55 problems

Let $G$ be a graph with vertex set $V$, $d=|V|$ vertices, maximum degree $Delta>1$, and $q$ colors. The Glauber dynamics conjecture. If $q\geq \Delta+\Omega(1)$, then the natural G…

Consider the Edwards–Anderson spin-glass model on a sparse random graph drawn from $\mathbold{G}(n,d/n)$, with inverse temperature $\beta>0$ satisfying … where $\mathbold{J}$ is a…

Consider the Ising model on a tree at zero external field, with Glauber dynamics started from the symmetric product measure $\nu={{\mathbb P}}_{1/2}$. Write $\nu P_t$ for its law a…

Subexponential relaxation-time conjecture. A weaker conjecture is

Critical logarithmic relaxation-time conjecture. The relaxation time is of order

The model is the ferromagnetic Ising Glauber dynamics on a finite tree or lattice region with $n$ vertices, at zero external field $h=0$ and inverse temperature $\beta>\beta_c$, wh…

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 the Sherrington–Kirkpatrick model be at an arbitrary temperature, and let the external field provide sufficiently strong magnetization. Consider Glauber dynamics initialized fr…

Let $W$ be a Gaussian orthogonal ensemble matrix, and consider the Sherrington–Kirkpatrick Gibbs measure … Here $\beta$ is the inverse temperature and $\theta\mathbf{1}$ is the ext…

For lozenge tilings of a hexagon of scale $N$, let the inverse spectral gap denote the reciprocal of the spectral gap of the Glauber dynamics, and let the mixing time be the time n…

Let the Sherrington--Kirkpatrick model have inverse temperature $\beta$, and let $\beta_{\mathrm{st}}(2)$ denote the stability threshold defining the replica-symmetric phase. Fast-…

The mean-field Potts model has state space $\Omega=[q]^n$, and the annealed single-site Glauber dynamics is initialized from $\mu_{\beta_0}$ for a constant…

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…

Consider the Edwards–Anderson spin-glass model on a sparse random graph drawn from $\mathbold{G}(n,d/n)$, with inverse temperature $\beta>0$ and standard Gaussian edge couplings…

Random-cluster initialization conjecture. On random regular graphs, the multimodality and associated bottlenecks of the random cluster model should be circumvented by initializing…

Initialization conjecture. The bottleneck phenomena that impede mixing from worst-case starting configurations should be avoidable by initializing the chain more judiciously.

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.

Conjecture on optimal mixing-time bounds. The mixing-time bounds established in the paper are optimal for $G(n,d/n)$.

The multi-component Ising model consists of particles partitioned into a fixed number of groups with fixed proportions, with interaction strengths determined by the groups. For its…

Possible-number-of-solutions conjecture. For any finite $k$, there exist $cbeta$, $h$, and $cP$ such that the equation above has any number of solutions in the set…

Consider zero-temperature Glauber dynamics on $\mathbb{Z}^2$ started from an i.i.d. initial condition whose initial density of positive spins is $\rho$. A phase transition at densi…