Finite Connections for Supercritical Bernoulli Bond Percolation in - PowerPoint PPT Presentation

Introduction Sketch of Proof Summary Finite Connections for Supercritical Bernoulli Bond Percolation in 2D M. Campanino 1 D. Ioffe 2 O. Louidor 3 1 Università di Bologna (Italy) 2 Technion (Israel) 3 Courant (New York University) Courant Probability Seminar, 11/6/2009

Introduction Sketch of Proof Summary Outline Introduction Percolation on Z d Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

Introduction Sketch of Proof Summary Outline Introduction Percolation on Z d Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

Introduction Sketch of Proof Summary Percolation • Take G = ( Z d , E d ) - the integer lattice with nearest neighbor edges. • Open each edge with probability p ∈ [ 0 , 1 ] independently. • Let B p be the underlying measure. Theorem (Broadbent, Hammersley 1957) For all d > 2 , there exists p c ( d ) ∈ ( 0 , 1 ) such that: � 0 if p < p c ( d ) B p ( 0 ← → ∞ ) = θ ( p ) > 0 if p > p c ( d )

Introduction Sketch of Proof Summary Basic Picture Sub-critical density p < p c ( d ) : • All clusters (connected components) are finite. • Radii of clusters have exponentially decaying distributions: → ∂ B ( R )) ≈ e − ξ p R . ∃ ξ p ∈ ( 0 , ∞ ) : B p ( 0 ← Russo-Menshikov (86), Barskey-Aizenman (87). Super-critical density p > p c ( d ) : • Unique infinite cluster B p -a.s. • Radii of finite clusters have exponentially decaying distributions: B p ( ∞ � 0 ↔ ∂ B ( R )) ≈ e − ζ p R . ∃ ζ p ∈ ( 0 , ∞ ) : Chayes 2 -Newman (87), C 2 -Grimmett-Kesten-Schonmann (89).

Introduction Sketch of Proof Summary Connectivities The point-to-point connectivity function is defined as: x , y ∈ Z d . τ p ( x , y ) = B p ( x ← → y ) ;

Introduction Sketch of Proof Summary Connectivities The point-to-point connectivity function is defined as: x , y ∈ Z d . τ p ( x , y ) = B p ( x ← → y ) ; Sub-critical case: Subadditivity (via FKG of B p ) and exponential decay of cluster radius distribution imply: Theorem Assume p < p c ( d ) . Then, for all x ∈ R d : n →∞ − 1 ξ p ( x ) = lim n log τ p ( 0 , ⌊ n x ⌋ ) is well-defined, convex and homogeneous function that is strictly positive on R d \ { 0 } . In other words ξ p is a norm on R d , called the inverse correlation norm.

Introduction Sketch of Proof Summary Connectivities - cont’d Super-critical case: FKG gives a uniform positive lower bound for all x , y ∈ Z d : → y ) � B p ( x ← → ∞ ) = θ 2 ( p ) > 0 . B p ( x ← → ∞ ) B p ( y ←

Introduction Sketch of Proof Summary Connectivities - cont’d Super-critical case: FKG gives a uniform positive lower bound for all x , y ∈ Z d : → y ) � B p ( x ← → ∞ ) = θ 2 ( p ) > 0 . B p ( x ← → ∞ ) B p ( y ← Therefore, define the finite (truncated) connectivity function: f x , y ∈ Z d . τ f ← → y ) = B p ( ∞ � x ↔ y ) ; p ( x , y ) = B p ( x

Introduction Sketch of Proof Summary Connectivities - cont’d Super-critical case: FKG gives a uniform positive lower bound for all x , y ∈ Z d : → y ) � B p ( x ← → ∞ ) = θ 2 ( p ) > 0 . B p ( x ← → ∞ ) B p ( y ← Therefore, define the finite (truncated) connectivity function: f x , y ∈ Z d . τ f ← → y ) = B p ( ∞ � x ↔ y ) ; p ( x , y ) = B p ( x Theorem ∈ { 0 , p c ( d ) , 1 } . Then, for all x ∈ R d : Assume p / n →∞ − 1 n log τ f ζ p ( x ) = lim p ( 0 , ⌊ n x ⌋ ) is well-defined, homogeneous and strictly positive on R d \ { 0 } . This is the finite (truncated) inverse correlation function.

Introduction Sketch of Proof Summary Logarithmic Scale Asymptotics In other words: f → y ) ≈ e − ξ p ( θ ) � y − x � 2 → y ) ≈ e − ζ p ( θ ) � y − x � 2 B p ( x ← and B p ( x ← for all x , y ∈ Z d as y − x → ∞ , where θ = ( x − y ) / � x − y � 2 .

Introduction Sketch of Proof Summary Logarithmic Scale Asymptotics In other words: f → y ) ≈ e − ξ p ( θ ) � y − x � 2 → y ) ≈ e − ζ p ( θ ) � y − x � 2 B p ( x ← and B p ( x ← for all x , y ∈ Z d as y − x → ∞ , where θ = ( x − y ) / � x − y � 2 . Some relations: • ξ p = ξ p ( e 1 ) . ζ p = ζ p ( e 1 ) . • If d = 2, p > p c ( 2 ) = 1 2 then ζ p = 2 ξ 1 − p . Chayes-Chayes-Grimmett-Kesten-Schonmann (89).

