77 problems

Nienhuis's critical-point conjecture. The point $x=x_c(n)$ is critical. This is the conjectured critical point for the loop $\mathrm{O}(n)$ model throughout $n\in[0,2]$. The source…

Combinatorial-map polynomiality conjecture. Solutions in this basis should have polynomial structure constants.

Let $L$ be an even circumference, let $P[c]$ denote the probability of connectivity state $c$, and let $A_{\mathrm{HT}}(L)$ denote the number of $L\times L$ alternating sign matric…

Let $D_{(r,s)}$ be the structure constants in the conformal-block decomposition of a torus one-point function, with contractible loop weight $n$, channel loop weight $w$, and exter…

Loop $O(n)$ exponential-decay conjecture. The loop $O(n)$ model exhibits exponential decay of correlations when $n>2$.

Zuber's polynomiality conjecture. The quantities $A_{(\pi)_p}$ and $\psi_{(\pi)_p}$ are polynomials in $p$.

On the honeycomb lattice, an $O(N)$ configuration $c$ is a family of disjoint finite self-avoiding loops. If $l(c)$ is the number of loops and $L(c)$ their total length, configurat…

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

Let $q$ be a generic parameter, and consider the higher-spin loop model with fusion level $\ell$. The unfused model has a level-$1$ (“free boson”) quantum KnizhnikZamolodchikov (…

Pascal-matrix formula conjecture. The quantity $Q(L,m)$ is an integer given by

Let $L$ be odd and consider a half-infinite cylinder of circumference $L$ in the $O(1)$ loop model. The modified connectivity state has one point not connected to any other point o…

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…

Let the disc polynomials be the polynomials associated with disc link patterns, and evaluate them at $z_i=1$. Let the loop fugacity be parametrized by the variable used in the disc…

Let the basis polynomials be indexed by link patterns, with spectral parameters $z_i$, and let the loop fugacity be the parameter of the loop model. Kasatani–Pasquier's positivity…

Same-loop connectivity conjectures. For $n=2,3,4$, respectively,

Distinct-loop connectivity conjecture. One has

Non-contractible-loop probability conjecture. The probability is

Normalized-probability integrality conjecture. The quantity

Nested-connectivity conjecture. The probability of the completely nested connectivity state is $1$ for even $L$, and the probability of the corresponding completely nested state wi…

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…