Planar reinforced k-out percolation

We investigate the percolation properties of a planar reinforced network model. In this model, at every time step, every vertex chooses k⩾1 incident edges, whose weight is then increased by 1. The choice of this k-tuple occurs proportionally to the product of the corresponding edge weights raised to some power α>0. Our investigations are guided by the conjecture that the set of infinitely reinforced edges percolates for k=2 and α≫1. First, we study the case α=∞, where we show the percolation for k=2 after adding arbitrarily sparse independent sprinkling and also allowing dual connectivities. We also derive a finite-size criterion for percolation without sprinkling. Then, we extend this finite-size criterion to the α<∞ case. Finally, we verify these conditions numerically.