  Numerical simulations for a number of water models have supported the
possibility of a metastable liquid-liquid critical point (LLCP) in the deep
super-cooled region. Here we consider a theoretical model for a supercooled
liquid water monolayer and its mathematical mapping onto a percolation problem.
The mapping allows us to identify the finite-size clusters at any state-point,
and the infinite cluster at the critical point, with the regions of correlated
hydrogen bonds (HBs). We show that the percolation line coincides with the
first-order liquid-liquid phase transition ending at the LLCP. At pressures
below the LLCP, the percolation line corresponds to the strong maxima of the
thermodynamic response functions and to the locus of maximum correlation length
(Widom line). At higher pressures, we find a percolation transition with a
positive slope and we discuss its possible relation with the thermodynamics.
