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 ...

Critical δZ2 p = pc T = Tc

Near criticality

Consider your favorite statistical physics model, for example:

◮ percolation ◮ FK percolation ◮ Ising model etc ...

Critical δZ2 Sub-critical δZ2 p < pc T > Tc p = pc T = Tc

Near criticality

Consider your favorite statistical physics model, for example:

◮ percolation ◮ FK percolation ◮ Ising model etc ...

Super-critical Critical δZ2 Sub-critical δZ2 p > pc T < Tc p < pc T > Tc p = pc T = Tc

Near criticality

Consider your favorite statistical physics model, for example:

◮ percolation ◮ FK percolation ◮ Ising model etc ...

δZ2 Sub-critical δZ2

Near criticality

Consider your favorite statistical physics model, for example:

◮ percolation ◮ FK percolation ◮ Ising model etc ...

δZ2 Sub-critical δZ2 T = Tc and h = 0 T = Tc and h > 0

Near criticality

Consider your favorite statistical physics model, for example:

◮ percolation ◮ FK percolation ◮ Ising model etc ...

δZ2 Sub-critical δZ2 T = Tc and h = 0 T = Tc and h > 0

What happens if T ≈ Tc or h ≈ 0 ??

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

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

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

1 p−pc|ν+o(1)

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

1 p−pc|ν+o(1)

Example (critical perco- lation):

L(p) = |

1 p−pc|4/3+o(1)

Theorem (Smirnov- Werner 2001):

The models we shall consider Percolation: Pp(ω) = po (1 − p)c FK Percolation (or random cluster model)

• = o(ω) = Nb of open ed

c = c(ω) = Nb of closed ed

The models we shall consider Percolation: Pp(ω) = po (1 − p)c FK Percolation (or random cluster model)

• = o(ω) = Nb of open ed

c = c(ω) = Nb of closed ed

The models we shall consider Percolation: Pp(ω) = po (1 − p)c FK Percolation (or random cluster model) Fix a parameter q ≥ 1

• = o(ω) = Nb of open ed

c = c(ω) = Nb of closed ed

The models we shall consider Percolation: Pp(ω) = po (1 − p)c FK Percolation (or random cluster model) Fix a parameter q ≥ 1 Pq,p(ω) ∼ po(1 − p)c q♯clusters

• = o(ω) = Nb of open ed

c = c(ω) = Nb of closed ed

The models we shall consider Percolation: Pp(ω) = po (1 − p)c FK Percolation (or random cluster model) Fix a parameter q ≥ 1 Pq,p(ω) ∼ po(1 − p)c q♯clusters

• = o(ω) = Nb of open ed

c = c(ω) = Nb of closed ed

Theorem (Kesten 1980)

The models we shall consider Percolation: Pp(ω) = po (1 − p)c FK Percolation (or random cluster model) Fix a parameter q ≥ 1 Pq,p(ω) ∼ po(1 − p)c q♯clusters

• = o(ω) = Nb of open ed

c = c(ω) = Nb of closed ed

Theorem (Kesten 1980) Theorem (Beffara, Duminil-Copin 2010)

pc(Z2) = 1

2

pc(q) =

√q 1+√q

Notion of correlation length (precise definition)

Definition

Fix ρ > 0. For any n ≥ 0, let Rn be the rectangle [0, ρn] × [0, n]. If p > pc, then define for all ǫ > 0 and all “boundary conditions” ξ around Rn, Lξ

ρ,ǫ(p) := inf n>0

• P ξ

p

• there is a left-right crossing in Rn
• > 1 − ǫ
• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 6 / 19

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

Estimating the correlation length, case of critical percolation n ρ n Rn : p = pc + δp ωpc

Estimating the correlation length, case of critical percolation n ρ n Rn : p = pc + δp ωpc ωpc+δp ≫ ωpc

Estimating the correlation length, case of critical percolation n ρ n Rn : p = pc + δp ωpc ωpc+δp ≫ ωpc Pivotal points

Estimating the correlation length, case of critical percolation n ρ n Rn : p = pc + δp ωpc ωpc+δp ≫ ωpc

In critical percolation: ♯(Pivotal points) ≈ n2 α4(n) ≈ n3/4

Pivotal points

Estimating the correlation length, case of critical percolation n ρ n Rn : p = pc + δp ωpc ωpc+δp ≫ ωpc

In critical percolation: ♯(Pivotal points) ≈ n2 α4(n) ≈ n3/4 One notices a change in the probability of left-right crossing when: |p − pc| n3/4 ≈ 1

Pivotal points

Estimating the correlation length, case of critical percolation n ρ n Rn : p = pc + δp ωpc ωpc+δp ≫ ωpc

