Lower large deviations for the maximal flow through a domain of {\mathbb{R}^d} in first passage percolation |
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Authors: | Rapha?l Cerf Marie Th??ret |
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Institution: | 1. D??partement de Math??matiques, Universit?? Paris Sud, Batiment 425, 91405, Orsay Cedex, France 2. D??partement de Math??matiques et Applications, ??cole Normale Sup??rieure, 45 rue d??Ulm, 75230, Paris Cedex 05, France
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Abstract: | We consider the standard first passage percolation model in the rescaled graph ${\mathbb{Z}^d/n}$ for d??? 2, and a domain ?? of boundary ?? in ${\mathbb{R}^d}$ . Let ??1 and ??2 be two disjoint open subsets of ??, representing the parts of ?? through which some water can enter and escape from ??. We investigate the asymptotic behaviour of the flow ${\phi_n}$ through a discrete version ?? n of ?? 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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