Universality and RSW for inhomogeneous bond percolation Ioan - PowerPoint PPT Presentation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Universality and RSW for inhomogeneous bond percolation Ioan Manolescu joint work with Geoffrey Grimmett Statistical Laboratory Department of Pure Mathemetics and Mathematical Statistics University of Cambridge 22 August 2011 Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Percolation � open with probability p e An edge e is closed with probability 1 − p e Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Percolation � open with probability p e An edge e is closed with probability 1 − p e Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Percolation � open with probability p e An edge e is closed with probability 1 − p e Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Percolation � open with probability p e An edge e is closed with probability 1 − p e Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Percolation � open with probability p e An edge e is closed with probability 1 − p e Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Homogeneous Bond Percolation Z 2 T p p p p p p < p c , a.s. no infinite component; p > p c , a.s. existence of an infinite component. Criticality: p c ( Z 2 ) = 1 p c ( T ) = 2 sin π 2 . 18 . Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Homogeneous Bond Percolation Z 2 T p p p p p p < p c , a.s. no infinite component; p > p c , a.s. existence of an infinite component. Criticality: p c ( Z 2 ) = 1 p c ( T ) = 2 sin π 2 . 18 . Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Homogeneous Bond Percolation Z 2 T p p p p p p < p c , a.s. no infinite component; p > p c , a.s. existence of an infinite component. Criticality: p c ( Z 2 ) = 1 p c ( T ) = 2 sin π 2 . 18 . Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Inhomogeneous bond percolation Z 2 with P � T with P △ p ( p h , p v ) p 2 p v p 1 p h p 0 Criticality for Z 2 : p v + p h = 1 . Criticality for T : κ △ ( p ) = p 0 + p 1 + p 2 − p 0 p 1 p 2 = 1 , ( p = ( p 0 , p 1 , p 2 ) ∈ [0 , 1) 3 ). Call M the above class of critical (inhomogeneous) models. Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Inhomogeneous bond percolation Z 2 with P � T with P △ p ( p h , p v ) p 2 p v p 1 p h p 0 Criticality for Z 2 : p v + p h = 1 . Criticality for T : κ △ ( p ) = p 0 + p 1 + p 2 − p 0 p 1 p 2 = 1 , ( p = ( p 0 , p 1 , p 2 ) ∈ [0 , 1) 3 ). Call M the above class of critical (inhomogeneous) models. Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Inhomogeneous bond percolation Z 2 with P � T with P △ p ( p h , p v ) p 2 p v p 1 p h p 0 Criticality for Z 2 : p v + p h = 1 . Criticality for T : κ △ ( p ) = p 0 + p 1 + p 2 − p 0 p 1 p 2 = 1 , ( p = ( p 0 , p 1 , p 2 ) ∈ [0 , 1) 3 ). Call M the above class of critical (inhomogeneous) models. Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Criticality For P critical we expect: A D Ω δ → D (Ω , A, B, C, D ), as δ → 0 P B C where D (Ω , A , B , C , D ) is conformally invariant and does not depend on the underlying model. Only known for site percolation on the triangular lattice (Cardy’s formula, Smirnov 2001) Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next Criticality For P critical we expect: A D Ω δ → D (Ω , A, B, C, D ), as δ → 0 P B C where D (Ω , A , B , C , D ) is conformally invariant and does not depend on the underlying model. Only known for site percolation on the triangular lattice (Cardy’s formula, Smirnov 2001) Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next The box-crossing property A model satisfies the box-crossing property if for all α there exists c ( α ) > 0 s.t. for all N : ∈ [ c ( α ) , 1 − c ( α )] P N αN Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The star–triangle transformation Use of star–triangle transformation What’s next The box-crossing property A model satisfies the box-crossing property if for all α there exists c ( α ) > 0 s.t. for all N : ∈ [ c ( α ) , 1 − c ( α )] P N αN The homogeneous models in M satisfy the box-crossing property. Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The box-crossing property The star–triangle transformation Critical exponents Use of star–triangle transformation What’s next Main result I Theorem All models in M satisfy the box-crossing property. Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The box-crossing property The star–triangle transformation Critical exponents Use of star–triangle transformation What’s next Exponents at criticality For a critical percolation A 4 ( N , n ) measure P p c , as n → ∞ , we expect: volume exponent: P p c ( | C 0 | = n ) ≈ n − 1 − 1 /δ , connectivity exponent: O P p c (0 ↔ x ) ≈ | x | − η , one-arm exponent: N P p c ( rad ( C 0 ) = n ) ≈ n − 1 − 1 /ρ , 2 j -alternating-arms exponents: P p c [ A 2 j ( N , n )] ≈ n − ρ 2 j , n Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The box-crossing property The star–triangle transformation Critical exponents Use of star–triangle transformation What’s next Exponents near ciritcality Percolation probability: P p c + ǫ ( | C 0 | = ∞ ) ≈ ǫ β as ǫ ↓ 0, Correlation length: ξ ( p c − ǫ ) ≈ ǫ − ν as ǫ ↓ 0, where − 1 1 n log P p c − ǫ ( rad ( C 0 ) ≥ n ) → n →∞ ξ ( p c − ǫ ) . Mean cluster-size: P p c + ǫ ( | C 0 | ; | C 0 | < ∞ ) ≈ | ǫ | − γ as ǫ → 0, Gap exponent: for k ≥ 1, as ǫ → 0, P p c + ǫ ( | C 0 | k +1 ; | C 0 | < ∞ ) ≈ | ǫ | − ∆ . P p c + ǫ ( | C 0 | k ; | C 0 | < ∞ ) Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation

Introduction and models Results The box-crossing property The star–triangle transformation Critical exponents Use of star–triangle transformation What’s next Scaling relations Kesten ’87. For models with the box-crossing property if ρ or η exist, then ηρ = 2 and 2 ρ = δ + 1 . Kesten ’87. For models with the box-crossing property rotation and translation invariance, β , ν , γ and δ may be expressed in terms of ρ and ρ 4 . Ioan Manolescu joint work with Geoffrey Grimmett Universality and RSW for inhomogeneous bond percolation