Abstract
In this paper, we show several rigorous results on the phase transition of Finitary Random Interlacements (FRI). For the high intensity regime, we show the existence of a critical fiber length, and find its exact asymptotic as intensity goes to infinity. At the same time, our result for the low intensity regime proves the global existence of a non-trivial phase transition with respect to the system intensity.
Highlights
We show several rigorous results on the phase transition of Finitary Random Interlacements (FRI)
For the high intensity regime, we show the existence of a critical fiber length, and find its exact asymptotic as intensity goes to infinity
Our result for the low intensity regime proves the global existence of a non-trivial phase transition with respect to the system intensity
Summary
The model of finitary random interlacements (FRI) was first introduced by Bowen [1] to solve a special case of the Gaboriau-Lyons problem. In addition to the local convergence as T → ∞, Bowen proved that the FRI on non-amenable graphs will a.s. have infinite connected cluster(s) for all sufficiently large T. We prove that, despite lacking global monotonicity, FRI is stochastically increasing with respect to T for all T ∈ (0, 1), which implies the existence and uniqueness of Tc for all sufficiently large u. For the low intensity regime, we prove a polynomial lower bound for the phase diagram, which at the same time proves the conjecture on the global existence of a non-trivial phase transition with respect to u.
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