Phase transition for the late points of random walk

2023/09/06 by Prévost, Alexis, Rodriguez, Pierre-François, Sousi, Perla
#60D05 #60G50 #82B26 #82C41 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2309.03192

Abstract

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_* ∈ (\frac12,1) 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>\frac12 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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