Magnetization field at criticality in the Ising model Christophe - - PowerPoint PPT Presentation
Magnetization field at criticality in the Ising model Christophe Garban ENS Lyon and CNRS Federico Camia (Vrije Universiteit Amsterdam) Joint work with Charles Newman (NYU) and Percolation and Interacting Systems, MSRI, February 2012 C. Garban
Christophe Garban
ENS Lyon and CNRS Joint work with
Federico Camia (Vrije Universiteit Amsterdam)
and
Charles Newman (NYU) Percolation and Interacting Systems, MSRI, February 2012
Magnetization field of the Ising model 1 / 25
Magnetization field of the Ising model 2 / 25
Magnetization field of the Ising model 2 / 25
What are the fluctuations
the magnetiza- tion field ?
Magnetization field of the Ising model 2 / 25
N N
Magnetization field of the Ising model 3 / 25
N N
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
Magnetization field of the Ising model 3 / 25
N N
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
1 N
σx → N(0, 1)
Magnetization field of the Ising model 3 / 25
N N
1 N
σx → N(0, 1)
1 N
σxδx → W, two- dimensional white noise
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
Magnetization field of the Ising model 3 / 25
N N
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
Magnetization field of the Ising model 4 / 25
N N
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
To each configuration σ ∈ {−1, 1}N2, one associates the Hamiltonian
i∼j σiσj
Magnetization field of the Ising model 4 / 25
N N
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
To each configuration σ ∈ {−1, 1}N2, one associates the Hamiltonian
i∼j σiσj
And we define:
Magnetization field of the Ising model 4 / 25
N N
1 N
σx → N(0, σ2
β) 1 N
σx δx → σβW
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
Magnetization field of the Ising model 5 / 25
N N
1 N
σx → N(0, σ2
β) 1 N
σx δx → σβW
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
N 2σ0− N 2σ0+
Magnetization field of the Ising model 5 / 25
N N
1 N
σx → N(0, σ2
β) 1 N
σx δx → σβW
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
σx−N 2E
β)
Magnetization field of the Ising model 5 / 25
N N
1 N
σx → N(0, σ2
β) 1 N
σx δx → σβW
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
σx−N 2E
β)
Magnetization field of the Ising model 5 / 25
To avoid boundary issues, consider our system on the torus Z2/NZ2. We have: Var(
=
E
To avoid boundary issues, consider our system on the torus Z2/NZ2. We have: Var(
=
E
N2
y
E
To avoid boundary issues, consider our system on the torus Z2/NZ2. We have: Var(
=
E
N2
y
E
N2
y
|y|−1/4 (by Onsager) ≍ N2 N2N−1/4 = N
15 4
To avoid boundary issues, consider our system on the torus Z2/NZ2. We have: Var(
=
E
N2
y
E
N2
y
|y|−1/4 (by Onsager) ≍ N2 N2N−1/4 = N
15 4
Hence it is natural to look at the random variable m(N) :=
σx N15/8 .
Question
Does m(N) have a (unique) scaling limit ?
Theorem (McCoy, Wu, 1967)
As N → ∞
where c = 21/12e3ζ′(−1)
Magnetization field of the Ising model 7 / 25
Theorem (McCoy, Wu, 1967)
As N → ∞
where c = 21/12e3ζ′(−1)
Proposition (Rotational invariance of the two-point function)
→
x2→∞ 1
Magnetization field of the Ising model 7 / 25
Rescaled lattice a Z2, a ≪ 1.
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
aZ2 1
Definition (Renormalized magnetization field)
Φa :=
δx σx a
15 8
ma := Φa, 1[0,1]2 ma
L := Φa, 1[0,L]2
Rescaled lattice a Z2, a ≪ 1.
+ + + + + + + − − − − − − − − + − − − + + + − − − − − + + + + − − − − + − + + − + − − + + + − − − − − + − + + − + + − − + − − + + − − +
aZ2 1
Definition (Renormalized magnetization field)
Φa :=
δx σx a
15 8
ma := Φa, 1[0,1]2 ma
L := Φa, 1[0,L]2
Question
The field Φa ∈ D′. Is it the case that Φa converges as a → 0 to some random distribution Φ ?
Theorem (Camia, G., Newman)
(i) The magnetization field Φa on a Z2 has a unique scaling limit as the mesh a → 0.
