440 problems

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

Consider the Tree Builder Random Walk in which, at time step $n$, a new leaf is attached with probability $p_n=n^{-\gamma}$, where $\gamma$ is a real parameter. For…

Critical-fugacity asymptotic conjecture.

Let $k\geq 2$, and let $\varphi$ be a random $k$-CNF formula with $m=m(n)$ clauses and $n$ variables, where $m/n\to\alpha$ as $n\to\infty$. The ratio $\alpha$ is the clause density…

First-order phase transition conjecture. The phase transition on random $\Delta$-regular graphs is of first order, and

Let $q\geq 1$ and consider the random-cluster model on the square lattice $\mathbb{Z}^2$, with critical threshold … Let $\theta^{\mathrm{w}}(p,q)$ denote the probability that the o…

Higher-dimensional phase-transition conjecture. Similar results hold in higher dimensions whenever

For each integer $k\ge 2$, let $\boldsymbol{F}(k,\alpha)$ denote a random $k$-SAT formula with clause density $\alpha$. Sharp satisfiability threshold conjecture. For every…

Let the Ising model have short-range interactions on the lattice in spatial dimension $d$, with spins taking values in $\\{-1,1\\}$. A higher-dimensional absence-of-transition conj…

Let the wall attraction in the half-space directed random polymer vary between a strong attractive-force regime, also called the subcritical or bound phase, and a weaker-attraction…

Let a uniform hypergraph stochastic block model (HSBM) be a random uniform hypergraph whose vertices are assigned to communities and whose hyperedges appear independently according…

For $\kappa\in\mathbb{R}$, let $\delta_{\mathrm{s}}(\kappa)$ denote the phase-transition threshold for the existence of $\kappa$-margin solutions, and let…

Let $\Delta\geq 3$ and $q>2$, and consider the random cluster model on a random $\Delta$-regular graph. Denote its critical point by $\beta_c(q,\Delta)$. Critical-point formula con…

Uniqueness conjecture. For $p\neq p_c$ there is a unique Gibbs measure.

Krapivsky–Redner–Leyvraz conjecture. The critical probability is

Critical-speed conjecture. There is a critical speed $s_c\in(0,1)$ such that the first bullet survives with positive probability if and only if its speed is greater than $s_c$.

Let $W_n:=Z_n/\mathbf{E}(Z_n)$ be the normalized partition function, and let $W_n\to W_\infty$ almost surely. Define $\bar\beta_c\in[0,\infty]$ so that weak disorder holds for…

A continuous rotator model, also called the classical Heisenberg model, is a spin system on a tree whose phase transitions can be studied through the branching number of the tree a…

Lifshitz-point conjecture. The triple point is a Lifshitz point, that is, a multicritical point at which a disordered, a homogeneous (uniform) ordered, and a spatially modulated (n…

On the honeycomb lattice, an $O(N)$ configuration $c$ is a family of disjoint finite self-avoiding loops. If $l(c)$ is the number of loops and $L(c)$ their total length, configurat…

Two-dimensional FK criticality conjecture. In two dimensions, at the critical value $p_c$, the FK measure exhibits nice large-scale properties only up to $q=4$. This is the random-…

Two-dimensional Potts criticality conjecture. In two dimensions, the critical model exhibits interesting phase-coexistence properties for $q\leq 4$, and not for larger $q$. This de…

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…

Consider a general ferromagnetic SOS model with coupling $J<0$, temperature $T>0$, and an integer height parameter $m>1$. A translation-invariant splitting Gibbs measure is a Gibbs…

Let $m=2$, $J<0$, and $\beta\leq\beta^1_{\rm cr}$. Let $\mu^*$ denote the symmetric Gibbs measure associated with the unique solution in the ferromagnetic SOS model. Uniqueness con…