Baire categorical aspects of first passage percolation (Q1714998)

The author considers a variation of percolation, without probabilities. Consider the graph with the integer lattice $\mathbb{Z}^d$ in some finite power or the real line as its vertex set and the lines that connect nearest neighbours as edge set $E$. Take some subset $A$ of $[0,\infty)$ with its subspace topology and take the topological power $A^E$. Every point in the power determines a percolation situation; the author asks questions that make sense when ``probability $1$'' is replaced by ``residual''.\par For example, the set of points in $A^E$ for which between any two points there is a geodesic is residual, if $A$ has no isolated points then those geodesics are unique for residually many points. \par By contrast, for residually many points there are very few geodesic rays. It depends a bit on the set $A$ whether this is useful: if $A=\mathbb{Q}_{\ge0}$ then the empty set is residual.