We discuss collisions of multiple independent simple random
walks on a graph. Whether the number of collisions is finite or infinite
serves as an indicator of the structure of the underlying graph. It is
easy to prove that two independent simple random walks collide infinitely
many times almost surely on transitive recurrent graphs, while some
intransitive recurrent graphs lack this property. In this talk, we
investigate the number of collisions of three random walks on a graph,
inspired by the recent article by Croydon and De Ambroggio. The graph of
interest is the random graph given as the trajectory of a four-dimensional
simple random walk. We will see that, with probability one, a realization
of the random graph has the infinite triple collision property.