Definition of the interlacement set I u Graph distance within the interlacement set On the internal distance in the interlacement set rí ˇ Jiˇ Cerný Serguei Popov ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set Definition of the interlacement set I u Graph distance within the interlacement set ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set ◮ Z d , d ≥ 3, so that SRW is transient ◮ informally speaking, random interlacements = stationary soup of doubly infinite SRW’s trajectories ◮ u is the “intensity” of the interlacement set, so I u 1 � I u 2 for u 1 > u 2 ◮ see the recent papers of Sznitman ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set Construction of I u on a finite set A ⊂ Z d : ◮ e A ( x ) := P x [ SRW escapes from A ] 1 A ( x ) � ◮ cap ( A ) := e A ( x ) x ∈ A ◮ place Poisson ( ue A ( x )) particles to x , independently for x ∈ A ◮ each particle performs a SRW ◮ (so that the total number of particles walking on A is Poisson ( u cap ( A )) ) ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set For example, ◮ A = S n = { x ∈ Z d : � x � ≤ n } ◮ e S n ( x ) = O ( n − 1 ) for x ∈ ∂ S n ◮ total number of particles on ∂ S n is Poisson ( u cap ( S n )) = O ( un d − 2 ) ◮ observe that P x [ SRW hits y ] ≃ � x − y � − ( d − 2 ) ◮ so, we have “just enough” particles (i.e., 0 < P u [ 0 ∈ I u S n ] < 1 uniformly) In fact, on the previous page we have the exact definition of I u on any given finite set (i.e., no need to take the limit n → ∞ here)! u In particular, P u [ 0 / ∈ I u S n ] = exp ( − g ( 0 , 0 ) ) ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set Definition of the interlacement set I u Graph distance within the interlacement set ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set I u y x ρ u ( x, y ) = 7 ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set ◮ let P u 0 = P [ · | 0 ∈ I u ] be the conditional law given that 0 ∈ I u ◮ for x , y ∈ I u we define ρ u ( x , y ) to be the internal distance between x and y within the interlacement set I u ◮ let Λ u ( n ) = { y ∈ I u : ρ u ( 0 , y ) ≤ n } be the ball of radius n in the internal distance Theorem For every u > 0 and d ≥ 3 there exists D u ⊂ R d such that for any ε > 0 ( 1 − ε ) nD u ∩ I u � ⊂ Λ u ( n ) ⊂ ( 1 + ε ) nD u � eventually. ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set ◮ the set D u is symmetric under rotations and reflections of Z d ◮ D u ⊂ { x ∈ R d : � x � 1 ≤ 1 } for all u ◮ it is straightforward to show that D u → { x ∈ R d : � x � 1 ≤ 1 } as u → ∞ ◮ it would be interesting, however, to be able to say something about the behaviour of D u when u → 0 (e.g., does the shape become close to the Euclidean ball, and what can be said about the size of D u as u → 0?) ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set Main tool: we prove that, for large enough C 0 [ for all x , y ∈ S n ∩ I u , ρ u ( x , y ) > Cn 2 ] < e − n δ P u (in fact, this also implies that I u is connected simultaneously for all u ) ˇ Cerný, Popov Internal distance in the interlacement set
Definition of the interlacement set I u Graph distance within the interlacement set Sketch of the proof (for d = 4): 3 3 3 3 B ( n ) 2 2 2 1 ˇ Cerný, Popov Internal distance in the interlacement set
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