4 problems

Let $G_m$ be the $m\times m$ grid, viewed as a subgraph of the Euclidean lattice $\mathbb{Z}^2$, with independent identically distributed positive random edge-lengths, and let…

Let $G$ be a finite graph, let $V$ be a uniformly random initial vertex, and let $L_{NUV}(G,V)$ denote the length of the nearest unvisited vertex walk started at $V$. Write…

Let $\Omega$ be a finite set, let $S\colon\Omega\times\Omega\to\mathbb R_+$ be a symmetric matrix with nonnegative entries, and let $u\colon\Omega\to\mathbb R_+$ be a nonnegative u…

The 2-way infinite 2-walk conjecture. The graph $G$ has a $2$-way infinite $2$-walk.