Papers › Phase transition for the late points of random walk
Phase transition for the late points of random walk
Alexis Prévost, Pierre-François Rodriguez, Perla Sousi
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Let X be a random walk on the torus of side length N in dimension d≥3 with uniform starting point, and t_(cov) be the expected value of its cover time, which is the first time that X has visited every vertex of the torus at least once. For α> 0, the set ℒ^α of α-late points consists of those points not visited by X at time αt_(cov). We prove the existence of a value α_* ∈(1/2,1) across which ℒ^α trivialises as follows: for all α> α_* and ϵ≥N⁻ᶜ there exists a coupling of ℒ^α and two occupation sets ℬ^(α_±) of i.i.d. Bernoulli fields having the same density as ℒ^(α±ϵ), which is asymptotic to N^(-(α±ϵ)d), with the property that the inclusion ℬ^(α_+) ⊆ℒ^α ⊆ℬ^(α_-) holds with high probability as N →∞. On the contrary, when α≤α_* there is no such coupling. Corresponding results also hold for the vacant set of random interlacements at high intensities. The transition at α_* corresponds to the (dis-)appearance of `double-points' (i.e. neighboring pairs of points) in ℒ^α. We further describe the law of ℒ^α for α>1/2 by adding independent patterns to ℬ^(α_±). In dimensions d ≥4 these are exactly all two-point sets. When d=3 one must also include all connected three-point sets, but no other.
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