8 problems

Random-regular-graph mixing conjecture. Random regular graphs exhibit different mixing behavior from the general maximum-degree setting, with the slowdown of the Glauber dynamics o…

Let $T$ be a tree with $n$ vertices, and let $\lambda$ be the fugacity of the hardcore model. The Glauber dynamics is the single-site Markov chain for this Gibbs distribution, and…

Let $T$ be a tree with $n$ vertices, and let $\lambda$ be the fugacity of the hardcore model. The Glauber dynamics is the single-site Markov chain for this Gibbs distribution, and…

Hardcore mixing conjecture. The natural Markov chain for sampling independent sets from the hardcore distribution mixes in polynomial time up to the uniqueness threshold.

Let the hardcore model be defined on a tree, and consider its Glauber dynamics. The hardcore-model mixing-time conjecture. In the non-reconstruction regime, the Glauber dynamics sh…

ODE characterization conjecture. There exists a unique $C_{\Delta,\lambda}>0$ for which this ODE and these boundary conditions admit a solution, and $F_{\Delta,\lambda}$ is twice c…

Sokal's uniqueness conjecture. Uniqueness on the $d$-regular tree for the hardcore model implies uniqueness on every graph of maximum degree $d$.

Mossel–Weitz–Wormald conjecture. Unless $\mathrm{NP}=\mathrm{RP}$, there does not exist a fully polynomial approximation scheme for the partition function of the hardcore model wit…