gaussian…

Let $U$ be Haar-distributed on the unitary group and let $j\in\mathbb{Z}_{+}$. The normalized trace $\operatorname{Tr} U^j/\sqrt{j}$ converges in distribution to a standard normal…

Let $\mathcal{N}_n^\circ[f]$ denote the centered linear spectral statistic associated with a test function $f$, and suppose its limiting variance exists. Johansson's conjecture. Th…

Let $(g_n)_{n\geqslant 1}$ be independent and identically distributed random elements with law $\mu$ on $\operatorname{GL}(V)$, where $V=\mathbb{R}^d$, and let … For $v\in V$ and…

Gaussian convergence conjecture. If

Athreya–Karlin conjecture. For the homogeneous generalized Friedman urn model, $\bm N_n$ is asymptotically normal.

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…

Consider the density fluctuation field for the one-dimensional asymmetric simple exclusion process in the longer time scale, after subtracting the velocity of the system. Static de…

Let $K$ be a convex set with volume one in ${\mathbb R}^d$. Choose $n$ random points independently and uniformly in $K$, and let $K_n$ be their convex hull. Write…

Let $\alpha\geq 1$ and let … For every real $x_0$, define the approximation error by comparing the distribution of $W_{\alpha}$ under Jack measure with the standard normal distribu…

Equality of random-cluster thresholds. The threshold $p_g^{*}$ is conjectured to equal $p_c$.

Consider the four quantum central limit results discussed in the source: the theorem for a pair of conjugate variables, its generalization to arbitrary canonical commutation relati…

Let $\mathcal{M}$ be a von Neumann algebra, let $\rho$ be a normal state on $\mathcal{M}$, and let $A_1,\ldots,A_a$ be bounded self-adjoint elements of $\mathcal{M}$. Let…

Let $M_n$ be the right boundary of the population at time $n$, defined by … Suppose the population model is one-dimensional, so $d=1$, and let $v$ be the almost-sure linear spreadi…

Let $(X_i)_{i\geq 1}$ be a phi-mixing process, let $\phi$ denote its mixing coefficients, and define … Assume that $\sigma_n^2\longrightarrow\infty$. Ibragimov–Linnik's conjecture.…

Let $\mathcal T_{\boldsymbol{n}_\kappa}$ be a random tree with prescribed degree-count vector $\boldsymbol{n}_\kappa$, and let $N_{m_\kappa}(\mathcal T_{\boldsymbol{n}_\kappa})$ de…

The preceding variance bound concerns triangulations in the regime where $p$ may approach $1$. Asymptotic equality conjecture. The result should hold with asymptotic equality, and…

Let $\mathcal{T}_n$ be a critical Galton--Watson tree with offspring distribution $\xi$ satisfying the standard extinction, criticality, and span-one conditions, and let…

Bivariate normal limit conjecture. The centralized-scaled pair tends, in distribution, to the bivariate normal distribution with covariance

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