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, 19th june, 2018

1/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

(f (x))x∈R2 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∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) :=

2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

(f (x))x∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) :=

Theorem (Sharp phase transition) Beffara-Gayet, 16 (ℓ = 0); Rivera-V, 17 (ℓ = 0) P [Crossℓ(R)]          ≤ exp(−c(ℓ)R) if ℓ < 0 ∈

• 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∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) := 1 P [Crossℓ(R)] ℓ

2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

(f (x))x∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) :=

2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

(f (x))x∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) :=

2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

(f (x))x∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) :=

2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

(f (x))x∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) :=

2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

(f (x))x∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) :=

2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

(f (x))x∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) :=

2/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

(f (x))x∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(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∈R2 planar Bargmann-Fock field

f > −ℓ

R 2R

Crossℓ(R) := 1 P [Crossℓ(R)] 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 Pp := (pδ1 + (1 − p)δ−1)⊗N F : {−1, 1}N → {−1, 1} measurable increasing If ω ∈ {−1, 1}N and i ∈ N, let ωi = (ω1, · · · , ωi−1,−ωi, ωi+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 Pp := (pδ1 + (1 − p)δ−1)⊗N F : {−1, 1}N → {−1, 1} measurable increasing If ω ∈ {−1, 1}N and i ∈ N, let ωi = (ω1, · · · , ωi−1,−ωi, ωi+1, · · · ) Theorem (Kolmogorov 0-1 law) If for every i ∈ N and every ω ∈ {−1, 1}N, F(ωi) = F(ω), then p → Pp [F = 1] is a Heaviside function. p 1 1 Pp[F = 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 Pp := (pδ1 + (1 − p)δ−1)⊗n F : {−1, 1}n → {−1, 1} increasing If ω ∈ {−1, 1}n, let ωi := (ω1, · · · , ωi−1,−ωi, ωi+1, · · · , ωn) The influence of i is I p

i (F)=Pp

• F(ωi) = F(ω)
• (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 Pp := (pδ1 + (1 − p)δ−1)⊗n F : {−1, 1}n → {−1, 1} increasing If ω ∈ {−1, 1}n, let ωi := (ω1, · · · , ωi−1,−ωi, ωi+1, · · · , ωn) The influence of i is I p

i (F)=Pp

• F(ωi) = F(ω)
• (Ben-Or–Linial, 90)

Theorem (Approximate 0-1 law, Russo, 82) For every ε > 0, there exists δ > 0 s.t. if max

1≤i≤n max p∈[0,1]I p i (F) ≤ δ

then p → Pp [F = 1] is ε-close to a Heaviside function.

p 1 1 Pp[F = 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) = Pp [changing ωi modifies F(ω)]

Russo, 82: maxi,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) = Pp [changing ωi modifies F(ω)]

Russo, 82: maxi,p I p

i (F) ≪ 1 ⇒ sharp transition

Some examples:

Maj : ω → sng(Σn

i=1ωi)

Dict : ω → ω1 max(Dict, Maj ) F Pp[F = 1] maxi,p Ip

i (F )

4/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Russo, 82: maxi,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: maxi,p I p

i (F) ≪ 1 ⇒ sharp transition

An application to Bernoulli percolation:

2R R

TR := a triangulation resricted to a 2R × R rectangle Let FR : {−1, 1}TR → {−1, 1} be the ±1 indicator 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: maxi,p I p

i (F) ≪ 1 ⇒ sharp transition

An application to Bernoulli percolation:

2R R

TR := a triangulation resricted to a 2R × R rectangle Let FR : {−1, 1}TR → {−1, 1} be the ±1 indicator 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: maxi,p I p

i (F) ≪ 1 ⇒ sharp transition

An application to Bernoulli percolation:

2R R

TR := a triangulation resricted to a 2R × R rectangle Let FR : {−1, 1}TR → {−1, 1} be the ±1 indicator 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: maxi,p I p

i (F) ≪ 1 ⇒ sharp transition

An application to Bernoulli percolation:

2R R

TR := a triangulation resricted to a 2R × R rectangle Let FR : {−1, 1}TR → {−1, 1} be the ±1 indicator function of the black crossing of this rectangle 1 = black ; −1 = white Theorem (Russo, 78; Seymour-Welsh, 78) P1/2 [FR = 1] ∈ [c, 1 − c] ; max

i,p I p i (FR) ≤ R−α

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 dpPp [F = 1]