In critical percolation: ♯(Pivotal points) ≈ n2 α4(n) ≈ n3/4 One notices a change in the probability of left-right crossing when: |p − pc| n3/4 ≈ 1

This suggests L(p) ≈ |p − pc|−4/3 Pivotal points

Estimating the correlation length, case of critical percolation n ρ n Rn : p = pc + δp ωpc

One notices a change in the probability of left-right crossing when: |p − pc| n3/4 ≈ 1

This suggests L(p) ≈ |p − pc|−4/3 Difficulty ! Pivotal points

Sharp threshold

To analyze the behavior of the correlation length, it is useful to rely on Russo’s formula: if φn(p) := Pp

• there is a left-right crossing in Rn
• , then

d dpφn(p) = Ep

• Number of pivotal points in ωp
• =
• x∈Rn

Pp

• x is a pivotal point
• This point of view also leads to the identity

|p − pc| L(p)2α4(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

• f pivotal points for FK percolation (q = 2) in a square ΛL of diameter L is
• f order:

L13/24 This suggests that L(p) should scale like L(p) ≈ | 1 p − pc(2)|24/13 .

• 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

• f pivotal points for FK percolation (q = 2) in a square ΛL of diameter L is
• f order:

L13/24 This suggests that L(p) should scale like L(p) ≈ | 1 p − pc(2)|24/13 . But this does not match with related results known since Onsager which suggest that L(p) should instead scale like |

1 p−pc | ≪ | 1 p−pc |24/13 !!

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 (ωp)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 (ωp)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 pc.

• 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 (ωp)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 pc.

Yet, this is only a logarithmic singularity, namely

d dpPp

• e is open
• ≍ log |p − pc|−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 (ωp)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 pc.

Yet, this is only a logarithmic singularity, namely

d dpPp

• e is open
• ≍ log |p − pc|−1.

2 As p increases, one can prove that “clouds” of several edges appear

simultaneously !

• 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 (ωp)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 pc.

Yet, this is only a logarithmic singularity, namely

d dpPp

• e is open
• ≍ log |p − pc|−1.

2 As p increases, one can prove that “clouds” of several edges appear

simultaneously !

3 The location of these clouds of edges highly depend on the current

configuration ωp (→ hint of an interesting self-organized mechanism).

• 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 (ωp)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 pc.

Yet, this is only a logarithmic singularity, namely

d dpPp

• e is open
• ≍ log |p − pc|−1.

2 As p increases, one can prove that “clouds” of several edges appear

simultaneously !

3 The location of these clouds of edges highly depend on the current

configuration ωp (→ hint of an interesting self-organized mechanism). Most remains unknown regarding the structure of these random clouds.

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 10 / 19

What we can prove

Theorem (Duminil-Copin, G., Pete, 2011)

Fix q = 2. For every ǫ, ρ > 0, there is a constant c = c(ǫ, ρ) > 0 s.t.

c 1 |p − pc| ≤ Lξ

ρ,ǫ(p) ≤ c−1

1 |p − pc|

• log

1 |p − pc|

for all p = pc, whatever the choice of the boundary condition ξ is.

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 11 / 19

Techniques behind the proof: Smirnov’s observable

a b

Techniques behind the proof: Smirnov’s observable

a b a b e γ

Techniques behind the proof: Smirnov’s observable

ea eb a b

wired arc free arc

a b

wired arc free arc γ

Techniques behind the proof: Smirnov’s observable

a b a b

Fp(e) := Ep,2

• e

i 2Wγ(e,b) 1e∈γ

• e

γ

Techniques behind the proof: Smirnov’s observable

a b a b

Fp(e) := Ep,2

• e

i 2Wγ(e,b) 1e∈γ

• e

γ

Techniques behind the proof: Smirnov’s observable

a b a b

Fp(e) := Ep,2

• e

i 2Wγ(e,b) 1e∈γ

• e

|Fp(e)| = Pp,2(e ∈ γ)

“Near-harmonicity” of Smirnov’s observable

Theorem (Smirnov, exact harmonicity at criticality)

For q = 2 and p = pc(2) = √ 2/(1 + √ 2), once restricted to a proper sub-lattice (NE pointing edges), the observable Fpc is harmonic: ∆Fpc(eX) = 0

X

E S W N

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 13 / 19

“Near-harmonicity” of Smirnov’s observable

Theorem (Smirnov, exact harmonicity at criticality)

For q = 2 and p = pc(2) = √ 2/(1 + √ 2), once restricted to a proper sub-lattice (NE pointing edges), the observable Fpc is harmonic: ∆Fpc(eX) = 0

X

E S W N

Theorem (Beffara, Duminil-Copin)

When p < pc, the observable Fp is now massive harmonic: namely ∆Fp(eX) = m(p) Fp(eX) , where the mass m(p) ≍ |p − pc|2.

