Critical percolation under conservative dynamics Christophe Garban ENS Lyon and CNRS Erik Broman (Uppsala University) Joint work with Jeffrey E. Steif (Chalmers University, Göteborg) and PASI conference, Buenos Aires, January 2012 C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 1 / 21
Overview • Dynamical percolation • Conservative dynamics on percolation C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 2 / 21
“standard” dynamical percolation Start with an initial configuration ω t = 0 at p = p c ( T ) = p c ( Z 2 ) = 1 / 2. And let evolve each edge (or site) independently at rate 1. This gives a Markov process ( ω t ) t ≥ 0 on critical percolation configurations. C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 3 / 21
Main results known Theorem (Schramm, Steif, 2005) On the triangular lattice T , there exist exceptional times t for which ω t 0 ← → ∞ . Furthermore, a.s. � 1 � 6 , 31 dim H ( Exc ) ∈ 36 C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 4 / 21
Main results (continued) Theorem (G. , Pete, Schramm, 2008) • On the square lattice Z 2 , there are exceptional times as well (with dim H ( Exc ) ≥ ǫ > 0 a.s.)
Main results (continued) Theorem (G. , Pete, Schramm, 2008) • On the square lattice Z 2 , there are exceptional times as well (with dim H ( Exc ) ≥ ǫ > 0 a.s.) • On the triangular lattice T • a.s. dim H ( Exc ) = 31 36 • There exist exceptional times Exc ( 2 ) such that 0
Strategy: noise sensitivity of percolation t n ω t + ǫ ω t
Strategy: noise sensitivity of percolation
Large scale properties are encoded by Boolean functions of the ‘inputs’ Let f n : {− 1 , 1 } O ( 1 ) n 2 → { 0 , 1 } be the Boolean function defined as follows b · n � 1 if left-right crossing f n ( ω ) := 0 else a · n C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 8 / 21
Large scale properties are encoded by Boolean functions of the ‘inputs’ Let f n : {− 1 , 1 } O ( 1 ) n 2 → { 0 , 1 } be the Boolean function defined as follows b · n � 1 if left-right crossing f n ( ω ) := 0 else a · n Theorem (Benjamini, Kalai, Schramm, 1998) For any fixed t > 0 : � � f n ( ω 0 ) , f n ( ω t ) n →∞ 0 − → Cov We say in such a case that ( f n ) n ≥ 1 is noise sensitive . C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 8 / 21
Main tool to study noise sensitivity: Fourier analysis Decompose f : {− 1 , 1 } m → { 0 , 1 } into “Fourier” series � ˆ f ( ω ) = f ( S ) χ S ( ω ) , S where χ S ( x 1 , . . . , x m ) := � i ∈ S x i .
Main tool to study noise sensitivity: Fourier analysis Decompose f : {− 1 , 1 } m → { 0 , 1 } into “Fourier” series � ˆ f ( ω ) = f ( S ) χ S ( ω ) , S where χ S ( x 1 , . . . , x m ) := � i ∈ S x i . ��� ��� �� � � � � f ( ω 0 ) f ( ω t ) = f ( S 1 ) χ S 1 ( ω 0 ) f ( S 2 ) χ S 2 ( ω t ) E E S 1 S 2 � � � f ( S ) 2 E � = χ S ( ω 0 ) χ S ( ω t ) S � f ( S ) 2 e − t | S | � = S Thus the covariance can be written: � � � � � 2 = f ( S ) 2 e − t | S | � E f ( ω 0 ) f ( ω t ) − E f ( ω ) S � = ∅
Fourier spectrum of critical percolation Let f n , n ≥ 1 be Boolean functions defined above. b · n One is interested in the shape of their Fourier spectrum. a · n � | S | = k � f n ( S ) 2 ? k . . . . . . C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 10 / 21
Fourier spectrum of critical percolation Let f n , n ≥ 1 be Boolean functions defined above. b · n One is interested in the shape of their Fourier spectrum. a · n � | S | = k � f n ( S ) 2 ? At which speed does the Spectral mass “spread” as the scale n goes to infinity ? k . . . . . . C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 10 / 21
Percolation undergoing conservative dynamics
Percolation undergoing conservative dynamics
Percolation undergoing conservative dynamics
Percolation undergoing conservative dynamics
Percolation undergoing conservative dynamics
The system evolves according to the symmetric exclusion process Let ( ω P t ) t ≥ 0 be a sample of a symmetric exclusion process with symmetric kernel P ( x , y ) , ( x , y ) ∈ Z 2 × Z 2 or ( x , y ) ∈ T × T We distinguish 2 cases: (a) Nearest neighbor dynamics: 1 P ( x , y ) = degree 1 x ∼ y (b) Medium-range dynamics: 1 P ( x , y ) ≍ for some exponent α > 0 � x − y � 2 + α C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 12 / 21
What we can and cannot :-( prove about these dynamics 1. Let’s start with the bad news: we don’t know if there are exceptional times for ω P t .
