3 problems

Let the graph be the $m$-dimensional torus with $n$ nodes, and let $\tau^\star$ denote the mixing time of the directed-edge process associated with consensus propagation. Consensus…

Fix $k\in\mathbb{N}$. For the Ising partition function $Z_{\mathbb{Z}_m^t}(\lambda,\beta)$ on the even torus $\mathbb{Z}_m^t$, let $L_j$ denote the sum of the weights of clusters o…

Let $C_m$ and $C_n$ be cycle graphs, and let $C_m\square C_n$ be their Cartesian product. Write $\gamma^*_e(G)$ for the minimum cardinality of an exponential dominating set in a gr…