113 problems

For any two hierarchical lattices arising in the stated Bernoulli-percolation setting, if their critical-exponent universality classes coincide, then their discrete renormalisation…

Consider the monomer–dimer model on the $d$-dimensional hypercubic lattice, with $d\ge 2$, chemical potential $\mu$, hard-core exclusion, and an attractive interaction of strength…

For every integer $r\ge 2$ and every integer $d\ge 1$, if $H$ is a finite simple $r$-uniform $d$-regular hypergraph, then the number of weak independent sets of $H$ satisfies…

For the discrete Derrida–Retaux models in the critical regime, after the model-specific rescaling of time and state, the resulting processes converge in distribution, in the releva…

For every finite graph $G=(V,E)$, every edge-parameter vector $(p_e)_{e\in E}\in[0,1]^E$, every $q\in[0,1]$, and every pair of distinct edges $e,f\in E$, the random-cluster measure…

Consider the directed random graph on $\mathbb{Z}^2$ in which each vertex independently chooses one of its four nearest-neighbour directions uniformly, places an arrow in that dire…

For every $\delta>0$ there exists $\varepsilon>0$ such that the following holds uniformly over every finite graph $G$, every percolation parameter $p\in[0,1]$, every origin vertex…

For every dimension $d\ge 3$, every prescribed slope $\theta\in\mathbb{R}^d$, and every general convex interaction, the corresponding finite-volume integer-valued $\nabla\phi$ heig…

For every infinite, locally finite, transitive graph $G=(V,E)$, let $\mathbb{P}_p$ denote Bernoulli bond percolation with parameter $p$, let $o\in V$, and define…

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 ${\mathcal U}$ be a critical update family with an infinite number of stable directions. Let $q$ denote the vacancy probability, let $\tau_0$ be the infection time at the origi…

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…