  The Berezinski-Kosterlitz-Thouless transition is a unique two dimensional
phase transition, separating two phases with exponentially and power-law
decaying correlations, respectively. In disordered systems, these correlations
propagate along favorable paths, with the transition marking the point where
global coherence is lost. Here we propose an experimental method to probe
locally these particular paths in superconducting thin films, which exhibit
this transition, and demonstrate theoretically that close to the transition the
coherence propagate along a ramified network, reminiscent of a percolation
transition. We suggest and calculate experimentally accessible quantities that
can shed light on the spatial correlations in the system as it approaches the
critical point.
