2 problems

Let $G=(V,E)$ be a graph, let $G^\times=G\mathbin{\square}\mathbbm{Z}$ be its Cartesian product with the one-dimensional lattice, and let $p\in(0,1)$. In Bernoulli percolation on…

Let $t(e)$ be Bernoulli edge weights, let $\mathbf{e}_1$ denote the first coordinate vector, and write $\mu_{\mathbf{e}_1}(a)=\mu^{(a)}(\mathbf{e}_1)$ for the shape function in dir…