Critical percolation under conservative dynamics

Transcription

1 Critical percolation under conservative dynamics Christophe Garban ENS Lyon and CNRS Joint work with and Erik Broman (Uppsala University) Jeffrey E. Steif (Chalmers University, Göteborg) PASI conference, Buenos Aires, January 2012 C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 1 / 21

2 Overview Dynamical percolation Conservative dynamics on percolation C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 2 / 21

3 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

4 Main results known Theorem (Schramm, Steif, 2005) On the triangular lattice T, there exist exceptional times t for which 0 ωt. Furthermore, a.s. [ 1 dim H (Exc) 6, 31 ] 36 C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 4 / 21

5 Main results (continued) Theorem (G., Pete, Schramm, 2008) On the square lattice Z 2, there are exceptional times as well (with dim H (Exc) ɛ > 0 a.s.)

6 Main results (continued) Theorem (G., Pete, Schramm, 2008) On the square lattice Z 2, there are exceptional times as well (with dim H (Exc) ɛ > 0 a.s.) On the triangular lattice T a.s. dim H (Exc) = There exist exceptional times Exc (2) such that 0

7 Strategy: noise sensitivity of percolation t n ω t ω t+ɛ

8 Strategy: noise sensitivity of percolation

9 Large scale properties are encoded by Boolean functions of the inputs b n Let f n : { 1, 1} O(1)n2 {0, 1} be the Boolean function defined as follows a n { 1 if left-right crossing f n (ω) := 0 else C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 8 / 21

10 Large scale properties are encoded by Boolean functions of the inputs b n Let f n : { 1, 1} O(1)n2 {0, 1} be the Boolean function defined as follows a n { 1 if left-right crossing f n (ω) := 0 else Theorem (Benjamini, Kalai, Schramm, 1998) For any fixed t > 0: [ ] Cov f n (ω 0 ), f n (ω t ) 0 n 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

11 Main tool to study noise sensitivity: Fourier analysis Decompose f : { 1, 1} m {0, 1} into Fourier series f (ω) = S ˆf (S) χ S (ω), where χ S (x 1,..., x m ) := i S x i.

12 Main tool to study noise sensitivity: Fourier analysis Decompose f : { 1, 1} m {0, 1} into Fourier series f (ω) = S ˆf (S) χ S (ω), where χ S (x 1,..., x m ) := i S x i. E [ f (ω 0 ) f (ω t ) ] = E [( )( ) ] f (S1 )χ S1 (ω 0 ) f (S2 )χ S2 (ω t ) S 1 S 2 = f (S) 2 E [ χ S (ω 0 )χ S (ω t ) ] S = S f (S) 2 t S e Thus the covariance can be written: E [ f (ω 0 ) f (ω t ) ] E [ f (ω) ] 2 = f (S) 2 t S e S

13 Fourier spectrum of critical percolation b n Let f n, n 1 be Boolean functions defined above. 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

14 Fourier spectrum of critical percolation b n Let f n, n 1 be Boolean functions defined above. 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

15 Percolation undergoing conservative dynamics

16 Percolation undergoing conservative dynamics

17 Percolation undergoing conservative dynamics

18 Percolation undergoing conservative dynamics

19 Percolation undergoing conservative dynamics

20 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: (b) Medium-range dynamics: P(x, y) P(x, y) = 1 degree 1 x y 1 x y 2+α for some exponent α > 0 C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 12 / 21

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.

22 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).

23 In other words, for medium-range exclusion dynamics (α > 0), we also obtain this picture t n ω t ω t+ɛ

24 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

25 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

26 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

27 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

28 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. i.i.d. basis: f n : { 1, 1} n2 {0, 1} (χ S ) S [m] C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 16 / 21

29 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. f n : { 1, 1} n2 {0, 1} 2. In the infinite volume case, L P is of course non-compact and it seems that it does not have pure-point spectrum. i.i.d. basis: exclusion basis: (χ S ) S [m] C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 16 / 21

30 The key identity We decompose f on the classical i.i.d. basis even though it does not diagonalize our exclusion process: E [ f (ω P 0 ) f (ω P t ) ] = E [( S )( ) ] f (S)χS (ω0 P ) f (S )χ S (ωt P ) S C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 17 / 21

