42 problems

Let $G$ be a random $\Delta$-regular bipartite graph on $2n$ vertices, and consider the anti-ferromagnetic $q$-state Potts model on $G$. Phase-transition conjecture. Whenever the p…

Let $\mathcal{G}$ be a class of finite graphs and let $\mathcal{V}$ be a subset of the complex plane. For a graph $G$ with edge weights $\{v_e\}$, let … For $G=(V,E)$ and distinct…

Two-dimensional Potts criticality conjecture. In two dimensions, the critical model exhibits interesting phase-coexistence properties for $q\leq 4$, and not for larger $q$. This de…

Potts monotonicity conjecture. If reconstruction is solvable for $M_1$, then it is also solvable for $M_2$.

Consider the symmetric Potts channel $M$ on $q\geq 3$ symbols, with $b$-ary tree branching factor $b$ and second eigenvalue $\lambda_2(M)$. Potts-model below-bound conjecture. Ther…

Rohde–Schramm's conjecture. The random cluster representation of the Potts model for $0<q<4$ is related to the SLE process.

Dotsenko's conjecture. The energy operator $\epsilon$ should correspond to the primary field $\phi_{21}$ for all $q\leq4$, implying

Chim–Zamolodchikov conjecture. The S-matrix elements with $\lambda$ in the range $[3/2,3]$ should correspond to the scaling tricritical models.

Let ${\cal B}$ denote the locus under consideration in the complex $q$-plane, and call a graph a global circuit when it has the corresponding global circuit structure. A necessary…

Let $d$ be the net parameter, let $\langle\!\langle G\rangle\!\rangle_d$ denote the corresponding net evaluation, and consider the unnormalized probability distribution…

Let $Q=\tau+2$ for DFib or $Q=2$ for the toric code, and let $\beta_c$ denote the Potts-model critical inverse temperature. For a net $G$, write $\langle G\rangle_\tau$ for its iso…

Let $\tau=\frac{1+\sqrt{5}}{2}$ and consider the DFib topological phase together with its connection to the $Q=\tau+2$ state Potts model. DFib critical-temperature conjecture. The…

Surface-free-energy conjecture. The function $f_\text{surf}(q)$ has a jump discontinuity at $q=q_0$, whereas $f_\text{bulk}(q)$ is continuous as a function of $q$.

Phase-diagram conjecture. Properties A–B hold throughout the entire Berker–Kadanoff phase, with $q_0(L)$ replaced by the boundary of the Berker–Kadanoff phase, both at finite size…

Consider the three-site Potts model on up-pointing triangles, with partition-function variable $w=(e^K-1)/q$. In the thermodynamic limit, let the zeros be viewed in the complex $w$…

Let $q$ be the number of Potts states, let $x=(e^K-1)/\sqrt q$, and let $P_G(q,x)$ be the partition-function polynomial for a finite self-dual square lattice or for a square lattic…

Let $L$ be the width of a triangular-lattice strip with cylindrical boundary conditions, let $B_k$ denote the Beraha numbers, and let $D(q)$ be the amplitude matrix. Cylindrical-st…

Let $B_2,B_3,\ldots$ denote the Beraha numbers, and let $q_0(\infty,\mathrm{tri})$ be the limiting threshold for chromatic roots of the infinite-size triangular lattice. The isolat…

Conjecture on the asymptotic exponent. The exponent $c$ is always an integer or a half-integer.

Conjecture on the large-$x$ expansion. The right-hand side of the coefficient formula for $P(i,j)$ is, to second-next-to-leading order in inverse powers of $x$,

Let $\nu=\nu(\beta,q,d)$ be Potts lattice gauge theory on $\mathbb{Z}^d$. Let $\sigma_N=[0,1]^2\times\{N\}\times\{0\}$ be a plaquette, and for a plaquette $\sigma$ let…

Let $\nu=\nu(\beta,q,d)$ be Potts lattice gauge theory on $\mathbb{Z}^d$ defined as a weak limit of finite-volume measures. For a rectangular boundary $\gamma$ in $\mathbb{Z}^d$, l…

Let $G\sim\mathcal{G}(n,n,d/n)$ be an Erdős–Rényi random bipartite graph, formed by independently including each edge between the two parts with probability $d/n$, and let…

Let $G$ be a random $\Delta$-regular bipartite graph, and let $Z_{G,q}(\beta)$ be the partition function of the anti-ferromagnetic $q$-state Potts model on $G$. Random-regular-grap…

Let $G$ be a random $\Delta$-regular bipartite graph on $2n$ vertices, let $\mu_{G,q;\beta}$ be the anti-ferromagnetic $q$-state Potts Gibbs distribution, and let an FPAUS mean a f…