  Our understanding of topological insulators is based on an underlying
crystalline lattice where the local electronic degrees of freedom at different
sites hybridize with each other in ways that produce nontrivial band topology,
and the search for material systems to realize such phases have been strongly
influenced by this. Here we theoretically demonstrate topological insulators in
systems with a random distribution of sites in space, i. e., a random lattice.
This is achieved by constructing hopping models on random lattices whose ground
states possess nontrivial topological nature (characterized e. g., by Bott
indices) that manifests as quantized conductances in systems with a boundary.
By tuning parameters such as the density of sites (for a given range of fermion
hopping), we can achieve transitions from trivial to topological phases. We
discuss interesting features of these transitions. In two spatial dimensions,
we show this for all five symmetry classes (A, AII, D, DIII and C) that are
known to host nontrivial topology in crystalline systems. We expect similar
physics to be realizable in any dimension and provide an explicit example of a
$Z_2$ topological insulator on a random lattice in three spatial dimensions.
Our study not only provides a deeper understanding of the topological phases of
non-interacting fermions, but also suggests new directions in the pursuit of
the laboratory realization of topological quantum matter.
