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 $\alpha >0$.

Our investigations are guided by the conjecture that the set of infinitely reinforced edges percolates for $k=2$ and $\alpha \gg 1$. First, we study the case $\alpha =\infty$, 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 $\alpha <\infty$ case. Finally, we verify these conditions numerically.