31 The key identity We decompose f on the classical i.i.d. basis even though it does not diagonalize our exclusion process: E [ f (ω0 P ) f (ωt P ) ] = E [( )( ) ] f (S)χS (ω0 P ) f (S )χ S (ωt P ) S S = f (S) f (S ) E [ χ S (ω0 P )χ S (ωt P ) ] S,S C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 17 / 21

32 The key identity We decompose f on the classical i.i.d. basis even though it does not diagonalize our exclusion process: E [ f (ω0 P ) f (ωt P ) ] = E [( )( ) ] f (S)χS (ω0 P ) f (S )χ S (ωt P ) S S = f (S) f (S ) E [ χ S (ω0 P )χ S (ωt P ) ] S,S = S = S f (S) f (S ) E [ χ S (ω P 0 )χ S (ω P t ) ] C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 17 / 21

33 The key identity We decompose f on the classical i.i.d. basis even though it does not diagonalize our exclusion process: E [ f (ω0 P ) f (ωt P ) ] = E [( )( ) ] f (S)χS (ω0 P ) f (S )χ S (ωt P ) S S = f (S) f (S ) E [ χ S (ω0 P )χ S (ωt P ) ] S,S = S = S = S = S f (S) f (S ) E [ χ S (ω P 0 )χ S (ω P t ) ] f (S) f (S ) P t (S, S ) where P t (S, S ) is the probability that the set S travels in time t towards the set S under the exclusion process. C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 17 / 21

34 E [ f n (ω P 0 ) f n (ω P t ) ] = = fn (S) f n (S ) P t (S, S ) S = S ˆfn, P t ˆf n We would like to prove that for large scale n, the vectors {ˆf n (S)} S and {P t ˆf n (S)} S are almost orthogonal. C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 18 / 21

35 E [ f n (ω P 0 ) f n (ω P t ) ] = = fn (S) f n (S ) P t (S, S ) S = S ˆfn, P t ˆf n We would like to prove that for large scale n, the vectors {ˆf n (S)} S and {P t ˆf n (S)} S are almost orthogonal. Unfortunately, we know much more on the vector {ˆf n (S) 2 } than on {ˆf n (S)}: C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 18 / 21

36 The spectral measure ν fn Definition Recall that f n is the Boolean function: Define the spectral measure of f n as follows: b n a n ν fn (S = S) := ˆf n (S) 2 In particular S can be considered as a random subset of [0, an] [0, bn]. C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 19 / 21

37 The spectral measure ν fn Definition Recall that f n is the Boolean function: Define the spectral measure of f n as follows: b n a n ν fn (S = S) := ˆf n (S) 2 In particular S can be considered as a random subset of [0, an] [0, bn]. We can prove the following: Proposition (asymptotic singularity) For any medium-range exponent α > 0 and any fixed t > 0: as n, the measures ν fn and P t ν fn are asymptotically mutually singular C. Garban (ENS Lyon and CNRS) Critical percolation under conservative dynamics 19 / 21

38 Why does this imply noise sensitivity? Fact If φ 2 and ψ 2 are the densities of two probability measures on R, then φ ψ 2 φ 2 ψ 2 In particular, if the two corresponding probability measures are almost singular with respect to each other then φψ is small.

39 Fact Why does this imply noise sensitivity? If φ 2 and ψ 2 are the densities of two probability measures on R, then φ ψ 2 φ 2 ψ 2 In particular, if the two corresponding probability measures are almost singular with respect to each other then φψ is small. Take { φ 2 ˆf n (S) 2 ψ 2 P t [ (ˆf n ) 2 ](S) ] ˆf n (S) 2, P t [(ˆf n ) 2 (S) this gives that is small. By Cauchy-Schwartz, one concludes that ˆfn (S), P t ˆf n (S) is small.

40 Singularity in the medium-range case (α > 0) (Recall P(x, y) 1 x y 2+α ) S fn ν fn S fn n 3/4 (GPS 2008) n

41 Singularity in the medium-range case (α > 0) (Recall P(x, y) 1 x y 2+α ) If the exponent α is small: S fn ν fn S fn n 3/4 (GPS 2008) n

42 Singularity in the medium-range case (α > 0) (Recall P(x, y) 1 x y 2+α ) If the exponent α is large... S fn ν fn S fn n 3/4 (GPS 2008) n Question What about the nearest-neighbor case?