4 problems

Concentration conjecture. The same sub-Gaussian concentration phenomenon should hold when the sum $$ is replaced by a more general Lipschitz function of the variables.

Let $\beta>0$, and for distinct edges write $\mathbb{P}_{\beta}[e_1,\ldots,e_k]$ for the probability that all listed edges occur in the forest. Negative association conjecture. For…

Stronger negative-association conjecture. The measures $\operatorname{USF}$ and $\operatorname{UCS}$ are negatively associated in one or both of these further senses.

Let $\mathcal G$ be a graph, and let $e$ and $f$ be distinct edges. Write $G=F(\mathcal G;\mathbf y)$ for the generating polynomial of its spanning forests, and let $\mathscr S$ be…