22 problems

Let ${\mathcal U}$ be a critical update family with an infinite number of stable directions. Let $q$ denote the vacancy probability, let $\tau_0$ be the infection time at the origi…

Stationary-measure decomposition conjecture. If $\mu$ is a stationary measure of a KCM, then there exists a probability measure $\mu_{*}$ on $\mathcal{E}$ such that

Exponential-relaxation conjecture. If there exists $\kappa>0$ such that

FA-1f convergence conjecture. For any $q\in(0,1)$,

Let the $G$-MCEM be a multicolour East model with a varying number of vacancy types, and let its spectral gap denote the corresponding relaxation rate. Spectral-gap monotonicity co…

Shape theorem conjecture. There exists a compact subset $\widehat S\subset\mathbb{R}^d_+$ such that

Pillai and Smith's conjecture.

Let $D(q)$ denote the self-diffusion coefficient in the Kob–Andersen model, with constraint parameter and dimension satisfying $k=d=2$, and let $q$ be the vacancy density. Sharp se…

Consider the square plaquette model in a box with all-plus boundary conditions, so that the unique ground state is the configuration with spin $+1$ at every site. At low temperatur…

Let $\mathcal U$ be a critical update family with finitely many stable directions and difficulty $\alpha$. Let $\tau_0$ denote the relaxation time at the origin under the equilibri…

Three-dimensional propagation conjecture. In the true dynamics, infection should be able to propagate in three directions, giving the scale $q^{-1/2}$ for the infection time, perha…

Let $\tau_{0}$ be the relaxation time at the origin in the mixed threshold FA model, let $q$ be the vacancy density, let $\nu$ denote the law of the random environment, and let…

Let ${\mathcal U}$ be a supercritical rooted update family, let $\tau_0$ be the time at which the origin becomes infected in the associated KCM, and let $q$ be the vacancy density.…

Let ${\mathcal U}$ be a supercritical update family on $\mathbb Z^2$, meaning that its set of stable directions does not meet some open semicircle in $S^1$. The associated kinetica…

Let ${\mathcal U}$ be a critical update family, let $T_{\rm rel}$ denote the relaxation time of the associated kinetically constrained model, and let $\tau_0$ be the infection time…

Let ${\mathcal U}$ be an $\alpha$-unrooted and unbalanced critical two-dimensional update family with difficulty $\alpha$. Let $q$ denote the vacancy density and let $\tau_0$ be th…

Let $D(\rho)$ denote the diffusion coefficient as a function of the density $\rho$. Analyticity conjecture. The diffusion coefficient $D(\rho)$ is an analytic function of $\rho^3$,…

Kordzakhia's limiting-shape conjecture. The influence region has a definite limiting shape $S\subset {{\mathbb R}}^2$ in the sense that, almost surely, $R_t$ converges to the shape…

Consider the kinetically constrained spin model on a finite $3$-ary rooted tree of depth $L$, at the critical density $p=p_c=\frac{8}{9}$ for the corresponding bootstrap percolatio…

Let $L$ be the system size, $t$ the observation time, $s$ the activity-counting parameter, and let $\psi_L(s)$ denote the finite-size dynamical free-energy function. Define … Let…

Thermodynamic-limit conjecture. The $limsup{Ntoinfty}$ in this definition can be replaced by $lim{Ntoinfty}$.

Let $p$ and $p'$ be the model parameters, let $p_c$ be the critical value, and let $\nu$ be the Bernoulli($p'$) product measure on $\mathcal T$. A local function is a function depe…