Near-critical percolation and minimal spanning tree in the plane

Transcription

1 Near-critical percolation and minimal spanning tree in the plane Christophe Garban ENS Lyon, CNRS joint work with Gábor Pete and Oded Schramm 37 th SPA, Buenos Aires, July 2014 C. Garban Near-critical percolation in the plane 0 / 39

2 Traveling Salesman Problem C. Garban Near-critical percolation in the plane 1 / 39

3 Traveling Salesman Problem TSP C. Garban Near-critical percolation in the plane 1 / 39

4 Traveling Salesman Problem C. Garban Near-critical percolation in the plane 2 / 39

5 Traveling Salesman Problem C. Garban Near-critical percolation in the plane 2 / 39

6 Traveling Salesman Problem C. Garban Near-critical percolation in the plane 2 / 39

7 Minimal Spanning Tree (MST) TSP C. Garban Near-critical percolation in the plane 3 / 39

8 Minimal Spanning Tree (MST) MST C. Garban Near-critical percolation in the plane 3 / 39

9 MAIN QUESTION: scaling limit of the planar MST? C. Garban Near-critical percolation in the plane 4 / 39

10 Minimal Spanning Tree on Z 2 C. Garban Near-critical percolation in the plane 5 / 39

11 Minimal Spanning Tree on Z C. Garban Near-critical percolation in the plane 5 / 39

12 Minimal Spanning Tree on Z C. Garban Near-critical percolation in the plane 5 / 39

13 MST on Z 2 seen from further away... C. Garban Near-critical percolation in the plane 6 / 39

14 Scaling limit of the uniform spanning Tree SLE κ process with κ = 8 Theorem by Lawler Schramm Werner (2003) C. Garban Near-critical percolation in the plane 7 / 39

15 Scaling limit of percolation Theorem (Smirnov, 2001) Critical site percolation on ηt is asymptotically (as η 0) conformally invariant. Convergence to SLE 6 C. Garban Near-critical percolation in the plane 8 / 39

16 A greedy algorithm to compute the MST Kruskal s algorithm: C. Garban Near-critical percolation in the plane 9 / 39

17 A greedy algorithm to compute the MST Kruskal s algorithm: C. Garban Near-critical percolation in the plane 9 / 39

18 A greedy algorithm to compute the MST Kruskal s algorithm: C. Garban Near-critical percolation in the plane 9 / 39

19 A greedy algorithm to compute the MST Kruskal s algorithm: C. Garban Near-critical percolation in the plane 9 / 39

20 A greedy algorithm to compute the MST Kruskal s algorithm: C. Garban Near-critical percolation in the plane 9 / 39

21 A greedy algorithm to compute the MST Kruskal s algorithm: C. Garban Near-critical percolation in the plane 9 / 39

22 A greedy algorithm to compute the MST Kruskal s algorithm: C. Garban Near-critical percolation in the plane 9 / 39

23 Kruskal s algorithm on Z C. Garban Near-critical percolation in the plane 10 / 39

24 Kruskal s algorithm on Z C. Garban Near-critical percolation in the plane 10 / 39

25 Kruskal s algorithm on Z C. Garban Near-critical percolation in the plane 10 / 39

26 Kruskal s algorithm on Z C. Garban Near-critical percolation in the plane 10 / 39

27 Kruskal s algorithm on Z C. Garban Near-critical percolation in the plane 10 / 39

28 Kruskal s algorithm on Z C. Garban Near-critical percolation in the plane 10 / 39

29 Kruskal s algorithm on Z C. Garban Near-critical percolation in the plane 10 / 39

30 Percolation Model on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 11 / 39

31 Percolation Model on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 11 / 39

32 Percolation Model on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 11 / 39

33 Percolation Model on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 11 / 39

34 Percolation Model on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 11 / 39

35 Monotone coupling in percolation Definition (Standard coupling) For all e Z 2, sample u e U([0, 1]). For any fixed p [0, 1], let ω p (e) := 1 ue p As such ω p P p for all p and ω p ω p if p p C. Garban Near-critical percolation in the plane 12 / 39