Introduction Sketch of Proof Summary Logarithmic Scale Asymptotics In other words: f → y ) ≈ e − ξ p ( θ ) � y − x � 2 → y ) ≈ e − ζ p ( θ ) � y − x � 2 B p ( x ← and B p ( x ← for all x , y ∈ Z d as y − x → ∞ , where θ = ( x − y ) / � x − y � 2 . Some relations: • ξ p = ξ p ( e 1 ) . ζ p = ζ p ( e 1 ) . • If d = 2, p > p c ( 2 ) = 1 2 then ζ p = 2 ξ 1 − p . Chayes-Chayes-Grimmett-Kesten-Schonmann (89). Want sharp asymptotics: → y ) ∼ ? → y ) ∼ ? f B p ( x ← and B p ( x ←

Introduction Sketch of Proof Summary Sharp Asymptotics - Subcritical Case For all d � 2, p < p c ( d ) , x , y ∈ Z d : → y ) ∼ A p ( θ ) � y − x � − ( d − 1 ) / 2 e − ξ p ( θ ) � y − x � 2 B p ( x ← as y − x → ∞ . 2 • Campanino-Chayes-Chayes (88) (y − x is on the axes). • Campanino-Ioffe (02) (all y − x). With the Gaussian correction this is called Ornstein-Zernike Behavior. after the work of L.Ornstein and F.Zernike.

Introduction Sketch of Proof Summary Sharp Asymptotics - Supercritical Case d � 3: For all p > p c ( d ) , x , y ∈ Z d , it is expected: f A p ( θ ) � y − x � − ( d − 1 ) / 2 → y ) ∼ � e − ζ p ( θ ) � y − x � 2 B p ( x ← as y − x → ∞ . 2 Verified for p ≫ p c ( d ) and y − x on axes. Braga-Procacci-Sanchis (04) .

Introduction Sketch of Proof Summary Sharp Asymptotics - Supercritical Case d � 3: For all p > p c ( d ) , x , y ∈ Z d , it is expected: f A p ( θ ) � y − x � − ( d − 1 ) / 2 → y ) ∼ � e − ζ p ( θ ) � y − x � 2 B p ( x ← as y − x → ∞ . 2 Verified for p ≫ p c ( d ) and y − x on axes. Braga-Procacci-Sanchis (04) . d = 2:

Introduction Sketch of Proof Summary Sharp Asymptotics - Supercritical Case d � 3: For all p > p c ( d ) , x , y ∈ Z d , it is expected: f A p ( θ ) � y − x � − ( d − 1 ) / 2 → y ) ∼ � e − ζ p ( θ ) � y − x � 2 B p ( x ← as y − x → ∞ . 2 Verified for p ≫ p c ( d ) and y − x on axes. Braga-Procacci-Sanchis (04) . d = 2: ?

Introduction Sketch of Proof Summary A Related Model - Nearest-Neighbor Ising

Introduction Sketch of Proof Summary A Related Model - Nearest-Neighbor Ising Exactly solvable in d = 2 ( Onsager (44) ). Explicit formulas for (truncated) k -point correlation functions at all temperatures ( Wu, McCoy, Tracy, Potts, Ward, Montroll (’70) ). Theorem (Cheng-Wu, Wu) If β < β c ( 2 ) then � σ x ; σ y � β � � σ x σ y � β ∼ A β ( θ ) � y − x � − 1 / 2 e − ξ β ( θ ) � y − x � 2 2 and if β > β c ( 2 ) then: � σ x ; σ y � T β � � σ x σ y � β − � σ x � β � σ y � β ∼ � A β ( θ ) � y − x � − 2 2 e − ζ β ( θ ) � y − x � 2 for all x , y ∈ Z 2 as y − x → ∞ . No OZ Behavior in d = 2 below the critical temperature!

Introduction Sketch of Proof Summary Sharp Asymptotics - Supercritical Case d � 3: For all p > p c ( d ) , x , y ∈ Z d , it is expected: f A p ( θ ) � y − x � − ( d − 1 ) / 2 → y ) ∼ � e − ζ p ( θ ) � y − x � 2 B p ( x ← as y − x → ∞ . 2 Verified for p ≫ p c ( d ) and y − x on axes. Braga-Procacci-Sanchis (04) . d = 2: ?

Introduction Sketch of Proof Summary Sharp Asymptotics - Supercritical Case d � 3: For all p > p c ( d ) , x , y ∈ Z d , it is expected: f A p ( θ ) � y − x � − ( d − 1 ) / 2 → y ) ∼ � e − ζ p ( θ ) � y − x � 2 B p ( x ← as y − x → ∞ . 2 Verified for p ≫ p c ( d ) and y − x on axes. Braga-Procacci-Sanchis (04) . d = 2: Theorem (Campanino, Ioffe, L. (09)) For all p > p c ( 2 ) = 1 / 2 , x , y ∈ Z 2 : f → y ) ∼ � A p ( θ ) � y − x � − 2 2 e − ζ p ( θ ) � y − x � 2 B p ( x ← as y − x → ∞ .

Introduction Sketch of Proof Summary Outline Introduction Percolation on Z d Logarithmic Asymptotics of Connectivities Sharp Asymptotics of Connectivities Sketch of Proof Setup Geometry of Finite Connections The Structure of a Cluster Asymptotics for No Intersection of Two Decorated RWs Summary

Introduction Sketch of Proof Summary Setup Dual lattice. • In d = 2 there is an isomorphic dual Z 2 ∗ . • Set: b ∗ is open ⇐ ⇒ b is close . • The dual model is Percolation with p ∗ = 1 − p . f • We assume p < p c ( 2 ) and find B p ( x ∗ → y ∗ ) . ← • However, we’ll express this event mainly using direct bonds. • For simplicity: x ∗ = 0 ∗ , y ∗ = 0 ∗ + ( N , 0 ) � N ∗ .