15 problems

Let a biased random walk on dynamical percolation be given in the setting of Proposition, and let an annealed invariance principle hold for its scaling limit. A quenched invariance…

In dynamical critical site percolation on the triangular lattice, call a time two-colour percolation time if there is both a white infinite cluster and a black infinite cluster at…

Consider the dynamical site-resampling setting for last-passage percolation, and let $E$ be the set of times at which non-trivial bigeodesics exist. A bigeodesic is a bi-infinite d…

Let $5malpha(5mathbb Z^d)$ be the critical dynamical-percolation speed exponent for the lattice $5mathbb Z^d$, and let $5malpha(5mathbb T_{b+1})$ be the corresponding exponent for…

Let $G$ be a nonamenable transitive unimodular graph, and let $u{pc}(5mu)$ denote the speed of the random walk on critical dynamical percolation on $G$ with refresh rate $5mu$. Su…

Let $G_\kappa(t)$ be the evolving scale-free network with vertex updates, and consider the alternative dynamical-percolation model in which edges are updated independently at rate…

Let $G$ be the hypercube $\{0,1\}^d$ and let the edge-percolation parameter be $p=c/d$ with $c>1$. Consider simple random walk (SRW) on the giant component of the percolated hyperc…

Supercritical mixing-time conjecture. If $p>p_c(\mathbb{Z}^d)$, then the mixing time is

Let $\mathsf{\mathscr{E}}$ be the exceptional time set and let $\mathsf{\mu}$ be the associated local-time measure, which is supported inside $\mathsf{\mathscr{E}}$. Full-support c…

For a fixed interval $[a,b]$, let $\mathsf{\overline{\mu}}_r[a,b]$ be the finite-radius approximations to the local-time measure and let $\mathsf{\overline{\mu}}[a,b]$ denote their…

Let $\mathsf{FETIC}$ be the law of the configuration at the first exceptional time, and let $\mathsf{IIC}$ denote the incipient infinite cluster measure. Singularity conjecture. Th…

Fix $q\in[1,4]$ and let $(\omega_t^{\mathrm{FK}_q})_{t\geq0}$ be the critical infinite-volume heat-bath dynamics on $\mathbb Z^2$. Let $\mathcal E_q$ be the random set of times at…

For a critical random-cluster model with cluster-weight $q$, let macroscopic pivotals mean pivotal edges whose influence persists at macroscopic scales, and let exceptional times m…

Let $\mathcal{E}^{(2)}\subset[0,\infty)$ be the random set of exceptional times of the second kind for dynamical percolation on the triangular lattice, namely times at which an inf…

Consider the Glauber dynamics for the critical two-dimensional Ising model on $\mathbb{Z}^2$. At the critical value, the Ising model does not percolate at any fixed time. Pete's co…