Pp [F = 1] (1 − Pp [F = 1]) ≥ c

• log
• max

i

I p

i (F)

• 6/9

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 dpPp [F = 1]

Pp [F = 1] (1 − Pp [F = 1]) ≥ c

• log
• max

i

I p

i (F)

• Gaussian case: ℓ ∈ R, K covariance matrix, X ∼ N
• (ℓ, · · · , ℓ), K
• Let µK

ℓ be the law of (sign(X1), · · · , sign(Xn)) ∈ {−1, 1}n

The influence of i is I ℓ,K

i

(F) = µK

ℓ [changing ωi modifies F(ω)]

6/9 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 dpPp [F = 1]

Pp [F = 1] (1 − Pp [F = 1]) ≥ c

• log
• max

i

I p

i (F)

• Gaussian case: ℓ ∈ R, K covariance matrix, X ∼ N
• (ℓ, · · · , ℓ), K
• Let µK

ℓ be the law of (sign(X1), · · · , sign(Xn)) ∈ {−1, 1}n

The influence of i is I ℓ,K

i

(F) = µK

ℓ [changing ωi modifies F(ω)]

Theorem (Keller–Mossel–Sen, 12; Cordero-Erausquin–Ledoux, 12; Rivera–V, 17)

d dℓµK ℓ [F = 1]

µK

ℓ [F = 1]

• 1 − µK

ℓ [F = 1]

≥ c

• log
• max

i

I ℓ,K

i

(F)

• 1/2

|| √ K||∞,op

6/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

• log
• max

i,ℓ I ℓ,K i

(F)

• 1/2

≫ || √ K||∞,op ⇒ sharp transition

7/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

• log
• max

i,ℓ I ℓ,K i

(F)

• 1/2

≫ || √ K||∞,op ⇒ sharp transition Now, two steps:

1 Application of this result to the Bargmann-Fock field

restricted to a lattice of small mesh ε: Estimates on the influences thanks to Beffara and Gayet. Problem: || √ K||∞,op blows up as ε ց 0

2 Appoximation estimates:

Natural idea: Prove that a discrete crossing at level ℓ implies a continuous crossing at level ℓ w.h.p. Problem: such results (see BG, BM) require ε to be very small Strategy: Prove that a discrete crosssing at level ℓ implies a continuous crossing at level 2ℓ w.h.p.: sprinkling procedure

7/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

• log
• max

i,ℓ I ℓ,K i

(F)

• 1/2

≫ || √ K||∞,op ⇒ sharp transition Discrete Bargmann-Fock percolation:

R ε 2R

Let X ε

R be the Bargmann-Fock field restricted to a

triangulation T ε

R of mesh ε in a 2R × R rectangle

Let F ε

R : {−1, 1}T ε

R → {−1, 1} be the ±1 indicator a

function of the black crossing of the rectangle K ε

R := covariance matrix of X ε R

7/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

• log
• max

i,ℓ I ℓ,K i

(F)

• 1/2

≫ || √ K||∞,op ⇒ sharp transition Discrete Bargmann-Fock percolation:

R ε 2R

Let X ε

R be the Bargmann-Fock field restricted to a

triangulation T ε

R of mesh ε in a 2R × R rectangle

Let F ε

R : {−1, 1}T ε

R → {−1, 1} be the ±1 indicator a

function of the black crossing of the rectangle K ε

R := covariance matrix of X ε R

Proposition (Sharp transition for the discrete Bargmann-Fock field) Beffara-Gayet, 16; Tassion, 15: F ε

R is non-degenerate at level

ℓ = 0; the maximum of influences is less than R−α . Rivera-V, 17: ||K ε

R||∞,op ≃ 1/ε

Therefore, there is a sharp transition at ℓ = 0 if log1/2(R) ≫ 1/ε .

7/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Sharp transition for discrete Bargmann-Fock percolaton if: log1/2(R) ≫ 1/ε Theorem (Beffara-Gayet, 16; Beliaev-Muirhead, 17) If 1/ε ≫ R then the discrete crossings approximate well the continuous crossings. Proposition (Sprinkling procedure, Rivera-V, 17) If 1/ε ≫ log1/4(R) then: P

• {discrete crossing at level ℓ} \ {continuous crossing at level 2ℓ}
• is very small.

8/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation

Thank you!

9/9 Hugo Vanneuville (ICJ, Universit´ e Lyon 1) Sharp phase transition for Bargmann-Fock percolation