37 problems

Spanning-forest Rayleigh conjecture. For any graph $G=(V,E)$, the SFGF $F_G(\mathbf{w})$ is Rayleigh.

Wagner's conjecture. Every regular matroid is S- and I-Rayleigh.

Let $\mu$ be a probability measure on the Boolean cube, and let its rank sequence be $(\mu(|\eta|=i))_{i=0}^n$. The sequence is ultra-log-concave (ULC) if…

Pemantle's conjecture. The ordinary convolution of a ULC($m$) sequence and a ULC($\ell$) sequence is ULC($m+\ell$), where $m$ and $\ell$ may be infinity. Ultra-log-concavity is clo…

Let $\mu$ be a CNA+ measure on the Boolean lattice $\mathcal{B}_n$, with coordinate indicators $X_1,\ldots,X_n$. Conditional-cover conjecture. The conditional distribution on…

Let $\mu$ be a CNA+ measure, and condition it on the total rank $\sum_eX_e$. Rank-cover conjecture. For each $k$, the conditional distribution given total rank $k+1$ stochastically…

Let $\mu$ be a measure on a Boolean lattice. It is ULC+ when every measure obtained from it by imposing an external field and projecting is ULC. ULC implication conjecture. If…

In the random-cluster (RC) model and the competing urns model, let $X_e$ denote the indicator associated with an edge or site, and let $S$ be any subset of the relevant index set.…

For a measure on a Boolean lattice, its rank sequence is the sequence of probabilities of the possible values of the total number of occupied coordinates. It is ULC (ultra-log-conc…

The properties CNA+, JNRD+ and h-NLC+ are stronger versions of CNA, JNRD and hereditary NLC, respectively, requiring the corresponding property after every imposition of an externa…

Uniform connected subgraph p-NC conjecture. For every $n$ for which the measure is defined, $\mathbb{P}_{\mathrm{UC}}$ satisfies the p-NC property.

Uniform forest p-NC conjecture. The measure $\mathbb{P}_{\mathrm{UF}}$ satisfies the p-NC property.

Random-cluster p-NC conjecture. The measures $\phi_{p,q}$ satisfy the p-NC property.

Conditional monotonicity conjecture. This conditional expectation is decreasing in $\bm x_I$, $\bm x_J$ and $\bm x_K$.

The edge-negative association conjecture. For every such graph $G$, the random forest $\Phi$ is edge-negatively associated. This is a longstanding problem concerning negative depen…

Let $G$ be a finite graph, and let $i$ and $j$ be distinct edges of $G$. Choose a forest $F$ of $G$ uniformly at random. Kahn–Grimmett–Winkler conjecture. The events that $F$ conta…

Let $q$ satisfy $0<q\leq 1$, and consider the probability distribution associated with the $q$-state Potts model. Potts-model negative dependence conjecture. The $q$-state Potts mo…

Pemantle's conjecture. Conditional negative association is equivalent to negative regression.

Let $G=(V,E)$ be a graph and let $0<q<1$. Consider the associated random cluster measure and its multiaffine probability-generating polynomial. Random-cluster Rayleigh conjecture.…

Let the exclusion process be the continuous-time Markov chain on $\{0,1\}^d$ in which the coordinates $i$ and $j$ swap independently at nonnegative rate $\lambda_{\{i,j\}}$. Let…

Mossel's concentration conjecture. There exist positive constants $c_0$ and $c$ such that, for every $a$, the product-measure concentration bound is replaced by

Farr's conjecture. If $G$ is a graph on $[m]$ and ${\cal I}$ is the collection of independent sets of $G$, then for any disjoint $I,J,K\subseteq[n]$,

Partitioned-urn conjecture. The variables $xi_{ij}$ are negatively associated conditional on

Pemantle's conjecture. CNC implies CNA. Pemantle also conjectures that NC+ implies NA+.

Let a probability measure on the Boolean hypercube satisfy certain negative correlation conditions, and let $\{p_k\}$ be the sequence of probabilities of picking a set of size $k$.…