45 problems

Let $\nu$ be a unit vector with $|\nu_x|\not=0$, and assume that the Fourier transform $\mathcal{F}J(\xi)$ is maximized at $\mathcal{F}J(\xi_{\perp \hat{\nu}_x})$. Let…

Consider the two-dimensional Ising model on the periodic torus … with nearest-neighbor ferromagnetic coupling $J$, Hamiltonian $\mathcal H_L$, periodic striped configurations…

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 $G$ be any graph equipped with a ferromagnetic Ising model, with couplings $J_{xy}$, and let $\tau_2$ denote the relaxation time of its Glauber dynamics. Peres's monotonicity c…

In the East model, configurations are spin configurations on the positive integers in which a site can flip only when its left neighbour is in state $+1$, with product Bernoulli(…

Consider Ising models on $\mathbb Z^d$ with nearest-neighbor ferromagnetic interactions and antiferromagnetic long-range interactions whose couplings decay as a power $s$ of the di…

Let $\zeta\in[0,1/3]$, and let $\mathcal{Z}_a$ be the set of solutions associated with the poles of the iterates of $\tau_z$ described above. Define … and let…

Let $0<p\le 1$ and set $J=0$ in the one-dimensional model, modified as explained after equation (1.1). Consider the two alternating boundary conditions $\cdots +-+- \cdots$ and…

Entropy conjecture. The exact location of the multicritical point is given by

Multicritical-point conjecture. The exact location of the multicritical point is determined by $x_0=x_0^{*}$ with $K=K_p$, equivalently

Let the models in the first two columns of Table 3, namely Families I and II, be the classified genus-1 isoradial Ising models, with parameter $k$. Criticality conjecture. All mode…

Let $\ell_{\mathbb{Z}^d,t}$ denote the uniform even subgraph measure on the lattice graph $\mathbb{Z}^d$ with parameter $t$, where $d\geq 2$ and $t\in(0,1)$. Consider the convex se…

Pressure per particle of the extended Ising model. There exists $\varphi^{\scriptscriptstyle \rm eIsing}(\beta,k,h)$ such that

Let $J$ be the interaction matrix of an Ising model, with largest and smallest eigenvalues denoted by $\lambda_{\max}(J)$ and $\lambda_{\min}(J)$, respectively. The spectral-gap ha…

Agnostic lower-bound conjecture. No agnostic algorithm can match the oracle case when

Let $J_V(ij)=\mathrm{J}(|i-j|)$ be the interactions of the long-range Ising models on $V=\mathbb{N}$ and $V=\mathbb{Z}$, and let $\beta_c(J_\mathbb{N})$ and $\beta_c(J_\mathbb{Z})$…

Let $c=e^{-2\beta H}$ and $d=e^{-2\beta J}$, and consider the one-dimensional 3-permaspin model with couplings $J$ and $H$. Let $z$ be the unique root of the cubic … whose argument…

Chomsky–Schützenberger conjecture. The connection between ${\sf L}_{\mathrm{1d}}$ and lattice paths is a consequence of the Chomsky–Schützenberger representation theorem: every con…

Constructive automata conjecture. For every context free ${\sf L}_{\mathcal{M}}$ there exists a constructive PDA that decides it. For every context sensitive…

Constructive Chomsky hierarchy conjecture. These two new levels are identical with context free and context sensitive, respectively.

Let $s$ be a direction and let $\beta_{\rm sat}(s)$ and $\beta_c$ denote the saturation and critical inverse-temperature thresholds, respectively. NSA conjecture. NSA holds for any…

The tensor Curie–Weiss model and tensor Erdős–Rényi Ising models are considered, with the Hamiltonian scaled according to the underlying hypergraph model. Extension conjecture. Sim…

Let $H$ be the Hamiltonian of an Ising model on a torus with $N$ vertices, and let $R_H$ denote the difference between the maximum and minimum values of $H$. Consider the normalize…

Unique critical-line conjecture. There exists a unique critical line in the $J$-$q$ plane separating the ordered phase from the disordered phase, and this line lies in the region b…

Universality conjecture. Universality holds for the generalized Ising model: its exponents should coincide with their $\l=0$ values.