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 13 / 19

Upper-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find a scale n so that the rectangle Rn is crossed horizontally with high probability (> 1 − ǫ).

Rn : ωp

Upper-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find a scale n so that the rectangle Rn is crossed horizontally with high probability (> 1 − ǫ).

Rn :

We want this probability under Pp to be > 1 − ǫ

ωp

Upper-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find a scale n so that the rectangle Rn is crossed horizontally with high probability (> 1 − ǫ). We want this probability under Pp to be > 1 − ǫ By duality, this is the same as having a VERTICAL crossing for the dual FK configuration (under Pp∗) with probability < ǫ

ωp∗ Rn :

p∗ < pc < p and |p − pc| ≍ |p∗ − pc|

Upper-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find a scale n so that the rectangle Rn is crossed horizontally with high probability (> 1 − ǫ). We want this probability under Pp to be > 1 − ǫ By duality, this is the same as having a VERTICAL crossing for the dual FK configuration (under Pp∗) with probability < ǫ

ωp∗ Rn :

p∗ < pc < p and |p − pc| ≍ |p∗ − pc|

Upper-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find a scale n so that the rectangle Rn is crossed horizontally with high probability (> 1 − ǫ). We want this probability under Pp to be > 1 − ǫ By duality, this is the same as having a VERTICAL crossing for the dual FK configuration (under Pp∗) with probability < ǫ

ωp∗ Rn : e |Fp∗(e)| ∆Fp∗(e) = m(p∗)Fp∗(e)

p∗ < pc < p and |p − pc| ≍ |p∗ − pc|

Upper-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find a scale n so that the rectangle Rn is crossed horizontally with high probability (> 1 − ǫ).

e |Fp∗(e)| ∆Fp∗(e) = m(p∗)Fp∗(e) Random Walk interpretation: at each step in the bulk, the mass

• f the particle is divided by 1+m

p∗ < pc < p and |p − pc| ≍ |p∗ − pc|

Upper-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find a scale n so that the rectangle Rn is crossed horizontally with high probability (> 1 − ǫ).

e |Fp∗(e)| ∆Fp∗(e) = m(p∗)Fp∗(e) Random Walk interpretation: at each step in the bulk, the mass

• f the particle is divided by 1+m

Recall m(p∗) ≍ |p∗ − pc|2 ≍ |p − pc|2

n

If scale n ≫ |p − pc|−1, then the RW does more than |p − pc|−2 steps and thus, in average its mass goes to zero. p∗ < pc < p and |p − pc| ≍ |p∗ − pc|

Lower-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find scales n so that the rectangle Rn is NOT crossed horizontally with high probability (i.e. with prob < 1 − ǫ).

Rn : ωp

Lower-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find scales n so that the rectangle Rn is NOT crossed horizontally with high probability (i.e. with prob < 1 − ǫ).

Rn :

We want this probability under Pp to be < 1 − ǫ

ωp

Lower-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find scales n so that the rectangle Rn is NOT crossed horizontally with high probability (i.e. with prob < 1 − ǫ).

We want this probability under Pp to be < 1 − ǫ By duality, this is the same as having a VERTICAL crossing for the dual FK configuration (under Pp∗) with probability > ǫ

ωp∗ Rn :

p∗ < pc < p and |p − pc| ≍ |p∗ − pc|

Lower-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find scales n so that the rectangle Rn is NOT crossed horizontally with high probability (i.e. with prob < 1 − ǫ).

We want this probability under Pp to be < 1 − ǫ By duality, this is the same as having a VERTICAL crossing for the dual FK configuration (under Pp∗) with probability > ǫ

ωp∗ Rn :

p∗ < pc < p and |p − pc| ≍ |p∗ − pc|

Lower-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find scales n so that the rectangle Rn is NOT crossed horizontally with high probability (i.e. with prob < 1 − ǫ).

p∗ < pc < p and |p − pc| ≍ |p∗ − pc| Introduce N :=Nb of lower points connected to the top

Lower-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find scales n so that the rectangle Rn is NOT crossed horizontally with high probability (i.e. with prob < 1 − ǫ).

p∗ < pc < p and |p − pc| ≍ |p∗ − pc| Introduce N :=Nb of lower points connected to the top Using the RW interpretation, show that Ep∗[N] > c √n

Lower-bound on the correlation length

Fix ρ, ǫ > 0. For any p > pc, we want to find scales n so that the rectangle Rn is NOT crossed horizontally with high probability (i.e. with prob < 1 − ǫ).

p∗ < pc < p and |p − pc| ≍ |p∗ − pc| Introduce N :=Nb of lower points connected to the top Using the RW interpretation, show that Ep∗[N] > c √n Using the RSW proof at pc from Duminil-Copin, Hongler, Nolin, Ep∗(N 2) ≤ Epc(N 2) < c−1n

