94 problems

Let $D$ denote the activity parameter of the high-density hard-core model on $\mathbb{Z}^3$, and let a Gibbs measure be an infinite-volume equilibrium measure for this model. Uniqu…

Random-current sharp-transition conjecture. The single random current on $\mathbb{Z}^d$ has a unique sharp percolative phase transition, namely

Let $d\geq 2$ and $q\geq 1$. For the random-cluster model on $\mathbb{L}^d$, let $\phi_{p,q}^0$ and $\phi_{p,q}^1$ be the infinite-volume random-cluster measures with free and wire…

Let $q\geq 3$, $r\in \\{1,\dots,q-1\\}$, and consider the fuzzy Potts model on the torus $\mathbb{T}^d$. Denote by $J_c$ the critical value corresponding to non-uniqueness of Gibbs…

Let $\operatorname{FP}_{p,5,2}(X)$ be the $p$th frame potential of a five-point configuration $X$ in $\mathbb{R}^2$, and let $X_5^\perp$, $X_5^{(h)}$, $Y(\alpha)$, and $Z(\alpha)$…

Decelle–Krzakala–Moore–Zdeborová conjecture. For the block model: (I) for all $q$, it is possible to detect communities better than random if $d\lambda^2>1$; (II) for $q\leq 4$, it…

Stability conjecture. If $n\leq 7$, then all level sets of $u$ are hyperplanes.

Two-transition conjecture. There exists $\bar N$ such that, for every $N\geq\bar N$, the model undergoes two phase transitions: for

Let $\rho(c)$ denote the giant-component fraction as a function of the scaling parameter $c$, and let $c_0$ be the critical value. Theorem5(i) establishes analyticity under conditi…

Pemantle–Steif conjecture. If

Let $T$ be an infinite tree, and consider the contact process on $T$ started from a single infection. Write $\lambda$ for the infection rate and let the survival probability be the…

Let $T$ be an infinite tree. Call a vertex $v$ essential if $T\setminus\{v\}$ has at least two infinite components. Say that $T$ has strongly exponential growth if there is some in…

Let $T$ be an infinite tree, and let $\rho$ be its root. For the contact process on $T$, define … … … and … where $\overline\mu$ is the upper invariant measure and $\delta_0$ is th…

Real sample covariance conjecture. For real sample covariance, Theorem 1 should still hold, with different limiting distributions but the same scaling. In particular, the critical…

Let $p_c(n)$ be the critical probability for the random subgraph of the $n$-cube, let $V=2^n$, and write … Assume $\epsilon\rightarrow0$ and…

MAX-SAT and SAT equivalence conjecture. For any $c<0$,

Satisfiability threshold conjecture. For each $k$ there exists a threshold density $c_k$, such that for any positive $\varepsilon$, for all $c<c_k-\varepsilon$, $F(n,cn)$ is almost…

Let $c^*_K$ be the constant introduced in the source's theorem for the optimal value ${\cal LP}(n,c)$ of the random K-LSAT linear program, and consider the corresponding feasibilit…

Fix an integer $k\geq 2$, and let $G(n,cn)$ be the sparse random graph with $n$ vertices and $cn$ edges. The graph is $k$-colorable when its vertices can be assigned colors…

A random K-SAT instance has $n$ Boolean variables and $m=cn$ clauses, where $c$ is the clause-to-variable ratio and $K$ is the number of literals per clause. The linear phase trans…

Consider the random location deaths (RLD) process: species are born with probability $p$, die with probability $q=1-p$, and, on a death event, the least fit species is removed with…

Tóth's conjecture. The quantity

Consider the minimum-energy problem for $k$ points on the unit interval with pairwise kernel $e^{-|x-y|^q}$, and write an odd number of points as … Let the global critical exponent…

Critical-fugacity asymptotic conjecture.

Height-three percolation conjecture. There exists $p_c\in(0,1)$ such that, for all $p<p_c$,