22 problems

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…

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.

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.

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…

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

Let $\mu$ be a probability measure on $\{0,1\}^n$. Say that $\mu$ has the stochastic covering property (SCP) if, for every coordinate $i$, the conditional law given $\eta_i=0$ stoc…

Let $\mu$ be a probability measure on $\{0,1\}^n$. Write $\mu_k$ for the conditional measure given $\sum_i\eta_i=k$, and say that $\mu$ has the normalized matching property (NMP) w…

Let $m$ balls be assigned independently to urns $1,\ldots,n$, and let $X_i$ indicate whether urn $i$ is occupied. The law of $(X_1,\ldots,X_n)$ is a competing urn measure. Its rank…

Let a weighted matroid measure be obtained by assigning positive external-field weights to the independent sets of a matroid. For a measure on $2k$ variables, define the antipodal…

Let $G$ be a finite graph and consider a random cluster measure on its edge sets with parameter $q<1$. Random-cluster conjecture. Every such random cluster measure is NA+. Equivale…

Let $\mu$ be a measure on a finite Boolean cube. The properties CNC, CNA, NC+, and NA+ are the correlation properties defined in the paper, with “+” denoting closure under external…

Let $\mu\in\mathfrak{P}_n$ be a probability measure on subsets of $[n]$, and for $0\leq k\leq n$ let $\mu_k$ be its conditional measure given that the sampled set has size $k$. Wri…

Let CNA and CNA+ denote the corresponding conditional negative-association properties, and let h-NLC and h-NLC+ denote the hereditary negative-lattice-condition properties; Rayleig…

Let $\mathfrak{P}_n$ be the probability measures on subsets of $[n]$, and let CNA+ denote the strongest negative-association property used in the source. Let ULC mean that the rank…

Let $\mathfrak{P}_n$ be the probability measures on subsets of $[n]$, and let a Rayleigh measure be a measure satisfying the Rayleigh negative-dependence inequalities. Let ULC mean…

Let a symmetric exclusion process start from a deterministic initial configuration. Write $\eta_t$ for its configuration at time $t$, and say that a distribution is negatively asso…