Abstract
We prove an asymptotic shape theorem for first-passage percolation on Cayley graphs of virtually nilpotent groups. By a theorem of Pansu, the asymptotic cone of a finitely generated nilpotent group is isometric to a simply connected nilpotent Lie group equipped with some left invariant Carnot-Caratheodory metric. Our main result is an extension of Pansu's theorem to random metrics, where the edges of the Cayley are i.i.d. random variable with some finite exponential moment. Based on a companion work by the second author, the proof relies on Talagrand's concentration inequality, and on Pansu's theorem. Adapting an argument of Benjamini, Kalai and Schramm, we prove a sublinear estimate on the variance for virtually nilpotent groups which are not virtually isomorphic to Z. We further discuss the asymptotic cones of first-passage percolation on general infinite connected graphs: we prove that the asymptotic cones are a.e. deterministic if and only the volume growth is subexponential.
Highlights
First passage percolation is a model of random perturbation of a given geometry
A fundamental result states that the random metric on Euclidean lattices when rescaled by 1/n, almost surely converges to a deterministic invariant metric on the Euclidean space [11, 20]. Underlying this theorem is the simple fact that the graph metric associated to the Euclidean grid when rescaled, converges to the euclidean space equipped with the 1-norm
It is natural to ask if when assigning random i.i.d. lengths to a Cayley graph of polynomial growth, the rescaled metric almost surely converges to a deterministic metric on the Lie group
Summary
We shall restrict to the simplest model, where random i.i.d lengths are assigned to the edges of a fixed graph. A fundamental result (the shape theorem) states that the random metric on Euclidean lattices when rescaled by 1/n, almost surely converges to a deterministic invariant metric on the Euclidean space [11, 20]. It is natural to ask if when assigning random i.i.d. lengths to a Cayley graph of polynomial growth, the rescaled metric almost surely converges to a deterministic metric on the Lie group. Establishing this was the original goal of this note.
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