From random walk trajectories to random interlacements
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1 From random walk trajectories to random interlacements Augusto Teixeira Department of Mathematics ETH Zurich IRTG SMCP Summer School 2009 Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
2 Acknowledgment to Prof. A.S. Sznitman Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
3 1 Motivation 2 Random interlacements (Q u ) 3 Percolation under Q u 4 Separation events Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
4 Motivation - Porous medium Random walk on (Z/NZ) d Large N, d 3, In time N d log N all covered In time N d proportion covered Abrupt connectivity transitions (simulations) Geometry of sites left: unknown Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
5 1 Fix a box in each torus (constant size) 2 Fix u 0. Run the random walk up to time un d Theorem - Windisch (2008) Law of visited sites in box converges 1 Poisson start points in boundary 2 Independent walks from them Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
6 Extended to Z d Random Interlacement - Sznitman (2007) Poisson soup of doubly infinite trajectories on Z d modulo time shift 1 Law Q u on {0, 1} Zd 2 u gives intensity 3 Ergodic under translations 4 Disconnection of discrete cylinder Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
7 Geometry of vacant sites Ergodicity = Q u [ infinite component] = 0 or 1 u Theorem - Sznitman (2007), Sidoravicius-Sznitman (2008) In Z d, (d 3) 0 < u < Theorem - T. (2009) In Z d, (d 3) Vacant infinite component is unique. Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
8 Separation events Theorem - T. (2009) (d 5) For u small, there is a δ such that Q u N c exp{ c N δ }. Consequences The infinite cluster is dense Typical components are either: small or infinite Could give some results in the torus Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
9 Strategies 1 - Renormalization q k k (q 2k 1 + c k) Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
10 2- Walking around sausages We only need to bound q 1 (probability of separation in a small box) If d 5, a random walk has several cut-times And the random walk can be regarded as a sausage Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
11 There will be less than 100 random walks in each of these grains With high probability they are far apart from each othe (d 5) There is not separation in this case: any two big sets can be linked by a path that travels through the skins of the sausages Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
12 Don t build dikes with sausages! Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
13 V. Sidoravicius, A.S. Sznitman Percolation for the vacant set of random interlacements Comm. Pure Appl. Math., 62(6), (2009) A.S. Sznitman Vacant set of random interlacements and percolation to appear in the Annals of Mathematics (2007) A. Teixeira On the uniqueness of the infinite cluster of the vacant set of random interlacements Annals of Applied Probability, 19, 1, (2009) A. Teixeira Interlacement percolation on transient weighted graphs Electronic Journal of Probability, 14, (2009) A.Teixeira On the size of a finite vacant cluster of random interlacements with small intensity submitted (2009) Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
14 Thanks! Augusto Teixeira (ETH Zürich) Random interlacements Chorin / 14
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