16 problems

For the corresponding hard-hexagon problem on the triangular lattice, let $\kappa(1)$ denote the exponential growth constant of the number of admissible particle configurations at…

Let $Z(M,N)$ be the hard-square partition function on an $M\times N$ torus, and let the roots of the width-$N$ transfer-matrix characteristic polynomial be grouped into strings ind…

Let $T_N$ be the transfer matrix for a periodic row of width $N$, let $D_N$ be its dimension, and define its characteristic polynomial by … Write the roots of $P_N(x)$ as $x_i(N)$.…

Let $G$ be a finite graph of maximum degree $\Delta$, let $P$ be the Glauber transition matrix at complex fugacity $w$, and let $\operatorname{spr}(P)$ denote its spectral radius o…

Let $G$ be a finite claw-free graph, meaning that it has no induced subgraph isomorphic to $K_{1,3}$. Let $Z(w)$ be the hard-core lattice-gas partition function of $G$. Hamidoune–S…

Let $G$ be a finite graph of maximum degree $\Delta$, let $Z(w)$ denote its hard-core lattice-gas partition function with equal fugacity $w$, and let $D_\Delta$ be a complex domain…

Let $G$ be a countably infinite graph of maximum degree … , the hard-core lattice gas on $G$ has a unique Gibbs measure whenever … for all vertices $x$; perhaps this remains true w…

Consider the four-dimensional finite-range lattice-gas model with blank particles and non-negative interaction energies, whose ground states include the all-blank configuration and…

Consider the four-dimensional finite-range lattice-gas model whose ground-state configurations include standard configurations, constant in the first two directions and correspondi…

Distinct-correlation-lengths conjecture. The existence of two distinct correlation lengths is a common feature of the conservative lattice gas in every dimension

Uniqueness conjecture. For every such $D$, the EPGM is unique, at least for sufficiently large $u$; this unique EPGM is called disordered.

Soft-core transition-time conjecture. There exists a time $t_{NG}(\beta)$ such that the time-evolved measure is Gibbs for all $t<t_{NG}(\beta)$ and non-Gibbs for all…

Let $E=\{-1,0,1\}$, and call a lattice configuration fully occupied when it contains only $+$ and $-$ spins, equivalently when it has no $0$ spins. A configuration is called bad fo…

Let $E=\{-1,0,1\}$, and let $\alpha\in\mathcal{M}_1(E)$ be asymmetric, meaning $\alpha(1)\neq\alpha(-1)$. For the hard-core model, write…

The AHR model is a two-component lattice gas with normal-mode couplings whose leading coefficients produce KPZ scaling. Nonlinear fluctuating hydrodynamics conjecture. Nonlinear fl…

Cubic-current height fluctuation conjecture. The height at the origin should satisfy