37 problems

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…

Bauke–Mertens conjecture. In most circumstances, the local statistics of energies in random spin systems with discrete spin space should be the same as in the random energy model.

Let $\sigma^t$ be the spin configuration at time $t$, and let $\mathcal C_L$ denote the event that the configuration restricted to the specified box of scale $L$ is constant, eithe…

Consider the homogeneous ferromagnetic zero-temperature stochastic Ising model in dimension two, with spins taking values in $\\{+1,-1\\}$ and with domain walls separating clusters…

Let $\mu$ be any subsequential limit of the distributions $\mu^t$ of the spin configuration $\sigma^t$, let $R_*(t)$ denote the relevant domain-wall length scale, and call a set of…

Converse integrability conjecture. Every integrable spin system is H-integrable.

Conjecture. If $H$ is of Heisenberg type, then

Let $E_{\text{min}}(S)$ and $E_{\text{max}}(S)$ denote the extremal energy levels in the sector with total spin quantum number $S$, and let the authors' bounds be the parabolic upp…

Non-degeneracy conjecture. For every $c\in (\kappa h_{\min},\kappa h_{\max})$, $U_n$ given $\operatorname{Conset}_n(h,c)$ lies asymptotically in $\operatorname{Gibbsnondeg}(c)$.

Let $J$ be an interaction matrix, and let $\mathcal{T}_v\mu_{N,\rho}$ denote the tilted canonical Ising measure with linear tilt $v$. Write…

Modulated-symmetry conjecture. Gauging the $p$-form symmetry gives a modulated symmetry in the dual theory, characterized by dipole algebra involving $(d-1-p)$-form and $q$-form di…

A one-dimensional, finite-range spin model is a spin system on a one-dimensional lattice whose interactions have finite range. It has a positive flipping rate when every allowed sp…

Polynomial relaxation-time conjecture. For all lattice dimensions $d\geq 2$, all $N\geq 2$, and arbitrary boundary conditions, the relaxation time is at most of order $L^d$ at any…

A probability measure $(\Omega^V,\mathbb{P})$ on configurations over an infinite graph is a factor of IID (FIID) measure if it is obtained as a factor of an IID process. A function…

Fix a nonzero parameter $\lambda$, and let $Z^{\sigma_\Lambda}_{G}(\beta)$ denote the partition function of the Ising model with boundary condition $\sigma_\Lambda$. Suppose it is…

Fix parameters $\beta$ and $\gamma$, not both zero, and let $Z^{\sigma_\Lambda}_{G}(\lambda)$ denote the partition function of the 2-spin system with boundary condition…

A sequence of graphs labeled by a spin system has a local weak limit describing the distribution of its finite neighborhoods. Local weak-limit conjecture. Many global properties of…

Let $S_1$, $S_2$, and $S_3$ be positive integers or half-integers. For adjacent edges $e=\{1,2\}$ and $e'=\{2,3\}$, let $P_{12}=P_{S_1+S_2}(\vec{S}_1+\vec{S}_2)$ and…

Let a similar finite-range, ferromagnetic, translation-invariant and reflection-invariant system be obtained by replacing the Ising spins in Theorem 1 with spin variables in the Si…

Consider zero-temperature Glauber dynamics on $\mathbb{Z}^2$ started from an i.i.d. initial condition whose initial density of positive spins is $\rho$. A phase transition at densi…

For the antiferromagnetic spin systems under consideration, let the tree uniqueness region be the parameter regime in which the corresponding infinite … . Rapid-mixing conjecture.…

Six-body-term removal conjecture. The modified action obtained from $S$ by removing the six-body term $S_{6}$ should lead to the same behaviour as the original action $S$.

Let $I_s[\mathcal{A}_\infty\|\mathscr{M}(\mathcal{A}_\infty)]$ denote the minimum information loss for the infinitely many components of a spin-$s$ observable, optimized over the r…

Let $\Lambda\uparrow\mathbb{Z}^d$ with $d\geq 3$, and consider the $\mathbb{H}^{2|2}$ model with external field $h>0$. Suppose the spin-spin correlations decay exponentially at a r…

Planar-minimizer conjecture. When applicable, $\mu$ reaches its minimum only on planar configurations, except for a degeneracy associated with triangle leaves. Equivalently, in the…