26 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 $(G_n)_n=(V_n,E_n)_n$ be a sequence of finite, connected, vertex-transitive graphs with vertex degree $\bDelta_n$ diverging as $n\to\infty$. For a fixed sequence and fixed $u$,…

Perron eigenvector conjecture. The vector $\Psi$ is the unique eigenvector of $H$ corresponding to its largest eigenvalue $2n$.

Link-pattern balance conjecture. For any $n=1,2,\ldots$, one has

For $0\le n<2$, let $c(n)$ denote the central charge associated with colored doodles, and write $n=-2\cos(\pi g)$ with $\frac12\le g<1$. Model I universality conjecture. Colored do…

Vertically symmetric ground-state conjecture. The vector

Half-turn-symmetric ground-state conjecture. The vector

Consider the dense O(1) loop model with periodic boundary conditions and odd system size $L=2n-1$, and let $S_{2n-1}^{(1)}$ be the sum of the components of its groundstate wavefunc…

Consider the dense O(1) loop model with periodic boundary conditions and odd system size $L=2n-1$, and normalise its groundstate wavefunction so that its smallest element is $1$. L…

Consider the dense O(1) loop model with periodic boundary conditions and even system size $L=2n$, and let $S_{2n}^{(1)}$ be the sum of the components of its groundstate wavefunctio…

Consider the dense O(1) loop model with periodic boundary conditions and even system size $L=2n$, and normalise its groundstate wavefunction so that its smallest element is $1$. Le…

Consider the dense O(1) loop model with open boundary conditions and system size $L$, and let $S_L^{(1)}$ be the sum of the components of its groundstate wavefunction, normalised s…

Consider the dense O(1) loop model with open boundary conditions and system size $L$, and normalise its groundstate wavefunction so that its smallest element is $1$. Let $N_8(2n)$…

Consider the loop $O(n)$ model with $n\in (0,2]$ and boundary condition $\beta\in\mathrm{LP}_N$. Let $\varsigma^{\delta}$ be the random connectivity in $\mathrm{LP}_N$ induced by t…

Critical polynomial decay conjecture. One has

Let $M$ be a combinatorial map and $W$ a choice of loop weights, and write $Z_{M,W}$ for the associated loop-model correlation function. Assume the critical limit exists. The corre…

Let $\mathcal{M}_{0,4}(r_1,r_2,r_3,r_4)$ be the set of maps with the specified vertex data, and order signatures componentwise: $\sigma ot greater than or equal tosigma'$ when…

Fully simple loop-model conjecture. After the symplectic transformation

Let $\rho$ be a link pattern, let $\mathcal D$ be a diagram of $\rho$, let $L_\rho$ be the associated scheme, and let $L_{\rho,\pi}$ denote its fixed-connectivity part for a connec…

Let $F_{2n+1}(x)$ be the generating polynomial associated with the odd-sized periodic ground state, and let ${\rm AV}_{2n+1}$ and ${\rm AHT}_{2n+1}$ denote the corresponding altern…

Let $F_{2n+1}(x)$ be the generating polynomial for the odd-sized periodic ground state, and let ${\rm AHT}_{m}$ denote the corresponding alternating-sign-triangle sum rule. Odd per…

Let $F_{2n+1}(x)$ be the generating polynomial associated with the odd-sized periodic ground state, let ${\rm A}_n$ denote the number of $n\times n$ alternating sign matrices, and…

For the $O(n)$ loop model on the semi-infinite hexagonal lattice with $n\in[-2,2]$, associate a fugacity $x_{\rm c}(n) = 1/\sqrt{2+\sqrt{2-n}}$ with occupied vertices and an additi…

Negative-root multiplicity conjecture. All real roots of $\psi_\pi(t)$ are negative integers, and $-p$ occurs with multiplicity $m_p(\pi)$. Equivalently,