83 problems

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…

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 $[q]^V$,…

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…

Let $q\geq 2$ and let $\beta_s$ denote the critical inverse temperature for the mean-field ferromagnetic Potts model. Consider the systematic scan dynamics at the critical temperat…

Potts-to-simple-CLE conjecture. The scaling limit of the critical $q$-state Potts model can be described by simple $\mathrm{CLE}_\kappa$. The conjecture identifies simple CLE as th…

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…

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

Potts critical-temperature conjecture. For the Potts model, the critical inverse temperature behaves as

Potts correlation-decay conjecture. Correlation decay can in fact be established in the regime

For the $q$-state Potts model, let $\theta^w(\beta)$ and $\theta^f(\beta)$ denote the wired and free FK-percolation probabilities at inverse temperature $\beta$, and let…

Consider the three-dimensional six-state Potts model in a box with boundary condition $i$ on the $i$-th face. Assume that the sharp simplex inequality holds, that the surface tensi…

Complex zero-free-domain conjecture. There exists a complex domain $D_r$ containing the real interval $(2r,\infty)$ such that

Generalized connected-sum monotonicity conjecture. If $|1+v_e|\leq 1$ for every edge $e$, then $|C_G(\lambda,\{v_e\})|$ is a decreasing function of $\lambda$ on…

Let $\Lambda$ be a lattice, let $W(\Lambda,q)$ denote the exponent of the ground-state entropy of the $q$-state Potts antiferromagnet, and define the reduced function…

For a family of strip lattices, let $f_n(z)$ be the finite-length partition functions and let the limiting zero set be the common limit of their zero sets as the length tends to in…

Let $q$ satisfy $0\leq q\leq4$, let $(_-)$ be the unphysical self-dual line, and let $\delta$ be the phase-diagram parameter. Assume the Berker–Kadanoff phase is described by the c…

Let $q$ satisfy $0\leq q\leq4$, let $(_-)$ be the unphysical self-dual line, and let $\delta$ be the parameter used for the Potts-model phase diagram. The Berker–Kadanoff phase is…

Let $\delta$ parameterize the antiferromagnetic critical curve $(_+)$, and let $x_T$ be the leading thermal scaling dimension; define the associated critical exponent by…

Let $\nu$ be the model parameter, let $\mathcal C:=\{(w,S_1)\}$ be the algebraic curve defined by the relation between $w$ and $S_1$, and let $U,V$ be coordinates on an auxiliary p…

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…

In the planar critical $q$-state Potts model, let $z_1,\ldots,z_n$ be distinct points and let $z_i^\delta$ be lattice approximations on $\delta\mathbb{Z}^2$. The properly normalize…

Consider the $D^{(2)}_3$ model with boundary conditions $(\varepsilon,p)=(0,1)$. Its symmetry algebra contains left and right $U_q(B_1)$ factors, with spins…

Let $G_n$ be a sequence of graphs converging locally to the $d$-regular tree, let $\beta$ and $B$ denote Potts-model parameters, and let $w$ and $B$ denote the corresponding random…

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 $\mathbb{G}$ be a random $d$-regular graph, and let $\mu_{\mathbb{G}}$ be the ferromagnetic $q$-state Potts distribution with parameter $\beta>0$, where $q$ and $d$ are integer…