On (Non-)Monotonicity and Phase Diagram of Finitary Random Interlacement

Abstract

In this paper, we study the evolution of a Finitary Random Interlacement (FRI) with respect to the expected length of each fiber. In contrast to the previously proved phase transition between sufficiently large and small fiber length, for all d≥3, FRI is NOT stochastically monotone as fiber length increases. At the same time, numerical evidence still strongly supports the existence and uniqueness of a critical fiber length, which is estimated theoretically and numerically to be an inversely proportional function with respect to system intensity.

Some rigorous results on the phase transition of finitary random interlacements

“…In this paper, 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 T c for all sufficiently large u. Meanwhile, we also show that the upper bound of T c found by [3,7] in the high intensity regime is actually sharp in the limit, and give an exact asymptotic of T c as u → ∞. Moreover, 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.…”

“…In this paper, we first show that for all d ≥ 3 and u > 0, FI u,T is stochastically increasing with respect to T for all T ∈ (0, 1]. It is worth noting that FI u,T has been proved not enjoying monotonicity for larger T 's (see Theorem 1,[3]). Theorem 3.1.…”

“…The fibers within this system each forms a geometrically killed simple random walk with an expected length T . Meanwhile, the starting point of which is sampled according to a Poisson point process with intensity measure proportional to system intensity u and inversely proportional to T + 1 [1,2,3,8]. See Section 2 for precise definitions.…”

“…Unlike the case of RI, this turns out to be a highly non-trivial problem, since it was recently proved that there is no global stochastic monotonicity with respect to T for FI u,T . See Theorem 1, [3] for details. On the other hand, numerical simulations in Section 5 of [3] provided evidences on the existence and uniqueness of T c .…”

“…See Theorem 1, [3] for details. On the other hand, numerical simulations in Section 5 of [3] provided evidences on the existence and uniqueness of T c . It was further conjectured in Conjecture 5 [3] that T c is asymptotically an inverse linear function with respect to u.…”