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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arxiv.org/abs/2112.13390

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

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