14 problems

Polynomial mixing-time conjecture. For any graph $G$, the Glauber dynamics on the $k$-colourings of $G$ for $k\geq\Delta+2$ has polynomial mixing time.

Let $G$ be any graph equipped with a ferromagnetic Ising model, with couplings $J_{xy}$, and let $\tau_2$ denote the relaxation time of its Glauber dynamics. Peres's monotonicity c…

Let $G$ be a finite graph of maximum degree $\Delta$, let $P$ be the Glauber transition matrix at complex fugacity $w$, and let $\operatorname{spr}(P)$ denote its spectral radius o…

Universal Type I conjecture. The model $I(G,p)$ is of Type $\mathcal{I}$ for every initial density $p\in(0,1)$.

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 $G$ be a graph of maximum degree $\Delta$, and consider the Glauber dynamics for sampling proper $q$-colorings of $G$. Jerrum's conjecture. The Glauber dynamics for sampling…

Expander-graph conjecture. There is a choice of corrupted set such that the conclusion to Theorem holds.

Let $(\xi_t)_{t\geq 0}$ be zero-temperature Glauber dynamics on $\mathbb{Z}^d$, and define … Here $\tilde{p}_c(d)$ is the threshold for fixation at opinion $1$. Fixation-threshold…

Let $\mathcal{C}_n^\star$ denote the critical set of configurations, let $K_n^\star$ denote the associated quantity governing the transition from the critical set, and let…

Let $Gamman^\star$ denote the energy barrier and let $\mathbb{P}n$ be the probability law of the random graph. Scaling conjecture for the critical height. There exists a…

Let $U$, $\varphi$ and their discretizations $(U_L,\varphi_L)_{L\ge1}$ be as above, and let $h_t^\eta$ denote the height function under Glauber dynamics started from an arbitrary i…

Let $T$ be a regular tree of even degree, and let the initial spin configuration have a product distribution whose Bernoulli marginals are biased. Consider zero-temperature Glauber…

Let $\{G_n\}$ be a sequence of transitive graphs, and consider the Glauber dynamics for the Ising model on these graphs. Let $\beta_c$ denote the critical temperature, let $\delta$…

Let $(G_n)$ be a sequence of transitive graphs, and let $t_{{\rm mix}}(n)$ denote the mixing time of Glauber dynamics on $G_n$. A sequence of chains has cut-off if there are times…