6 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…

Fix $t$ and let $m$ tend to infinity. Consider the partition function of homomorphisms from $\mathbb{Z}_m^t$ to a fixed graph equipped with fixed vertex and edge weights. The clust…

Let $t$ be fixed, let $m$ tend to infinity, and consider weighted homomorphisms from the torus graph $\mathbb{Z}_m^t$ to a fixed graph. The cluster expansion is the expansion for t…

Torus bandwidth conjecture. For every integer $k$, the bandwidth of the torus graph satisfies

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…