104 problems

Let $H$ be a finite edge-weighted graph, possibly with loops, whose adjacency matrix $A_H$ is entrywise nonnegative and has at most one positive eigenvalue, counted with multiplici…

Let $D$ denote the activity parameter of the high-density hard-core model on $\mathbb{Z}^3$, and let a Gibbs measure be an infinite-volume equilibrium measure for this model. Uniqu…

Let $q\ge2$, $k\ge2$, $\theta\ge1$, and $\alpha\in\mathbb{R}$. Let $\nu(\theta,\alpha)$ denote the number of positive solutions of the system of equations. Universal bound conjectu…

Let $q=3$, let $\theta\le\tilde{\theta}_1$, and let $\alpha=\alpha_1(\theta)$. Consider the system of equations in the variables $(u,v)$ given by. Uniqueness conjecture. The system…

Two-transition conjecture. There exists $\bar N$ such that, for every $N\geq\bar N$, the model undergoes two phase transitions: for

Non-integer BP solution conjecture. The non-integer solution extends beyond the small $T_c+\varepsilon$ vicinity of $T_c$ and transitions smoothly as $T\to\infty$ into the fully ho…

Let $q$ be the number of Potts symbols, let $\theta=\tanh(\beta)$, and let $d_0$ denote the random offspring degree under the tree measure $Q$. Reconstruction is the problem of rec…

Let $d\geq 3$ be the maximum degree, let $\ell$ be the activity, and let $\ell_c(d)=(d-1)^{d-1}/(d-2)^d$ be the critical activity for decay of correlations on the infinite $d$-regu…

Let ${\mathbb T}$ be a periodically tiled torus with lattice of periods $\Lambda$, and let ${\mathbb T}_n={\mathbb T}/n\Lambda$ be such that ${\mathbb T}_n^*$ is bipartite. Let…

Let $\mu_{s,t}$ denote the measure on domino tilings of the plane associated with tilt $(s,t)$. A measure is conditionally uniform when, given a tiling outside any finite region, t…

In the dimer model on an $n \times n$ torus, let $p_a$, $p_b$, $p_c$, and $p_d$ denote the limiting densities of the four edge types in the thermodynamic limit as…

Let $n$ denote the size of a triangulation, and let the average number of its nodes of degree $d$ be divided by $n$, where $d$ is an integer larger than $2$. Polynomial degree-dist…

Recursive upper-bound conjecture. There exists a geometric constant $C<\infty$ such that

Refined flat-facet conjecture. The event that there are such planes for which $L_i\cap\Gamma(\sigma)$ are flat facets in the following sense has probability tending to $1$ as…

Let $L$ be the sector label and let $\varepsilon_m$ denote the quantity introduced in the paper for sector $m$. Ground-state energy ordering conjecture. … for all $m$ such that…

For $0\le n<2$, let $c(n)$ denote the central charge associated with colored doodles. Model II universality conjecture. Colored doodles are in the universality class of models with…

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…

Faceted-droplet hypothesis. As $l\to\infty$, with probability tending to $1$, there exist six distinct two-dimensional planes $L_i=L_i(\sigma)\subset\mathbb{T}^3$, $i=1,\ldots,6$,…

Let $q_0(L)$ denote the relevant real crossing for a triangular-lattice strip of width $L$ with fully periodic boundary conditions. Triangular-lattice 3p ± 1 conjecture. For all wi…

Let $q_0(L)$ denote the maximum real value in the limiting curve for a square-lattice strip of width $L$ with fully periodic boundary conditions, and let $q_{\rm c}(\mathrm{sq})=3$…

Let $q_0(L)$ denote the maximum real value in the limiting curve for a square-lattice strip of width $L$ with fully periodic boundary conditions. Odd-width square-lattice conjectur…

Reflection-permutation conjecture. The reflection permutation $\pi$ gives the true minimum for all $(q,n)$, namely

Let $V_N$ denote the relative phase-space volume of the union of KAM islands for an $N$-particle system at fixed energy density and fixed forcing parameters. Vanishing-of-islands c…

CFT scaling-limit conjecture. The scaling limits of two-dimensional statistical mechanics models undergoing a continuous phase transition are given by the correlation functions of…

Let $u(k,q)$ denote the size of the smallest set of uniqueness for the positive function class $\left(\mathcal{B}^{k}_{q}\right)_+$. In particular, consider the case $q=2$. Minimum…