Abstract
We consider the standard first passage percolation model in the rescaled graph \({\mathbb{Z}^d/n}\) for d ≥ 2, and a domain Ω of boundary Γ in \({\mathbb{R}^d}\) . Let Γ1 and Γ2 be two disjoint open subsets of Γ, representing the parts of Γ through which some water can enter and escape from Ω. We investigate the asymptotic behaviour of the flow \({\phi_n}\) through a discrete version Ω n of Ω between the corresponding discrete sets \({\Gamma^{1}_{n}}\) and \({\Gamma^{2}_{n}}\) . We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of \({\phi_n/ n^{d-1}}\) below a certain constant are of surface order.
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Cerf, R., Théret, M. Lower large deviations for the maximal flow through a domain of \({\mathbb{R}^d}\) in first passage percolation. Probab. Theory Relat. Fields 150, 635–661 (2011). https://doi.org/10.1007/s00440-010-0287-6
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DOI: https://doi.org/10.1007/s00440-010-0287-6
Keywords
- First passage percolation
- Maximal flow
- Minimal cut
- Large deviations
Mathematics Subject Classification (2000)
- 60K35