36 Abrubt phase transition Seen from far away it looks as follows: Sub-critical (p <p c ) Critical (p c ) Super-critical (p >p c ) δz 2 δz 2 Theorem (Kesten, 1980) p c (Z 2 ) = 1 2 C. Garban Near-critical percolation in the plane 13 / 39

37 Kruskal s algorithm on Z 2 C. Garban Near-critical percolation in the plane 14 / 39

38 Kruskal s algorithm on Z 2 C. Garban Near-critical percolation in the plane 14 / 39

39 Kruskal s algorithm on Z 2 C. Garban Near-critical percolation in the plane 14 / 39

40 Kruskal s algorithm on Z 2 p < p c (Z 2 ) = 1/2 C. Garban Near-critical percolation in the plane 14 / 39

41 Kruskal s algorithm on Z 2 p < p c (Z 2 ) = 1/2 C. Garban Near-critical percolation in the plane 14 / 39

42 Kruskal s algorithm on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 14 / 39

43 Kruskal s algorithm on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 14 / 39

44 Kruskal s algorithm on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 14 / 39

45 Kruskal s algorithm on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 14 / 39

46 Kruskal s algorithm on Z 2 ω p, p = C. Garban Near-critical percolation in the plane 14 / 39

47 Minimal Spanning Tree in the plane Theorem (Aizenman, Burchard, Newman, Wilson, 1999) The Minimal Spanning Tree on ηz 2 is tight as η 0 (for a metric on the space of planar spanning trees inspired by the Hausdorff distance) On the triangular lattice, we will prove the convergence as η 0 This requires a detailed analysis of near-critical percolation: Sub-critical (p <p c ) Critical (p c ) Super-critical (p >p c ) δz 2 δz 2

48 References A) 2010, Pivotal, cluster and interface measures for critical planar percolation, G., Pete, Schramm, J.A.M.S B) 2013, The scaling limits of near-critical and dynamical percolation, G., Pete, Schramm, arxiv: C) 2013, The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane, G., Pete, Schramm, Arxiv: C. Garban Near-critical percolation in the plane 16 / 39

49 Near-critical geometry in general Ising model near its critical point: T = Tc C. Garban T = Tc δt Near-critical percolation in the plane 17 / 39

50 Near-critical percolation (mean-field case) Erdos-Renyi random graphs G(n, p), p [0, 1] (p-percolation on the complete graph n ). p < 1/n log n p = 1/n p > 1/n n 2/3 O(n) It is well-known that everything happens in the near-critical window p = 1 n + λ 1 n 4/3 where λ is the near-critical parameter C. Garban Near-critical percolation in the plane 18 / 39

51 A quotation and a Theorem Alon and Spencer (2002): With λ = 10 6, say we have feudalism. Many components (castles) are each vying to be the largest. As λ increases the components increase in size and a few large components (nations) emerge. An already large France has much better chances of becoming larger than a smaller Andorra. The largest components tend to merge and by λ = 10 6 it is very likely that a giant component, the Roman Empire, has emerged. With high probability this component is nevermore challenged for supremacy but continues absorbing smaller components until full connectivity One World is achieved. Theorem (Addario-Berry, Broutin, Goldschmidt, Miermont, 2013) Let MST n be the Minimal Spanning Tree on n (MST n, 1 n 1/3 d graph) law MST where the convergence in law holds under the Gromov-Hausdorff topology.

52 Near-critical percolation in the plane Site percolation on the triangular lattice T : feudalism p < 1/2 p = 1/2 1 n Roman empire p > 1/2 Renormalise the lattice as follows: ηt where η corresponds to the mesh of the rescaled lattice. η 0??

