Eviatar B. Procaccia, Sarai Hernandez-Torres: The chemical distance in random interlacements in the low-intensity regime

Abstract: In this talk, I will present a new proof of the sharpness of the phase transition for Random interlacements is a Poissonian soup of doubly-infinite random walk trajectories on \mathbb{Z}^d, with a parameter u >0 controlling the intensity of the Poisson point process. The model defines a percolation on the edges of \mathbb{Z}^d with long-range correlations.

We consider the time constant \rho_u associated to the chemical distance in random interlacements at low intensity u>0. For dimension d \geq 5, it is conjectured that u^{1/2}\rho_u converges to the Euclidean norm as u ↓ 0. In this high-dimensional case, we prove a sharp upper bound (of order u^{-1/2}) and an almost sharp lower bound (of order u^{-1/2 + \varepsilon} ) for the time constant as the intensity decays to zero.


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