  Understanding the flow of yield stress fluids in porous media is a major
challenge. In particular, experiments and extensive numerical simulations
report a non-linear Darcy law as a function of the pressure gradient. In this
letter, we consider a tree-like porous structure for which the problem of the
flow can be resolved exactly thanks to a mapping with the directed polymer (DP)
with disordered bond energies on the Cayley tree. Our results confirm the
non-linear behavior of the flow and expresses its full pressure-dependence via
the density of low-energy paths of DP restricted to vanishing overlap. These
universal predictions are confirmed by extensive numerical simulations.
