Sharp phase transition for Bargmann-Fock percolation Hugo - PowerPoint PPT Presentation

Sharp phase transition for Bargmann-Fock percolation Hugo Vanneuville (ICJ, Universit´ e Lyon 1) joint work with Alejandro Rivera (IF, Universit´ e de Grenoble) Random waves in Oxford, 19 th june, 2018 1/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field E [ f ( x ) f ( y )] = exp( −| x − y | 2 / 2) . Connections properties of the sets { f > − ℓ } ? { f > − ℓ } is coloured black, its biggest component is coloured blue ℓ = − 0 . 1 ℓ = 0 ℓ = 0 . 1 Simulations by Alejandro Rivera 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R Theorem (Sharp phase transition) Beffara-Gayet, 16 ( ℓ = 0 ); Rivera-V, 17 ( ℓ � = 0 )  ≤ exp( − c ( ℓ ) R ) if ℓ < 0     P [ Cross ℓ ( R )] � � ∈ c , 1 − c if ℓ = 0    ≥ 1 − exp( − c ( ℓ ) R ) if ℓ > 0  Generalizations: Beffara-Gayet, Beliaev-Muirhead, Beliaev-Muirhead- Wigman, Rivera-V ( ℓ = 0); Muirhead-V ( ℓ � = 0) 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R P [ Cross ℓ ( R )] 1 0 ℓ 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R Corollary (The critical threshold is ℓ = 0) Alexander, 96; Beffara-Gayet, 17 ( ℓ ≤ 0 ); Rivera-V, 17 ( ℓ > 0 ) If ℓ ≤ 0 then a.s. there is no unbounded connected component in { f > − ℓ } . If ℓ > 0 then a.s. there is a unique unbounded connected component in { f > − ℓ } . Early works: Molchanov-Stepanov, 83 ( | ℓ | very large) 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

( f ( x )) x ∈ R 2 planar Bargmann-Fock field f > − ℓ Cross ℓ ( R ) := R 2 R P [ Cross ℓ ( R )] 1 0 ℓ 2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Sharp transitions for Boolean functions p ∈ [0 , 1], equip {− 1 , 1 } N with P p := ( p δ 1 + (1 − p ) δ − 1 ) ⊗ N F : {− 1 , 1 } N → {− 1 , 1 } measurable increasing If ω ∈ {− 1 , 1 } N and i ∈ N , let ω i = ( ω 1 , · · · , ω i − 1 , − ω i , ω i +1 , · · · ) 3/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Sharp transitions for Boolean functions p ∈ [0 , 1], equip {− 1 , 1 } N with P p := ( p δ 1 + (1 − p ) δ − 1 ) ⊗ N F : {− 1 , 1 } N → {− 1 , 1 } measurable increasing If ω ∈ {− 1 , 1 } N and i ∈ N , let ω i = ( ω 1 , · · · , ω i − 1 , − ω i , ω i +1 , · · · ) Theorem (Kolmogorov 0-1 law) If for every i ∈ N and every ω ∈ {− 1 , 1 } N , F ( ω i ) = F ( ω ) , then p �→ P p [ F = 1] is a Heaviside function. P p [ F = 1] 1 p 0 1 3/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Sharp transitions for Boolean functions Equip the hypercube {− 1 , 1 } n with P p := ( p δ 1 + (1 − p ) δ − 1 ) ⊗ n F : {− 1 , 1 } n → {− 1 , 1 } increasing If ω ∈ {− 1 , 1 } n , let ω i := ( ω 1 , · · · , ω i − 1 , − ω i , ω i +1 , · · · , ω n ) The influence of i is I p � F ( ω i ) � = F ( ω ) � i ( F )= P p (Ben-Or–Linial, 90) 3/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Sharp transitions for Boolean functions Equip the hypercube {− 1 , 1 } n with P p := ( p δ 1 + (1 − p ) δ − 1 ) ⊗ n F : {− 1 , 1 } n → {− 1 , 1 } increasing If ω ∈ {− 1 , 1 } n , let ω i := ( ω 1 , · · · , ω i − 1 , − ω i , ω i +1 , · · · , ω n ) The influence of i is I p � F ( ω i ) � = F ( ω ) � i ( F )= P p (Ben-Or–Linial, 90) Theorem (Approximate 0-1 law, Russo, 82) p ∈ [0 , 1] I p i ( F ) ≤ δ For every ε > 0 , there exists δ > 0 s.t. if max 1 ≤ i ≤ n max then p �→ P p [ F = 1] is ε -close to a Heaviside function. P p [ F = 1] 1 p 0 1 3/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

F : {− 1 , 1 } n → {− 1 , 1 } increasing The influence of i is I p i ( F ) = P p [changing ω i modifies F ( ω )] Russo, 82: max i , p I p i ( F ) ≪ 1 ⇒ sharp transition 4/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

F : {− 1 , 1 } n → {− 1 , 1 } increasing The influence of i is I p i ( F ) = P p [changing ω i modifies F ( ω )] Russo, 82: max i , p I p i ( F ) ≪ 1 ⇒ sharp transition Some examples: max i,p I p P p [ F = 1] F i ( F ) Maj : ω �→ sng (Σ n i =1 ω i ) Dict : ω �→ ω 1 max ( Dict, Maj ) 4/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Russo, 82: max i , p I p i ( F ) ≪ 1 ⇒ sharp transition 5/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Russo, 82: max i , p I p i ( F ) ≪ 1 ⇒ sharp transition An application to Bernoulli percolation: T R := a triangulation resricted to a 2 R × R 2 R rectangle Let F R : {− 1 , 1 } T R → {− 1 , 1 } be the ± 1 indicator R function of the black crossing of this rectangle 1 = black ; − 1 = white 5/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Russo, 82: max i , p I p i ( F ) ≪ 1 ⇒ sharp transition An application to Bernoulli percolation: T R := a triangulation resricted to a 2 R × R 2 R rectangle Let F R : {− 1 , 1 } T R → {− 1 , 1 } be the ± 1 indicator R function of the black crossing of this rectangle 1 = black ; − 1 = white 5/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Russo, 82: max i , p I p i ( F ) ≪ 1 ⇒ sharp transition An application to Bernoulli percolation: T R := a triangulation resricted to a 2 R × R 2 R rectangle Let F R : {− 1 , 1 } T R → {− 1 , 1 } be the ± 1 indicator R function of the black crossing of this rectangle 1 = black ; − 1 = white 5/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Russo, 82: max i , p I p i ( F ) ≪ 1 ⇒ sharp transition An application to Bernoulli percolation: T R := a triangulation resricted to a 2 R × R 2 R rectangle Let F R : {− 1 , 1 } T R → {− 1 , 1 } be the ± 1 indicator R function of the black crossing of this rectangle 1 = black ; − 1 = white Theorem (Russo, 78; Seymour-Welsh, 78) i , p I p i ( F R ) ≤ R − α P 1 / 2 [ F R = 1] ∈ [ c , 1 − c ] ; max Theorem (Kesten, 80) The critical point of percolation on this lattice is 1 / 2 . 5/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

A quantitative result: the KKL theorem F : {− 1 , 1 } n → {− 1 , 1 } increasing Theorem ( Bourgain, Friedgut, Kahn, Kalai, Katznelson, Linial, Talagrand; 82 → 96) d dp P p [ F = 1] � � �� I p � � P p [ F = 1] (1 − P p [ F = 1]) ≥ c � log max i ( F ) � � i � 6/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation