Lower large deviations for the maximal flow through a domain of $${\mathbb{R}^d}$$ in first passage percolation

Raphaël Cerf (1) , Marie Théret (2)

1 UP11 - Université Paris-Sud - Paris 11
2 Modélisation stochastique

We consider the standard first passage percolation model in the rescaled graph Z d /n for d ≥ 2, and a domain Ω of boundary Γ in R d. Let Γ 1 and Γ 2 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 φ n through a discrete version Ω n of Ω between the corresponding discrete sets Γ 1 n and Γ 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 φ n /n d−1 below a certain constant are of surface order.

Keywords: AMS 2000 subject classifications: 60K35 First passage percolation; maximal flow; minimal cut; large deviations

Domains: Mathematics [math]

Raphaël Cerf, Marie Théret. Lower large deviations for the maximal flow through a domain of $${\mathbb{R}^d}$$ in first passage percolation. Probability Theory and Related Fields, 2011, 150 (3-4), pp.635-661. ⟨10.1007/s00440-010-0287-6⟩. ⟨hal-03856495⟩