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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.12484 |
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| _version_ | 1866917724951674880 |
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| author | Amir, Gideon Heydenreich, Markus Hirsch, Christian |
| author_facet | Amir, Gideon Heydenreich, Markus Hirsch, Christian |
| contents | 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 $α> 0$. Our investigations are guided by the conjecture that the set of infinitely reinforced edges percolates for $k = 2$ and $α\gg 1$. First, we study the case $α= \infty$, 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 $α< \infty$ case. Finally, we verify these conditions numerically. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12484 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Planar reinforced $k$-out percolation Amir, Gideon Heydenreich, Markus Hirsch, Christian Probability 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 $α> 0$. Our investigations are guided by the conjecture that the set of infinitely reinforced edges percolates for $k = 2$ and $α\gg 1$. First, we study the case $α= \infty$, 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 $α< \infty$ case. Finally, we verify these conditions numerically. |
| title | Planar reinforced $k$-out percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2407.12484 |