Planar reinforced $k$-out percolation

Abstract

We investigate the percolation properties of a planar reinforced network model. In this model, at every time step, every vertex chooses $k\ge 1$ 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 $\hat{I}\pm >0$. Our investigations are guided by the conjecture that the set of infinitely reinforced edges percolates for $k=2$ and $\hat{I}\pm \gg 1$. First, we study the case $\hat{I}\pm =\infty$, 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 $\hat{I}\pm <\infty$ case. Finally, we verify these conditions numerically.

14 pages, 3 figures