2024/07/17 by Gideon Amir, Markus Heydenreich, Amir, Gideon +3
Computer Science · Mathematics · #Algorithms and Data Compression #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2407.12484
openalex publication_date 2024/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.