The time constant for Bernoulli percolation is Lipschitz continuous strictly above p_c

2021/01/28 by Cerf, Raphaël, Dembin, Barbara
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2101.11858

Abstract

We consider the standard model of i.i.d. first passage percolation on ℤ^d given a distribution G on [0,+∞] (+∞ is allowed). When G([0,+∞]) < p_c(d), it is known that the time constant μ_G exists. We are interested in the regularity properties of the map G↦μ_G. We first study the specific case of distributions of the form G_p=pδ₁+(1-p)δ_∞ for p>p_c(d). In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter p. We show that the function p↦ μ_Gp is Lipschitz continuous on every interval [p₀,1], where p₀>p_c(d).

Related

• Regularity of the time constant for a supercritical Bernoulli percolation
• Lipschitz-continuity of time constant in generalized First-passage percolation
• First-order behavior of the time constant in Bernoulli first-passage percolation
• Continuity of the time constant in a continuous model of first passage percolation
• First-order behavior of the time constant in non-isotropic continuous first-passage percolation