3 problems

Let $G$ be a Cayley graph, let $(\psi_e)_{e\in E(G)}$ be independent unit-rate Poisson processes on the edges, and let $\beta>0$. At time $\beta$, swapping the particles across eac…

Transience conjecture. There exists a constant $c=c(\mathsf{GW})>0$ such that $\mathrm{FM}(\mathsf{T},\eta,\mathsf{SRW})$ is transient for $\mathsf{GW}$-almost all infinite realiza…

Peres's conjecture. The sufficient condition for a phase transition in the classical XY model on $G$ is that there exists a $p\in(0,1)$ such that, after independent bond percolatio…