Ising model

To each configuration σ ∈ {−1, 1}N2, one associates the Hamiltonian

Hh(σ) := −

i∼j σiσj − h σi

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 16 / 19

Ising model

To each configuration σ ∈ {−1, 1}N2, one associates the Hamiltonian

Hh(σ) := −

i∼j σiσj − h σi

And we define:

Pβ,h(σ) ∝ e−β Hh(σ)

N N

+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 16 / 19

Ising model

To each configuration σ ∈ {−1, 1}N2, one associates the Hamiltonian

Hh(σ) := −

i∼j σiσj − h σi

And we define:

Pβ,h(σ) ∝ e−β Hh(σ)

N N

+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +

The Ising model is intimately related with FK percolation (q = 2) via the following identity: If h = 0,

Eβ(σxσy) = Pp,q=2[x ↔ y] with 1 − p = e−2β

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 16 / 19

Classical near-critical results

Theorem (Kesten - Smirnov/Werner)

For site percolation on the triangular lattice, θ(p) := P

• 0 ↔ ∞
• = |p − pc|5/36+o(1)

as p ց pc

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 17 / 19

Classical near-critical results

Theorem (Kesten - Smirnov/Werner)

For site percolation on the triangular lattice, θ(p) := P

• 0 ↔ ∞
• = |p − pc|5/36+o(1)

as p ց pc

Theorem (Onsager, 1944)

For Ising model on Z2: σ0+

β ≍ |β − βc|1/8

as β ց βc

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 17 / 19

Average magnetization under small external field

Theorem (Camia, G., Newman)

Consider Ising model on Z2 at βc with a positive external magnetic field h > 0, then σ0βc,h ≍ h

1 15

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 18 / 19

Average magnetization under small external field

Theorem (Camia, G., Newman)

Consider Ising model on Z2 at βc with a positive external magnetic field h > 0, then σ0βc,h ≍ h

1 15

Rough idea of proof:

◮ Lower bound: prove that the correlation length L(h) ≍ h−8/15 and

conclude using σ0βc,h αFK

1 (L(h)) ≍ L(h)−1/8 ≍ h1/15

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 18 / 19

Average magnetization under small external field

Theorem (Camia, G., Newman)

Consider Ising model on Z2 at βc with a positive external magnetic field h > 0, then σ0βc,h ≍ h

1 15

Rough idea of proof:

◮ Lower bound: prove that the correlation length L(h) ≍ h−8/15 and

conclude using σ0βc,h αFK

1 (L(h)) ≍ L(h)−1/8 ≍ h1/15 ◮ Upper bound:

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 18 / 19

Average magnetization under small external field

Theorem (Camia, G., Newman)

Consider Ising model on Z2 at βc with a positive external magnetic field h > 0, then σ0βc,h ≍ h

1 15

Rough idea of proof:

◮ Lower bound: prove that the correlation length L(h) ≍ h−8/15 and

conclude using σ0βc,h αFK

1 (L(h)) ≍ L(h)−1/8 ≍ h1/15 ◮ Upper bound: rely on a kind of strong “convexity” property satisfied by

the Ising model, namely the GHS inequality.

• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 18 / 19

Theorem (GHS inequality, Griffiths, Hurst, Sherman, 1970)

Zβ,h :=

σ e−βH(σ)+h P σx is such that

∂3

h

• log Zβ,h
• ≤ 0
• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 19 / 19

Theorem (GHS inequality, Griffiths, Hurst, Sherman, 1970)

Zβ,h :=

σ e−βH(σ)+h P σx is such that

∂3

h

• log Zβ,h
• ≤ 0

∂3

h

• log(
• e−βcH+h P σx)
• ≤ 0
• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 19 / 19

Theorem (GHS inequality, Griffiths, Hurst, Sherman, 1970)

Zβ,h :=

σ e−βH(σ)+h P σx is such that

∂3

h

• log Zβ,h
• ≤ 0

∂3

h

• log(
• e−βcH+h P σx)
• ≤ 0

⇔ ∂2

h

• σ( σx)e−βcH+h P σx
• σ e−βcH
• ≤ 0
• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 19 / 19

Theorem (GHS inequality, Griffiths, Hurst, Sherman, 1970)

Zβ,h :=

σ e−βH(σ)+h P σx is such that

∂3

h

• log Zβ,h
• ≤ 0

∂3

h

• log(
• e−βcH+h P σx)
• ≤ 0

⇔ ∂2

h

• σ( σx)e−βcH+h P σx
• σ e−βcH
• ≤ 0

⇔ ∂2

h

• Eβc,h
• σx
• ≤ 0
• C. Garban (ENS Lyon and CNRS)

Near-critical Ising model 19 / 19