We consider first-passage percolation (FPP) on a book graph with multiple pages, where upper half-planes of $\mathbb{Z}^2$ are glued together along a common axis. FPP was introduced by Hammersley and Welsh in 1965 as a model for fluid flow through a random medium. In this model, independent non-negative random variables, called passage times, are assigned to the edges of the graph. A geodesic between two distinct points is defined to be a path that attains the minimum total passage time among all paths connecting the two points.

One of the central topics in FPP is the study of the geometry of geodesics. In this talk, we consider the notion of a bigeodesic, namely an infinite self-avoiding path indexed by $\mathbb Z$ such that every finite subpath is a finite geodesic. It has been proved,
under suitable assumptions on the passage-time distribution, that there is no bigeodesic in the upper half-plane of ${Z}^2$, and it is widely believed that the same is true for $\mathbb {Z}^2$ itself. In contrast, we show that a book graph with sufficiently
many pages admits a bigeodesic. This striking difference from the case of $\mathbb {Z}^2$ arises from the geometry of the book graph, where the spine has a strong influence on the geometry of geodesics.

This talk is based on joint work with Tzu-Han Chou (NUS) and Wai-Kit Lam (NTU).