  We consider a four-terminal setup of a two-dimensional topological insulator
(quantum spin Hall insulator) with local tunneling between the upper and lower
edges. The edge modes are modeled as helical Luttinger liquids and the
electron-electron interactions are taken into account exactly. Using
perturbation theory in the tunneling, we derive the cumulant generating
function for the inter-edge current. We show that different possible transport
channels give rise to different signatures in the current noise and current
cross-correlations, which could be exploited in experiments to elucidate the
interplay between electron-electron interactions and the helical nature of the
edge states.
