2,031 problems

Ibragimov–Iosifescu conjecture for φ-mixing sequences in probability theory is the collective name for 2 closely related conjectures by Ildar Ibragimov and ro:Marius Iosifescu.

Cohen–Lenstra conjecture. The proportion of discriminants $D\in S_X^{\pm}$ for which the Sylow $p$-subgroup of the class group of $\mathbb{Q}(\sqrt D)$ is isomorphic to $B$ satisfi…

Let $p$ be a prime, and consider the sequence of residue classes … More generally, these terms arise from sequences of values of a power, polynomial, or rational function modulo…

Let $\mathcal{R}$ be the family of random variables of the form $X=\sum_{k=1}^n a_k\varepsilon_k$, where $n\geq 1$, $a_k\in\mathbb{R}$, $\sum_{k=1}^n a_k^2=1$, and the…

Let $\sigma_n$ be a uniformly random permutation in $S_n$, and let $L(\sigma_n)$ denote the length of its longest increasing subsequence. Hammersley's conjecture. The limit … exist…

Let $A_n$ be an $n\times n$ random matrix whose entries are independent and identically distributed random variables taking values in $\{\pm1\}$. Let $s_n(A_n)$ denote its smallest…

Derrida–Retaux conjecture. Under the assumption $p_c>0$ and some integrability conditions on $\vartheta$, there exists a constant $C>0$ such that

Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb E(X_i)=\mu_i\leq 1$ for each $i=1,\ldots,n$, and let $\delta>0$. Feige's conjecture. … This strengthe…

Brunet–Derrida conjecture. The selection mechanism shifts the speed of the population by a correction term of order

Helson's conjecture.

Let $\sigma^{\mathrm{Unif}}_n$ be a uniformly random permutation of $\{1,\dots,n\}$, and let $\operatorname{LIS}(\sigma)$ denote the length of its longest increasing subsequence. U…

Let $X$ and $Y$ be independent identically distributed symmetric Bernoulli sequences, and let $\ell_n$ denote the length of their longest common subsequence among the first $n$ sym…

Let $G=(V,E)$ be a graph, let $boldsymbol{x}=xe:e otin E$ be positive edgeweights, and let the random cluster model on $G$ have parameter $q$. For distinct edges $e,f otin E$, writ…

Sharp upper-bound conjecture. Without any further assumption on the model, the sharp upper bound for $p(\beta)$ should be the bound in equation (e117). The source compares this pro…

Let the once-reinforced random walk (ORRW) with reinforcement parameter $c>0$ be run on the integer lattice $\mathbb{Z}^d$. Sidoravicius's recurrence conjecture. The walk is recurr…

Let $T_m(N;\alpha)$ denote the coupon-collection time for the interlacing family of coupon distributions studied in the paper, and suppose the random variable is normalized in the…

Let $(T)$, $(T')$, and $(T)_\gamma$ for $\gamma\in(0,1)$ be Sznitman's ballisticity conditions for a uniformly elliptic, independent and identically distributed random walk in rand…

Let $L_n$ be the length of the longest common subsequence of two random sequences. Waterman's conjecture. The variance grows linearly: … This conjecture addresses the order of fluc…

Samuels' conjecture.

Let $D\subset\mathbb{C}$ be a simply connected domain, let $x,y$ be distinct boundary points, and let $\eta$ be a chordal curve from $x$ to $y$. A random curve from $x$ to $y$ is r…

Let $(X_1,\dots,X_n)$ be a centered Gaussian random vector, and let $\alpha_j$ be non-negative real numbers for $j=1,\ldots,n$. Gaussian product inequality conjecture. … This is a…

Eaton's conjecture. For $x>\sqrt{2}$,

Let $G(n,d,p)$ be the spherical random geometric graph and let $G(n,p)$ be the Erdős–Rényi random graph. For probability distributions on a finite set, define total variation dista…

Brownian-sheet persistence conjecture. The probability that $\pi$ is below $\tau$ in the strong Bruhat order satisfies

Positive rates conjecture. Every one-dimensional PCA with strictly positive transition rates is ergodic.