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 , 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.
Highlights
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We investigate the phase transitions in the Finitary Random Interlacement (FRI) introduced by Bowen in his study on Gaboriau–Lyons problem [1]
The percolation phase transition is closely related to the trade-off mechanism with respect to the parameter T: As T increases, there will be on average fewer and fewer fibers starting from each vertex
Summary
Since F I u,T d may be nonmonotonic with respect to T, the existence of a subcritical and a supercritical phase is insufficient to guarantee a critical value in between It is conjectured in [5] that there is a unique critical value Tc (u, d) such that F I u,T d percolates when T > Tc and has no infinite cluster almost surely when T < Tc. The percolation phase transition is closely related to the trade-off mechanism with respect to the parameter T: As T increases, there will be on average fewer and fewer fibers starting from each vertex.
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