Near-critical Ising model Christophe Garban ENS Lyon and CNRS 8th - PowerPoint PPT Presentation

Near-critical Ising model Christophe Garban ENS Lyon and CNRS 8th World Congress in Probability and Statistics Istanbul, July 2012 C. Garban (ENS Lyon and CNRS) Near-critical Ising model 1 / 19

Plan 1 Near-critical behavior, case of percolation ◮ Notion of correlation length L ( p ) C. Garban (ENS Lyon and CNRS) Near-critical Ising model 2 / 19

Plan 1 Near-critical behavior, case of percolation ◮ Notion of correlation length L ( p ) 2 Near-critical Ising model as the temperature varies ◮ Joint work with H. Duminil-Copin and Gábor Pete. C. Garban (ENS Lyon and CNRS) Near-critical Ising model 2 / 19

Plan 1 Near-critical behavior, case of percolation ◮ Notion of correlation length L ( p ) 2 Near-critical Ising model as the temperature varies ◮ Joint work with H. Duminil-Copin and Gábor Pete. 3 Near-critical Ising model as the external magnetic field varies ◮ Joint work with F. Camia and C. Newman. C. Garban (ENS Lyon and CNRS) Near-critical Ising model 2 / 19

Near criticality Consider your favorite statistical physics model, for example: ◮ percolation ◮ FK percolation ◮ Ising model etc ...

Near criticality Consider your favorite statistical physics model, for example: ◮ percolation ◮ FK percolation ◮ Ising model etc ... p = p c Critical T = T c δ Z 2

Near criticality Consider your favorite statistical physics model, for example: ◮ percolation ◮ FK percolation ◮ Ising model etc ... p = p c p < p c Critical Sub-critical T = T c T > T c δ Z 2 δ Z 2

Near criticality Consider your favorite statistical physics model, for example: ◮ percolation ◮ FK percolation ◮ Ising model etc ... p > p c p = p c p < p c Super-critical Critical Sub-critical T < T c T = T c T > T c δ Z 2 δ Z 2

Near criticality Consider your favorite statistical physics model, for example: ◮ percolation ◮ FK percolation ◮ Ising model etc ... Sub-critical δ Z 2 δ Z 2

Near criticality Consider your favorite statistical physics model, for example: ◮ percolation ◮ FK percolation ◮ Ising model etc ... T = T c and h > 0 T = T c and h = 0 Sub-critical δ Z 2 δ Z 2

Near criticality Consider your favorite statistical physics model, for example: ◮ percolation ◮ FK percolation ◮ Ising model etc ... T = T c and h > 0 T = T c and h = 0 Sub-critical δ Z 2 δ Z 2 What happens if T ≈ T c or h ≈ 0 ??

Notion of correlation length (informal) p = p c + δp

Notion of correlation length (informal) L ( p ) p = p c + δp

Notion of correlation length (informal) L ( p ) p = p c + δp 1 p − p c | ν + o (1) L ( p ) = |

Notion of correlation length (informal) L ( p ) p = p c + δp 1 p − p c | ν + o (1) L ( p ) = | Example (critical perco- lation): (Smirnov- Theorem Werner 2001): p − p c | 4 / 3+ o (1) 1 L ( p ) = |

The models we shall consider Percolation: FK Percolation (or random cluster model) P p ( ω ) = p o (1 − p ) c o = o ( ω ) = Nb of open ed c = c ( ω ) = Nb of closed ed

The models we shall consider Percolation: FK Percolation (or random cluster model) P p ( ω ) = p o (1 − p ) c o = o ( ω ) = Nb of open ed c = c ( ω ) = Nb of closed ed

The models we shall consider Percolation: FK Percolation (or random cluster model) P p ( ω ) = p o (1 − p ) c Fix a parameter q ≥ 1 o = o ( ω ) = Nb of open ed c = c ( ω ) = Nb of closed ed

The models we shall consider Percolation: FK Percolation (or random cluster model) P p ( ω ) = p o (1 − p ) c Fix a parameter q ≥ 1 o = o ( ω ) = Nb of open ed P q,p ( ω ) ∼ p o (1 − p ) c q ♯ clusters c = c ( ω ) = Nb of closed ed

The models we shall consider Percolation: FK Percolation (or random cluster model) P p ( ω ) = p o (1 − p ) c Fix a parameter q ≥ 1 o = o ( ω ) = Nb of open ed P q,p ( ω ) ∼ p o (1 − p ) c q ♯ clusters c = c ( ω ) = Nb of closed ed Theorem (Kesten 1980)

The models we shall consider Percolation: FK Percolation (or random cluster model) P p ( ω ) = p o (1 − p ) c Fix a parameter q ≥ 1 o = o ( ω ) = Nb of open ed P q,p ( ω ) ∼ p o (1 − p ) c q ♯ clusters c = c ( ω ) = Nb of closed ed Theorem (Kesten Theorem (Beffara, 1980) Duminil-Copin 2010) √ q p c ( Z 2 ) = 1 p c ( q ) = 1+ √ q 2