53 looking for the right ZOOMING We shall now zoom around p c as follows: p = p c + λ r(η) C. Garban Near-critical percolation in the plane 21 / 39

54 looking for the right ZOOMING We shall now zoom around p c as follows: p = p c + λ r(η) λ < 0 λ = 0 λ > 0 C. Garban Near-critical percolation in the plane 21 / 39

55 looking for the right ZOOMING We shall now zoom around p c as follows: p = p c + λ r(η) Theorem (Kesten, 1987) λ < 0 λ = 0 λ > 0 The right zooming factor is given by r(η) := η 2 α 4 (η, 1) 1 = η 3/4+o(1) C. Garban Near-critical percolation in the plane 21 / 39

56 Heuristics behind these scalings p = 1/n + λn 4/3 versus p c + λη 3/4+o(1) C 1 C 2 O(n 2/3 ) O(n 2/3 ) O(n 4/3 ) C. Garban Near-critical percolation in the plane 22 / 39

57 Heuristics behind these scalings p = 1/n + λn 4/3 versus p c + λη 3/4+o(1) C 1 C 2 η O(n 2/3 ) O(n 2/3 ) O(n 4/3 ) = η 3/4+o(1) C. Garban Near-critical percolation in the plane 22 / 39

58 Scaling limit? Definition Define ω nc η (λ) to be the percolation configuration on ηt of parameter p = p c + λ r(η) For all η > 0, we define this way a monotone càdlàg process λ R ω nc η (λ) {0, 1} ηt Question Does the process λ R ω nc η (λ) converge (in law) as η 0 to a limiting process λ ω nc (λ)? For which topology?? Find an appropriate Polish space (E, d) whose points ω E are naturally identified to percolation configurations.

59 The first natural idea which comes to mind This configuration on ηt may be coded by the distribution X η := η σ x δ x x ηt {X η } η is tight in H 1 ε and converge to the Gaussian white noise on R 2. C. Garban Near-critical percolation in the plane 24 / 39

60 The first natural idea which comes to mind This configuration on ηt may be coded by the distribution X η := η σ x δ x x ηt {X η } η is tight in H 1 ε and converge to the Gaussian white noise on R 2. Theorem (Benjamini, Kalai, Schramm, 1999) This setup is NOT appropriate to handle percolation: natural observables for percolation are highly discontinuous under the topology induced by H 1 ε and in fact are not even measurable in the limit. C. Garban Near-critical percolation in the plane 24 / 39

61 Some other historical approaches 1 Aizenman 1998 and Aizenman, Burchard Camia, Newman The topological space (H, T ) of Schramm-Smirnov, 2011 C. Garban Near-critical percolation in the plane 25 / 39

62 The Schramm-Smirnov space H 1 Q Q 3 Let (Q, d Q ) be the space of all quads. On might consider the space {0, 1} Q

63 The Schramm-Smirnov space H 1 Q Q 3 Let (Q, d Q ) be the space of all quads. On might consider the space {0, 1} Q In fact, one considers instead H {0, 1} Q which preserves the partial order on Q : Q > Q Schramm-Smirnov prove that H can be endowed with a natural topology T ( Fell s topology) for which, (H, T ) is compact, Hausdorff and metrizable

64 The critical slice ω P Definition (λ = 0) For each mesh η > 0, one may view ω η P η as a random point in the compact space (H, d H ). Theorem (Smirnov 2001, CN 2006, GPS 2013) ω η P η converges in law in (H, d H ) to a continuum percolation ω P this handles the case λ = 0 C. Garban Near-critical percolation in the plane 27 / 39

65 Away from the critical slice Recall: Question Let λ > 0 be fixed. p = p c + λ r(η) Does ω nc η (λ) converge in law in H to a limiting object? C. Garban Near-critical percolation in the plane 28 / 39

66 Main results Theorem (G., Pete, Schramm 2013) Fix λ R. ω nc η (λ) (d) ω nc (λ) The convergence in law holds in the space (H, d H ). Theorem (G., Pete, Schramm 2013) The càdlàg process λ ω nc η (λ) converges in law to λ ω nc (λ) for the Skorohod topology on H. C. Garban Near-critical percolation in the plane 29 / 39

