21 problems

Loop-model modules with through-lines conjecture. For $j>0$ and $2j$ through-lines, the scaling limit of $\mathscr W_{j,1}$ is the direct sum displayed above. This extends the prop…

Quotient loop-model module without through-lines conjecture. For generic $\mathfrak q$, the scaling limit of the quotient loop-model module without through-lines is the direct sum…

Let $q\ge2$, $k\ge2$, $\theta\ge1$, and $\alpha\in\mathbb{R}$. Let $\nu(\theta,\alpha)$ denote the number of positive solutions of the system of equations. Universal bound conjectu…

Let $q=3$, let $\theta\le\tilde{\theta}_1$, and let $\alpha=\alpha_1(\theta)$. Consider the system of equations in the variables $(u,v)$ given by. Uniqueness conjecture. The system…

Fix $q\in(1,4)$, set $r=1/q$, and let $\omega^\delta$ be the fuzzy Potts configuration on $\delta\mathbb{Z}^2$. For lattice approximations $z_i^\delta$ of…

Consider FK percolation for the 3-state and 4-state Potts models, with inverse temperatures $\beta^{q=3}$ and $\beta^{q=4}$ satisfying … These shifts correspond to the expected cor…

Let $q \geq 2$ be an integer, let $\beta \in [0,1]$, and consider the $q$-state antiferromagnetic Potts model on graphs of maximum degree $\Delta$. Define … Weak spatial mixing (WS…

Take a polygon $\Omega$ whose boundary is composed of edges of a hexagonal lattice of mesh $h$, and let $A_1,A_2,A_3,A_4,A_5,A_6$ be boundary vertices. Assign each hexagon one of f…

Monotonicity conjecture. For each $m=1,\dots,q-2$, the function $\theta\mapsto\alpha_m(\theta)$ is monotone increasing. This would give the monotonicity observed in the plotted exa…

Monotonicity conjecture. The function $\alpha_-(\theta)$ is monotone decreasing for all $\theta\geq\theta_{\rm c}$, while $\alpha_+(\theta)$ is decreasing for…

Let $G$ be a $d$-regular graph, let $q\ge d+1$, let $\beta>0$, and let $U^q_G(\beta)$ denote the objective associated with the Potts-model linear-programming relaxation. Davies–Jen…

Let $d\ge 3$ and $q\ge 3$. Consider the nearest-neighbor Potts model on $\mathbb{Z}^d$. Discontinuity conjecture. The phase transition of this model is discontinuous. The claim is…

Partition-LP tightness conjecture. The minimization LP is tight and yields, for every $d$-regular graph $G$,

Regular-graph Potts bounds. The conjectured inequalities are

Let $G=(V,E)$ be a graph, let $q$ be a non-negative-integer variable, and let $x,y$ be real variables. Define the $q$-dichromate by … where $\operatorname{\mathcal{C}}(A)$ is the s…

Consider a family of non-planar connected graphs $G_{n,L}$ with $n$ identical layers of width $L$ and periodic boundary conditions in the $n$-direction, together with a curve…

Linear-coefficient conjecture. Both displayed expansions hold. The coefficients have been checked computationally in the ranges reported in the paper, and the relations among the a…

Free-energy expansion conjecture. The two displayed asymptotic expansions hold. This has been verified for $m\le 10$ using the explicitly computed polynomials $C_1,\ldots,C_{10}$.…

Let $G$ be a finite planar self-dual lattice, or let $G$ be the square lattice with free or periodic boundary conditions in the thermodynamic limit. For $q\geq 1$, introduce $v$ by…

Mézard–Montanari's conjecture. If $q\leq 4$ and $d<d_{\max}$, then there is reconstruction if and only if

Equality conjecture. We believe that