Continuity of the time constant in a continuous model of first passage percolation
Abstract
For a given dimension d ≥ 2 and a finite measure ν on (0, +∞), we consider ξ a Poisson point process on R d × (0, +∞) with intensity measure dc ⊗ ν where dc denotes the Lebesgue measure on R d. We consider the Boolean model Σ = ∪ (c,r)∈ξ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y ∈ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside Σ and at infinite speed inside Σ. By a standard application of Kingman sub-additive theorem, one easily shows that T (0, x) behaves like µ x when x goes to infinity, where µ is a constant named the time constant in classical first passage percolation. In this paper we investigate the regularity of µ as a function of the measure ν associated with the underlying Boolean model.
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