Abstract
We consider independent anisotropic bond percolation on \({\mathbb {Z}}^d\times {\mathbb {Z}}^s\) where edges parallel to \({\mathbb {Z}}^d\) are open with probability \(p<p_c({\mathbb {Z}}^d)\) and edges parallel to \({\mathbb {Z}}^s\) are open with probability q, independently of all others. We prove that percolation occurs for \(q\ge 8d^2(p_c({\mathbb {Z}}^d)-p)\). This fact implies that the so-called Dimensional Crossover critical exponent, if it exists, is greater or equal than 1. In particular, using known results, we conclude the proof that, for \(d\ge 11\), the crossover critical exponent exists and equals 1.
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Acknowledgements
The authors would like to thank two anonymous referees for their valuable comments. Remy Sanchis was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and by Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG), grant PPM 00600/16. Pablo A. Gomes was partially supported by FAPESP, grant 2020/02636-3. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
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Gomes, P.A., Sanchis, R. & Silva, R.W.C. A note on the dimensional crossover critical exponent. Lett Math Phys 110, 3427–3434 (2020). https://doi.org/10.1007/s11005-020-01336-3
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DOI: https://doi.org/10.1007/s11005-020-01336-3