656 problems

For a planar lattice $\Lambda$ in Euclidean space, let $K_\Lambda$ denote the lattice-dependent constant governing the asymptotic critical-temperature scaling under iterative trian…

Let CLE$_4$ denote the conformal loop ensemble with parameter $4$. The double-dimer model is a random collection of loops, and the loop $O(n)$ model is parametrized here by $n=2$ a…

Let $Q=Q(G)$ be a bipartite quadrangulation of the torus, so $Q\text{ belongs to }\mathcal{Q}_0$, and let $q_c$ denote the critical value of the $q$-state antiferromagnetic Potts m…

Consider the one-dimensional long-range Ising model with interaction exponent $\alpha$; a phase transition means the existence of more than one Gibbs measure. Kac–Thompson conjectu…

For an integer $q\geq 3$ and real $\beta>0$, let $G$ be a random $d$-regular graph and let $c\mu_{G,\beta}$ be the ferromagnetic Potts-model distribution on configurations $[q]^V$,…

In an integrable model, let the long-time asymptotic stationary state be the state reached after the system evolves for arbitrarily long times from a nonequilibrium initial conditi…

Recurrence conjecture. The model is recurrent in $d=2$ even with weak reinforcement.

Large-$N$ critical energy conjecture. In the large-$N$ limit,

A self-avoiding walk (SAW) is a contiguous sequence of moves on a lattice that does not cross itself and does not visit the same point more than once. In two dimensions, consider t…

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…

Consider the six-vertex model with domain-wall boundary conditions on an $N$ by $N$ lattice in the disordered region, and let $Z_N$ be its partition function. Let $C$ and $\kappa$…

Let $p_e$ be the point where the 1RSB and replica-symmetric (RS) error exponents coincide, with $p_e>p_{\rm 1RSB}$. Let $p_{\rm RS}$ denote the spinodal point below which the RS so…

Let $a$ be a spin taking values modulo $N$, let $\omega$ be an $N$th root of unity, and define the order parameter by … Here $k$ is the universal chiral-Potts-model constant, and…

Let $q$ be the number of states in the Potts model, and let $(_+)$ denote the specified antiferromagnetic critical curve in the model's phase diagram. Baxter's conjecture. The curv…

Consider the arboreal gas $\mathcal{A}$ on a two-dimensional graph, whose configurations are forests and whose edge parameter is conventionally $x_e=\beta$ for every edge $e$. Arbo…

Thermodynamic limit of the random-cluster partition function. As $n\rightarrow\infty$,

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 $d>4$, let $\Lambda_n$ be the finite torus used for the critical Ising model with periodic boundary conditions, and set $\beta_c=\beta_c(d)$. Let $\bar{Y}_n$ be the normalized…

Stationary-measure conjecture. The process $(\widetilde H_t)_{t\in[0,\tau]}$ represents the stationary measure of the KPZ equation on the interval $[0,\tau]$ with Neumann boundary…

For a finite domain $V\subset\mathbb{Z}^d$, define strong spatial mixing as follows: there exist constants $C,m\in(0,\infty)$ such that for every $\Delta\subset V$ and…

Let $\theta\sim\mathbb{P}_{\beta,\mathbb{Z}^2}^{\operatorname{Vil}}$ be the Villain model at inverse temperature $\beta$, and let $\phi$ be a Gaussian free field with inverse tempe…

Consider the line with $i_{13}=24$, which is a singlet level, and its finite-size amplitude fit. Let the Jordan cell referred to in the source contain two fields. Jordan-cell ident…

Let ${\mathcal U}$ be a critical update family with an infinite number of stable directions. Let $q$ denote the vacancy probability, let $\tau_0$ be the infection time at the origi…

Feynman's conjecture. BEC is signalled by the decisive appearance of a macroscopic amount of “infinite” cycles, that is, cycles whose lengths diverge with the number of particles.

Loop $O(n)$ exponential-decay conjecture. The loop $O(n)$ model exhibits exponential decay of correlations when $n>2$.