19 problems

Consider a dilute random structure with density parameter $\alpha$, whose cluster expansion can be rearranged as a power series in $\alpha$. Above the percolation threshold, the pe…

Let $K$-SAT instances be represented as dilute spin-glass models, with a fictive temperature taken to $T\to0$ so that the ground-state entropy describes the logarithm of the number…

Fix $k\in\mathbb{N}$. For the Ising partition function $Z_{\mathbb{Z}_m^t}(\lambda,\beta)$ on the even torus $\mathbb{Z}_m^t$, let $L_j$ denote the sum of the weights of clusters o…

Fix $t$ and let $m$ tend to infinity. Consider the partition function of homomorphisms from $\mathbb{Z}_m^t$ to a fixed graph equipped with fixed vertex and edge weights. The clust…

Let $t$ be fixed, let $m$ tend to infinity, and consider weighted homomorphisms from the torus graph $\mathbb{Z}_m^t$ to a fixed graph. The cluster expansion is the expansion for t…

Extension conjecture. The conclusions established for the restricted class of Gibbs states and noise channels should hold in these more general cases as well.

In the setting of hard-core, tempered, radially symmetric pair potentials, let local stability be the pointwise condition assumed in the paper, and let stability mean that the pote…

Let $V$ be a stable pair potential with stability constant $B$, and let ${\rm\bf m}$ be the Penrose partition scheme. For $n\geq 2$, let $T_n$ be the set of trees on $n$ labelled v…

The virial expansion expresses thermodynamic quantities in powers of the density, while the activity expansion expresses them in powers of the activity. The convergence domain refe…

Cluster-expansion accuracy conjecture. The cluster expansion formalism can be used to provide accurate effective pair and three-body coarse-grained potentials at high $T$ and low…

Modified scaling conjecture. The limits obtained using $E_1$, $E_2$, or $E$ are equal and satisfy

Modified cluster-expansion limit conjecture. Using $E_1$, $E_2$, or $E$, the following limit exists:

Universal-limit conjecture. The limits obtained using $E_1$, $E_2$, or $E$ are equal, and

Cluster-expansion limit conjecture. Using $E_1$, $E_2$, or $E$, the following limit exists:

For the Federbush expansion of the matching entropy, write the coefficients as $a_k$. For the square lattice $\mathbb{Z}^2$, the computed values include … Equivalently, for the mod…

Ruelle-potential decomposition conjecture. Any stable and tempered pair potential can always be written as a sum of a non-negative potential plus an absolutely summable stable pote…

Ruelle-type decomposition conjecture. The class of Ruelle potentials actually coincides with the whole class of stable and tempered potentials.

Let $A=\{\tau_1,\tau_2\}$ be a two-polymer set, let $v(A)$ denote its weight, and let $\operatorname{Dom}(\tau_i)$ denote the domain associated with $\tau_i$. Suppose that the free…

Let $(\mathbb X,\mu)$ be the underlying measure space, let $u$ be a pair potential, and set $\zeta(x,y)={\rm e}^{-u(x,y)}-1$. Let $b$ be a nonnegative function satisfying the stabi…