67 Main results Theorem (G., Pete, Schramm 2013) Fix λ R. ω nc η (λ) (d) ω nc (λ) The convergence in law holds in the space (H, d H ). Theorem (G., Pete, Schramm 2013) The càdlàg process λ ω nc η (λ) converges in law to λ ω nc (λ) for the Skorohod topology on H. Theorem (Nolin, Werner 2007) Fix λ 0. All the subsequential scaling limits of ω nc η k (λ) (d) ω (λ) are such that their interfaces are singular w.r.t the SLE 6 curves! C. Garban Near-critical percolation in the plane 29 / 39

68 Two possible approaches Recall the case λ = 0 (critical case). One has ω η P η and we wish to prove a scaling limit result. tightness, uniqueness?? main ingredient for uniqueness: Cardy/Smirnov s formula! C. Garban Near-critical percolation in the plane 30 / 39

69 Two possible approaches Recall the case λ = 0 (critical case). One has ω η P η and we wish to prove a scaling limit result. tightness, uniqueness?? main ingredient for uniqueness: Cardy/Smirnov s formula! 1 This suggests the following approach to handle the case λ 0: for all p p c (T) = 1/2, find a massive harmonic observable F p : F p (x) m(p)f p (x) The mass m(p) should then scale as p p c 8/3. C. Garban Near-critical percolation in the plane 30 / 39

70 Two possible approaches Recall the case λ = 0 (critical case). One has ω η P η and we wish to prove a scaling limit result. tightness, uniqueness?? main ingredient for uniqueness: Cardy/Smirnov s formula! 1 This suggests the following approach to handle the case λ 0: for all p p c (T) = 1/2, find a massive harmonic observable F p : F p (x) m(p)f p (x) The mass m(p) should then scale as p p c 8/3. 2 A perturbative approach. C. Garban Near-critical percolation in the plane 30 / 39

71 Naïve Strategy to build λ ω nc (λ) ω (0) P λ λ = 0 C. Garban Near-critical percolation in the plane 31 / 39

72 Naïve Strategy to build λ ω nc (λ) ω (0) P λ P(ω (0)) λ = 0 C. Garban Near-critical percolation in the plane 31 / 39

73 Naïve Strategy to build λ ω nc (λ) PPP on R 2 R + of intensity µ P (dx)dλ ω (0) P λ P(ω (0)) λ = 0 C. Garban Near-critical percolation in the plane 31 / 39

74 Naïve Strategy to build λ ω nc (λ) PPP on R 2 R + of intensity µ P (dx)dλ ω (0) P λ P(ω (0)) λ = 0 λ > 0 C. Garban Near-critical percolation in the plane 31 / 39

75 Naïve Strategy to build λ ω nc (λ) PPP on R 2 R + of intensity µ P (dx)dλ ω (0) P ω (λ) λ P(ω (0)) λ = 0 λ > 0 C. Garban Near-critical percolation in the plane 31 / 39

76 Difficulty 1: too many pivotal points The mass measure µ is highly degenerate ( ) introduce a cut-off ε > 0 and try to define µ ε, a mass measure on the pivotal points P ε. ε

77 Difficulty 1: too many pivotal points The mass measure µ is highly degenerate ( ) introduce a cut-off ε > 0 and try to define µ ε, a mass measure on the pivotal points P ε. ε Theorem (GPS 2013) There is a measurable map µ ε from H to the space of locally finite measures such that Q i α Q j (ω η, µ ε (ω η )) (d) (ω, µ ε (ω )) as η 0

78 Difficulty 2: Stability question as ε 0 λ ω nc,ε η (λ) STABILITY problem as ε 0? x, λ 1 y, λ 2 x, t 2 y, t 1 Theorem (GPS 2013) There is a function ψ : [0, 1] [0, 1], with ψ(0) = 0 so that unif. in 0 < η < ε, E [ d Sk (ω η ( ), ω ε η( )) ] ψ(ε) C. Garban Near-critical percolation in the plane 33 / 39

79 Scaling invariance of our limiting object Theorem Near-critical percolation behaves as follows under the scaling z α z: ( ) λ α ω (λ) nc (d) ( ) = λ ω (α nc 3/4 λ) H Φ H 2 3/4 λ C. Garban Near-critical percolation in the plane 34 / 39