Notion of correlation length (precise definition) Definition Fix ρ > 0. For any n ≥ 0, let R n be the rectangle [ 0 , ρ n ] × [ 0 , n ] . If p > p c , then define for all ǫ > 0 and all “boundary conditions” ξ around R n , � � L ξ P ξ � � ρ,ǫ ( p ) := inf there is a left-right crossing in R n > 1 − ǫ p n > 0 C. Garban (ENS Lyon and CNRS) Near-critical Ising model 6 / 19

Estimating the correlation length, case of critical percolation R n : ρ n n

Estimating the correlation length, case of critical percolation R n : ρ n n p = p c + δp ω p c

Estimating the correlation length, case of critical percolation R n : ρ n n p = p c + δp ω p c ω p c + δp ≫ ω p c

Estimating the correlation length, case of critical percolation R n : ρ n n p = p c + δp ω p c Pivotal points ω p c + δp ≫ ω p c

Estimating the correlation length, case of critical percolation R n : ρ n n p = p c + δp ω p c Pivotal points ω p c + δp ≫ ω p c In critical percolation: ♯ ( Pivotal points ) ≈ n 2 α 4 ( n ) ≈ n 3 / 4

Estimating the correlation length, case of critical percolation R n : ρ n n p = p c + δp ω p c Pivotal points ω p c + δp ≫ ω p c One notices a change in the probability of In critical percolation: left-right crossing when: | p − p c | n 3 / 4 ≈ 1 ♯ ( Pivotal points ) ≈ n 2 α 4 ( n ) ≈ n 3 / 4

Estimating the correlation length, case of critical percolation R n : ρ n n p = p c + δp ω p c Pivotal points ω p c + δp ≫ ω p c One notices a change in the probability of In critical percolation: left-right crossing when: | p − p c | n 3 / 4 ≈ 1 ♯ ( Pivotal points ) ≈ n 2 α 4 ( n ) ≈ n 3 / 4 This suggests L ( p ) ≈ | p − p c | − 4 / 3

Estimating the correlation length, case of critical percolation R n : ρ n n p = p c + δp ω p c Pivotal points One notices a change in the probability of left-right crossing when: | p − p c | n 3 / 4 ≈ 1 Difficulty ! This suggests L ( p ) ≈ | p − p c | − 4 / 3

Sharp threshold To analyze the behavior of the correlation length, it is useful to rely on � � Russo’s formula: if φ n ( p ) := P p there is a left-right crossing in R n , then d � � dp φ n ( p ) = E p Number of pivotal points in ω p � � � = x is a pivotal point P p x ∈ R n This point of view also leads to the identity | p − p c | L ( p ) 2 α 4 ( L ( p )) ≍ 1 C. Garban (ENS Lyon and CNRS) Near-critical Ising model 8 / 19

What about the correlation length for FK-Ising percolation ? C. Garban (ENS Lyon and CNRS) Near-critical Ising model 9 / 19

What about the correlation length for FK-Ising percolation ? In a work in progress with H. Duminil-Copin, we establish that the number of pivotal points for FK percolation ( q = 2) in a square Λ L of diameter L is of order: L 13 / 24 This suggests that L ( p ) should scale like 1 p − p c ( 2 ) | 24 / 13 . L ( p ) ≈ | C. Garban (ENS Lyon and CNRS) Near-critical Ising model 9 / 19

What about the correlation length for FK-Ising percolation ? In a work in progress with H. Duminil-Copin, we establish that the number of pivotal points for FK percolation ( q = 2) in a square Λ L of diameter L is of order: L 13 / 24 This suggests that L ( p ) should scale like 1 p − p c ( 2 ) | 24 / 13 . L ( p ) ≈ | But this does not match with related results known since Onsager which p − p c | 24 / 13 !! 1 1 suggest that L ( p ) should instead scale like | p − p c | ≪ | So what is wrong here !? C. Garban (ENS Lyon and CNRS) Near-critical Ising model 9 / 19

Monotone couplings of FK percolation, q = 2 Grimmett constructed in 1995 a somewhat explicit monotone coupling of FK percolation configurations ( ω p ) p ∈ [ 0 , 1 ] . This monotone coupling differs in several essential ways from the standard monotone coupling ( q = 1): C. Garban (ENS Lyon and CNRS) Near-critical Ising model 10 / 19

Monotone couplings of FK percolation, q = 2 Grimmett constructed in 1995 a somewhat explicit monotone coupling of FK percolation configurations ( ω p ) p ∈ [ 0 , 1 ] . This monotone coupling differs in several essential ways from the standard monotone coupling ( q = 1): 1 The edge-intensity has a singularity near p c . C. Garban (ENS Lyon and CNRS) Near-critical Ising model 10 / 19

Monotone couplings of FK percolation, q = 2 Grimmett constructed in 1995 a somewhat explicit monotone coupling of FK percolation configurations ( ω p ) p ∈ [ 0 , 1 ] . This monotone coupling differs in several essential ways from the standard monotone coupling ( q = 1): 1 The edge-intensity has a singularity near p c . Yet, this is only a logarithmic singularity, namely d � � ≍ log | p − p c | − 1 . e is open dp P p C. Garban (ENS Lyon and CNRS) Near-critical Ising model 10 / 19