  Given a connected graph $G$ with some subset of its vertices excited and a
fixed target vertex, in the geodesic-biased random walk on $G$, a random walker
moves as follows: from an unexcited vertex, she moves to a uniformly random
neighbour, whereas from an excited vertex, she takes one step along some fixed
shortest path towards the target vertex. We show, perhaps counterintuitively,
that the geodesic-bias can slow the random walker down exponentially: there
exist connected, bounded-degree $n$-vertex graphs with excitations where the
expected hitting time of a fixed target is at least $\exp (\sqrt[4]{n} / 100)$.
