We consider the statistical inference problem of recovering an unknown
perfect matching, hidden in a weighted random graph, by exploiting the
information arising from the use of two different distributions for the weights
on the edges inside and outside the planted matching. A recent work has
demonstrated the existence of a phase transition, in the large size limit,
between a full and a partial recovery phase for a specific form of the weights
distribution on fully connected graphs. We generalize and extend this result in
two directions: we obtain a criterion for the location of the phase transition
for generic weights distributions and possibly sparse graphs, exploiting a
technical connection with branching random walk processes, as well as a
quantitatively more precise description of the critical regime around the phase
transition.Comment: 19 pages, 8 figure
