141 problems

Critical-fugacity asymptotic conjecture.

Nienhuis's connective-constant conjecture. The connective constant of the hexagonal lattice equals

Let $d\geq 2$ and consider Bernoulli bond percolation on $\mathds{Z}^d$, with critical probability … An edge is open when its Bernoulli($p$) state is $1$, and an open cluster is a…

Yang–Baxter conjecture. The operators $X_i(u)$ obey the Yang–Baxter equation.

Positivity conjecture. For bipartite lattices, the coefficients satisfy $C_k\geq 1$.

Perturbation conjecture. Almost surely:

Higher-dimensional Weil conjecture. There exists a $d$-dimensional super lattice model $(V_1,\dots,V_d,R)$ in $\operatorname{Super}_{\mathbb C}$ such that, for every $n\ge 1$, the…

Two-dimensional Weil conjecture. For every endomorphism $M\in\operatorname{End}_{\mathcal C_{\mathbb F_q}}(X)$ there exist super Boltzmann data $(V_1,V_2,R)$ such that, for every f…

On the square lattice, consider the monomer–monomer correlation as a function of the separation and orientation of two monomers. Fisher–Stephenson's conjecture. The monomer–monomer…

Let $G$ be the infinite nearest-neighbor graph on the integer lattice in $d$ dimensions, and let ${\bf T}$ be the uniform spanning forest obtained as a distributional limit of unif…

Distinct-variable conjecture. There are no coinciding variables $x_i$.

Let $m$ be a discrete map from ${\bf Z}^2$ to ${\bf S}^2$, with degree defined through its extension to the one-point compactification, and let ${\cal F}(\cdot,|m^c)$ denote the Ka…

The model is defined by the nodal and antinodal Hamiltonians discussed above, with the approximations introduced in Section 5 used to obtain its effective description. Low-energy a…

Monotonicity conjecture. $H(d)$ is a strictly decreasing function of $d$. The preceding discussion identifies the two-dimensional value as $H(2)=1$; the conjecture concerns the beh…

Negative anomalous dimension conjecture. It has been conjectured that

Let a high-dimensional lattice have a particle density per site and an optimal magnetic flux per plaquette, namely a flux minimizing the ground-state energy. Flux phase conjecture.…

Let $n$ and $g$ be parameters, and let $\mathcal{G}_{n,g}$ denote the fundamental vector conjugate $A_{n,g}$ graph. Consider solutions of the Yang–Baxter equation based on…

Threshold conjecture. The critical $X=X_0$ is effectively a threshold beyond which the signal becomes effective.

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 $d$-dimensional lattice $\Lambda$, let $k_d$ denote its degree and let $z_{\Lambda}$ denote its spanning-tree growth constant. Large-dimensional spanning-tree entropy conject…

Let ${\cal B}$ be the limiting zero set for a strip of width $L$, let $\#E$ be its number of endpoints, let $\#C$ be its number of connected components, and let $\#Q$ be its number…

Let $w_0({\cal L})$ denote the relevant critical-point parameter for a regular two-dimensional lattice ${\cal L}$ of coordination number $r$. The inverse-coordination conjecture. T…

The width-two strip conjecture. The family $2_{\rm P}\times n_{\rm F}$ possesses this property for every finite length $n$. The observation at $w=-1/2$ supports the conjecture, but…

Let $f(w)$ be the spanning-forest free energy per site on a regular lattice of dimension $d\geq2$ and coordination number $r$, and let $f_0$ denote the entropy per site for spannin…

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…