2. What is percolation? ETH Zürich, Spring semester 2018

Transcription

1 2. What is percolation? ETH Zürich, Spring semester 2018

2 Percolation: applied motivations

3 Percolation: applied motivations Geology: How would water flow through these rocks?

4 Percolation: applied motivations Geology: How would water flow through these rocks? Ecology: How do forest fires propagate?

5 Percolation: mathematical motivations

6 Percolation: mathematical motivations A central tool to understand other models in statistical physics.

7 Percolation: mathematical motivations A central tool to understand other models in statistical physics. Level lines of random functions.

8 Percolation: mathematical motivations A central tool to understand other models in statistical physics. Level lines of random functions. Picture by Dmitry Belyaev.

9 Percolation: mathematical motivations A central tool to understand other models in statistical physics. Level lines of random functions. Picture by Dmitry Belyaev. Related to recent developments in continuous random geometry.

10 Percolation: mathematical motivations A central tool to understand other models in statistical physics. Level lines of random functions. Picture by Dmitry Belyaev. Related to recent developments in continuous random geometry. Picture by Gabor Pete.

11 Percolation: main motivation! Harry Kesten Percolation theory for mathematicians, July 1982.

12 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons.

13 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1.

14 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons:

15 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons.? Parameter: 0 ď p ď 1. Random coloring of the hexagons:

16 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons.? Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p,

17 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons.? Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p.

18 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p.

19 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p.

20 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p.

21 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p.

22 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p. p 1 2

23 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p. p 1 10

24 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p. p 9 10

25 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p. p 0

26 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p. p 1

27 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p.

28 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. x y Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p. Red path: a path made of red hexagons.

29 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p. Red path: a path made of red hexagons. Red Cluster: red connected component. Island

30 Bernoulli site percolation [Broadbent and Hammersley, 1957] We tile a lozenge with hexagons. Parameter: 0 ď p ď 1. Random coloring of the hexagons: A given hexagon is colored: red with probability p, blue with probability 1 p. Red path: a path made of red hexagons. Red Cluster: red connected component. Island

31 A porous stone?

32 QUESTION 1: Is there a red path from top to bottom in a large lozenge?

33 p 0

34 p 1

35 p 0.1

36 p 0.2

37 p 0.3

38 p 0.4

39 p 0.4

40 p 0.45

41 p 0.5

42 p 0.5

43 p 0.5

44 p 0.5

45 p 0.55

46 p 0.6

47 p 0.7

48 p 0.8

49 p 0.9

50 p 1

51 p ă 1 2 p 1 2 p ą 1 2

52 p ă 1 2 p 1 2 p ą 1 2 Rigorous answer to Question 1 Theorem [Kesten, 1980] For percolation with parameter p, we have lim nñ8» Prob p n n $ fi 0 if p ă & 1 2 ffi fl 1 if p % 1 if p ą 1 2

53 A forest?

54 QUESTION 2: How far can we go when starting from a single hexagon in the center?

55 p 0

56 p 0.3

57 p 0.4

58 p 0.45

59 p 0.5

60 p 0.6

61 p 0.7

62 p 0.8

63 p 0.9

64 p 1

65 p ă 1 2 p 1 2 p ą 1 2

66 p ă 1 2 p 1 2 p ą 1 2 Rigorous answer to Question 2 Theorem [Kesten, 1980] For percolation with parameter p, we have $» fi & ffi Prob p n fl % n ď e cppqn if p ă 1, [exponential decay] 2 ď 1 if p 1, [polynomial decay] n cppq 2 ě cppq if p ą 1. [uniform positivity] 2

67 p ă 1 2 p 1 2 p ą 1 2 Rigorous answer to Question 2 Theorem [Kesten, 1980] For percolation with parameter p, we have $» fi & ffi Prob p n fl % n Remark: For p 1 2, Probp «ď e cppqn if p ă 1, [exponential decay] 2 ď 1 if p 1, [polynomial decay] n cppq 2 ě cppq if p ą 1. [uniform positivity] 2 n n ff» 1 n5{48 [Lawler, Schramm, Werner 02]

68 Some percolation processes: Percolation on hexagons.

69 Some percolation processes: Percolation on hexagons. Percolation on Z d, d ě 2.

70 Some percolation processes: Percolation on hexagons. Percolation on Z d, d ě 2. Voronoi percolation in R d.

71 Some percolation processes: Percolation on hexagons. Percolation on Z d, d ě 2. Voronoi percolation in R d. Boolean percolation in R d.

72 Some percolation processes: Percolation on hexagons. Percolation on Z d, d ě 2. Voronoi percolation in R d. Boolean percolation in R d.

73 Some percolation processes: Percolation on hexagons. Percolation on Z d, d ě 2. Voronoi percolation in R d. Boolean percolation in R d. Phase transition (p density of red points). 0 p c 1 p All the red clusters are bounded. One giant red cluster. p c: critical parameter (depends on the model).