80 Some other properties 1 Conformal covariance C. Garban Near-critical percolation in the plane 35 / 39

81 Some other properties 1 Conformal covariance 2 Obtain scaling limits of (i) Invasion percolation (ii) Gradient percolation (iii) Dynamical percolation C. Garban Near-critical percolation in the plane 35 / 39

82 Some other properties 1 Conformal covariance 2 Obtain scaling limits of (i) Invasion percolation (ii) Gradient percolation (iii) Dynamical percolation 3 Two natural Markov processes on H Theorem t ω (t) is a reversible Markov process for the measure P. λ ω nc (λ) is a non-reversible time-homogeneous Markov process. C. Garban Near-critical percolation in the plane 35 / 39

83 Some other properties 1 Conformal covariance 2 Obtain scaling limits of (i) Invasion percolation (ii) Gradient percolation (iii) Dynamical percolation 3 Two natural Markov processes on H Theorem t ω (t) is a reversible Markov process for the measure P. λ ω nc (λ) is a non-reversible time-homogeneous Markov process.!! These are NOT Feller processes. C. Garban Near-critical percolation in the plane 35 / 39

84 Main theorem for the scaling limit of the MST Theorem (GPS 2013) 1 On the rescaled triangular lattice ηt, MST η converges in law to MST (under the topology used in ABNW 1999) 2 The UNIVERSALITY of this limit only requires the universality of the critical slice of percolation C. Garban Near-critical percolation in the plane 36 / 39

85 Very rough idea of proof Take λ λ 4/3 C. Garban Near-critical percolation in the plane 37 / 39

86 Very rough idea of proof Take λ Take ɛ small λ 4/3 C. Garban Near-critical percolation in the plane 37 / 39

87 Very rough idea of proof Take λ Take ɛ small λ 4/3 C. Garban Near-critical percolation in the plane 37 / 39

88 Very rough idea of proof Take λ Take ɛ small Take λ small λ 4/3 C. Garban Near-critical percolation in the plane 37 / 39

89 Very rough idea of proof Take λ Take ɛ small Take λ small λ 4/3 MST λ,λ,ɛ C. Garban Near-critical percolation in the plane 37 / 39

90 A.s. properties of MST Theorem (GPS 2013) 1 Rotational invariance 2 The Hausdorff dimension of the branches a.s. lies in (1 + ε, 7/4 ε) 3 There are no points of degree 5 4 There are no pinching points C. Garban Near-critical percolation in the plane 38 / 39

91 Some open questions Show that MST is not conformally-invariant Find the Hausdorff dimension d of branches (d??) Show that MST SLE 8!!! C. Garban Near-critical percolation in the plane 39 / 39

92 Some open questions Show that MST is not conformally-invariant Find the Hausdorff dimension d of branches (d??) Show that MST SLE 8!!! C. Garban Near-critical percolation in the plane 39 / 39

93 The end C. Garban Near-critical percolation in the plane 39 / 39

94 The end C. Garban Near-critical percolation in the plane 39 / 39

95 Back to the MST in the plane C. Garban Near-critical percolation in the plane 39 / 39

96 Back to the MST in the plane u(e) := u x u y C. Garban Near-critical percolation in the plane 39 / 39

97 Back to the MST in the plane u(e) := u x u y C. Garban Near-critical percolation in the plane 39 / 39

98 Back to the MST in the plane u(e) := u x u y C. Garban Near-critical percolation in the plane 39 / 39

99 Back to the MST in the plane u(e) := u x u y C. Garban Near-critical percolation in the plane 39 / 39

100 Back to the MST in the plane u(e) := u x u y C. Garban Near-critical percolation in the plane 39 / 39

101 Back to the MST in the plane u(e) := u x u y C. Garban Near-critical percolation in the plane 39 / 39

102 Back to the MST in the plane ? u(e) := u x u y C. Garban Near-critical percolation in the plane 39 / 39