Convergence Towards an Asymptotic Shape in First-Passage Percolation on Cone-Like Subgraphs of the Integer Lattice.

In first-passage percolation on the integer lattice, the shape theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $${\mathbb {Z}}^d$$ lattice, where $$d\ge 2$$ . In particular, we identify the asymptotic shapes associated with these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for $$L^p$$ - and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment. [ABSTRACT FROM AUTHOR]