What we can and cannot :-( prove about these dynamics 1. Let’s start with the bad news: we don’t know if there are exceptional times for ω P t . 2. If the dynamics is medium-range with exponent α > 0 (recall P ( x , y ) ≍ � x − y � − 2 − α ), then we get quantitative bounds on the noise sensitivity of the crossing events f n under ω P t . More precisely: Theorem (Broman, G., Steif, 2011) If P is any transition kernel with exponent α > 0 , then on Z 2 , site , Z 2 , bond or T , at the critical point, one has Cov ( f n ( ω P 0 ) , f n ( ω P t )) − n →∞ 0 → Furthermore, one can choose t = t n ≥ n − β ( α ) .
In other words, for medium-range exclusion dynamics ( α > 0), we also obtain this “picture” t n ω t + ǫ ω t
Which approach for this problem ? Two strategies: 1. Either the noise sensitivity results for the iid case transfer to these conservative dynamics ? 2. Or an “appropriate” spectral approach ? C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 15 / 21
Which approach for this problem ? Two strategies: 1. Either the noise sensitivity results for the iid case transfer to these conservative dynamics ? 2. Or an “appropriate” spectral approach ? strategy 1. is “hopeless” since Fact There exist Boolean functions ( f n ) n which are highly noise sensitive to i.i.d. noise but which are stable to symmetric exclusion P- dynamics. C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 15 / 21
What about the spectral approach ? Natural attempt: decompose our Boolean function f on a basis of eigenvectors which diagonalize the generator L = L P of our P -exclusion process. C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 16 / 21
What about the spectral approach ? Natural attempt: decompose our Boolean function f on a basis of eigenvectors which diagonalize the generator L = L P of our P -exclusion process. But there are difficulties: 1. In the finite volume case, such a basis obviously exists, but it highly depends on P and it is not very “explicit”. 2. In the infinite volume case, L P is of course non-compact and it seems that it does not have pure-point spectrum. C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 16 / 21
What about the spectral approach ? Natural attempt: decompose our Boolean function f on a basis of eigenvectors which diagonalize the generator L = L P of our P -exclusion process. But there are difficulties: f n : {− 1 , 1 } n 2 → { 0 , 1 } 1. In the finite volume case, such a basis obviously exists, but it highly depends on P and it is not very “explicit”. 2. In the infinite volume case, L P i.i.d. basis: i.i.d. basis: is of course non-compact and it seems that it does not have pure-point spectrum. ( χ S ) S ⊂ [ m ] C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 16 / 21
What about the spectral approach ? Natural attempt: decompose our Boolean function f on a basis of eigenvectors which diagonalize the generator L = L P of our P -exclusion process. But there are difficulties: f n : {− 1 , 1 } n 2 → { 0 , 1 } 1. In the finite volume case, such a basis obviously exists, but it highly depends on P and it is not very “explicit”. 2. In the infinite volume case, L P i.i.d. basis: “exclusion” is of course non-compact and it basis: seems that it does not have pure-point spectrum. ( χ S ) S ⊂ [ m ] C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 16 / 21
The key identity We decompose f on the classical “i.i.d.” basis even though it does not diagonalize our exclusion process: ��� ��� �� � � � � f ( ω P 0 ) f ( ω P f ( S ) χ S ( ω P f ( S ′ ) χ S ′ ( ω P t ) = 0 ) t ) E E S S ′ C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 17 / 21
The key identity We decompose f on the classical “i.i.d.” basis even though it does not diagonalize our exclusion process: ��� ��� �� � � � � f ( ω P 0 ) f ( ω P f ( S ) χ S ( ω P f ( S ′ ) χ S ′ ( ω P t ) = 0 ) t ) E E S S ′ � � � � f ( S ) � f ( S ′ ) E χ S ( ω P 0 ) χ S ′ ( ω P = t ) S , S ′ C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 17 / 21
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