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Percolation on isoradial graphs Ioan Manolescu Joint work with Geoffrey Grimmett University of Geneva 15 August 2013 Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 1 / 25 General Percolation

  1. Isoradial graphs – details Isoradial Graphs G isoradial graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  2. Isoradial graphs – details Isoradial Graphs G isoradial graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  3. Isoradial graphs – details Isoradial Graphs G isoradial graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  4. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  5. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  6. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph G � diamond graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  7. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph G � diamond graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  8. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph G � diamond graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  9. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph G � diamond graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  10. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph G � diamond graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  11. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph G � diamond graph Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  12. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph G � diamond graph Track system Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  13. Isoradial graphs – details Isoradial Graphs G isoradial graph G ∗ dual isoradial graph G � diamond graph Track system Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

  14. Isoradial graphs – details Conditions Conditions for isoradial graphs. Bounded angles condition : There exist ǫ 0 > 0 such that for any edge e , θ e ∈ [ ǫ 0 , π − ǫ 0 ]. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 12 / 25

  15. Isoradial graphs – details Conditions Conditions for isoradial graphs. Bounded angles condition : There exist ǫ 0 > 0 such that for any edge e , θ e ∈ [ ǫ 0 , π − ǫ 0 ]. t 0 t − 1 t 1 s 2 Square grid property : s 1 Families of ”parallel” tracks ( s i ) i ∈ Z and ( t j ) j ∈ Z . The number of intersections on s i between s 0 t j and t j +1 is uniformly bounded by a con- stant I . (same for t ). Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 12 / 25

  16. Isoradial graphs – details Conditions Examples: Penrose tilings and no square grid Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

  17. Isoradial graphs – details Conditions Examples: Penrose tilings and no square grid Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

  18. Isoradial graphs – details Conditions Examples: Penrose tilings and no square grid Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

  19. Isoradial graphs – details Conditions Examples: Penrose tilings and no square grid Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

  20. Isoradial graphs – details Conditions Examples: Penrose tilings and no square grid Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

  21. Isoradial graphs – details Conditions Examples: Penrose tilings and no square grid Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

  22. Star–triangle transformation Star–triangle transformation A A p 2 p 1 O B B p 0 C C κ △ ( p ) = p 0 + p 1 + p 2 − p 0 p 1 p 2 = 1 . Take ω , respectively ω ′ , according to the measure on the left, respectively right. The families of random variables � � � ω ′ � ω x ← → y : x , y = A , B , C , x ← → y : x , y = A , B , C , have the same joint law. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 14 / 25

  23. Star–triangle transformation Coupling T (1 − p 0 ) p 1 p 2 p 0 (1 − p 1 ) p 2 p 0 p 1 (1 − p 2 ) p 0 p 1 p 2 P P P P T and similarly for all single edges T S S and similarly for all pairs of edges S (1 − p 0 ) p 1 p 2 p 0 (1 − p 1 ) p 2 p 0 p 1 (1 − p 2 ) p 0 p 1 p 2 P P P P where P = (1 − p 0 )(1 − p 1 )(1 − p 2 ) . Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 15 / 25

  24. Star–triangle transformation Path transformation A A O B C B C A A A O B C B C B C A A O B C B C Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 16 / 25

  25. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  26. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  27. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  28. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  29. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  30. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  31. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  32. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  33. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  34. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  35. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  36. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  37. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  38. From Z 2 to isoradial square lattice. Proof for box-crossing property Track exchange Two parallel tracks s 1 and s 2 with no intersection between them. We may exchange s 1 and s 2 using star–triangle transformations. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

  39. From Z 2 to isoradial square lattice. Proof for box-crossing property Initial Principal Secondary Probability configuration outcome outcome of secondary outcome θ 2 p π − θ 1 p θ 2 p θ 1 p π − θ 2 θ 1 θ 2 Open paths are p π − θ 1 p π − θ 2+ θ 1 preserved (unless p θ 1 p θ 2 − θ 1 θ 1 the deleted edge was θ 2 part of the path). p θ 2 p π − θ 2+ θ 1 p π − θ 2 p θ 2 − θ 1 θ 1 θ 2 p θ 2 p π − θ 2+ θ 1 p π − θ 2 p θ 2 − θ 1 θ 1 θ 2 p π − θ 1 p π − θ 2+ θ 1 p θ 1 p θ 2 − θ 1 θ 1 Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 18 / 25

  40. From Z 2 to isoradial square lattice. Proof for box-crossing property Strategy Proposition If two isoradial square lattices have same transverse angles for the vertical/horizontal tracks, and one has the box-crossing property, then so does the other. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 19 / 25

  41. From Z 2 to isoradial square lattice. Proof for box-crossing property Strategy Proposition If two isoradial square lattices have same transverse angles for the vertical/horizontal tracks, and one has the box-crossing property, then so does the other. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 19 / 25

  42. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  43. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  44. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  45. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  46. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  47. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  48. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  49. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  50. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  51. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  52. From Z 2 to isoradial square lattice. Proof for box-crossing property Transport of horizontal crossings Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

  53. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  54. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  55. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  56. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  57. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  58. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  59. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  60. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  61. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  62. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  63. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  64. Proof for box-crossing property From square lattices to general graphs Track stacking Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  65. Proof for box-crossing property From square lattices to general graphs Track stacking P gen ( C h [ B ( ρ N , N )]) ≥ P sq ( C h [ B ( I ρ N , N )]) P sq ( C v [ B ( N , N )]) 2 Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

  66. Proof for box-crossing property Arm exponents Transport of the arm exponents . . . . . . using the same strategy as for the box-crossing property. Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 22 / 25

  67. Proof for box-crossing property Arm exponents Square lattices Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

  68. Proof for box-crossing property Arm exponents Square lattices Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

  69. Proof for box-crossing property Arm exponents Square lattices Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

  70. Proof for box-crossing property Arm exponents Square lattices Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

  71. Proof for box-crossing property Arm exponents Square lattices Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

  72. Proof for box-crossing property Arm exponents Square lattices Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

  73. Proof for box-crossing property Arm exponents Square lattices c 1 P reg ( A k ( n )) ≤ P irreg ( A k ( n )) Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

  74. Proof for box-crossing property Arm exponents Square lattices c 1 P reg ( A k ( n )) ≤ P irreg ( A k ( n )) Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

  75. Proof for box-crossing property Arm exponents Square lattices c 1 P reg ( A k ( n )) ≤ P irreg ( A k ( n )) ≤ c 2 P reg ( A k ( n )) . Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

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