Theorem (Camia, G., Newman)
(i) The magnetization field Φa on a Z2 has a unique scaling limit as the mesh a → 0. (ii) On a finite domain Ω with boundary conditions ξ, the magnetization field Φa
|Ω∩aZ2 has a scaling limit Φξ Ω.
Theorem (Camia, G., Newman)
(i) The magnetization field Φa on a Z2 has a unique scaling limit as the mesh a → 0. (ii) On a finite domain Ω with boundary conditions ξ, the magnetization field Φa
|Ω∩aZ2 has a scaling limit Φξ Ω.
(iii) In particular, ma has a unique limiting law m = mξ which depends on the boundary conditions (usually +, −, or free).
Theorem (Camia, G., Newman)
(i) The magnetization field Φa on a Z2 has a unique scaling limit as the mesh a → 0. (ii) On a finite domain Ω with boundary conditions ξ, the magnetization field Φa
|Ω∩aZ2 has a scaling limit Φξ Ω.
(iii) In particular, ma has a unique limiting law m = mξ which depends on the boundary conditions (usually +, −, or free). (iv) The scaling limit is NOT Gaussian.
Theorem (Camia, G., Newman)
(i) The magnetization field Φa on a Z2 has a unique scaling limit as the mesh a → 0. (ii) On a finite domain Ω with boundary conditions ξ, the magnetization field Φa
|Ω∩aZ2 has a scaling limit Φξ Ω.
(iii) In particular, ma has a unique limiting law m = mξ which depends on the boundary conditions (usually +, −, or free). (iv) The scaling limit is NOT Gaussian.
(i) Conformal covariance of ΦΩ (ii) Tail behavior of m etc ...
Theorem (Camia, G., Newman)
Consider Ising model on Z2 at βc with a positive external magnetic field h > 0, then σ0 ≍ h
1 15
Magnetization field of the Ising model 10 / 25
Theorem (Camia, G., Newman)
Consider Ising model on Z2 at βc with a positive external magnetic field h > 0, then σ0 ≍ h
1 15
This is the analog of (i) Onsager: σ0+
β ≍ |β − βc|1/8 as β → βc.
and (ii) Kesten - Smirnov/Werner: θ(p) = |p − pc|5/36+o(1) for site percolation on the triangular lattice.
Magnetization field of the Ising model 10 / 25
Magnetization field of the Ising model 11 / 25
Fact
lim sup
a→0
Eβc
< ∞
Magnetization field of the Ising model 11 / 25
Fact
lim sup
a→0
Eβc
< ∞
Fact
lim sup
a→0
E
H−2
Since BH−2(0, 1) is compact in H−3, one can take subsequential scaling limits Φak → Φ∗ in H−3.
Magnetization field of the Ising model 11 / 25
Fact
lim sup
a→0
Eβc
< ∞
Fact
lim sup
a→0
E
H−2
Since BH−2(0, 1) is compact in H−3, one can take subsequential scaling limits Φak → Φ∗ in H−3. It remains to characterize the subsequential scaling limits Φ∗
Magnetization field of the Ising model 11 / 25
Theorem (Smirnov, Chelkak-Smirnov)
aZ2
−
≡ ≡ + a
Theorem (Smirnov, Chelkak-Smirnov)
aZ2 a → 0
−
≡ ≡ + a CLE3 ≈ SLE3
Theorem (Smirnov, Chelkak-Smirnov)
aZ2 a → 0
−
≡ ≡ + a CLE3 ≈ SLE3
Question
Can one recover the magnetization m = lima→0 ma as a functional of the CLE3 ?
Definition
Pp,q
1 Zp,q po(ω)(1 − p)c(ω) q#clusters
Magnetization field of the Ising model 13 / 25
Definition
Pp,q
1 Zp,q po(ω)(1 − p)c(ω) q#clusters ma =
a15/8σx
Magnetization field of the Ising model 13 / 25
Definition
Pp,q
1 Zp,q po(ω)(1 − p)c(ω) q#clusters ma =
a15/8σx =
a15/8σx +
a15/8σx + . . . = ξ1 Areaa(C1) + ξ2 Areaa(C2) + . . . where (ξi)i≥1 are i.i.d coin flips.
Magnetization field of the Ising model 13 / 25
Definition
Pp,q
1 Zp,q po(ω)(1 − p)c(ω) q#clusters ma =
a15/8σx =
a15/8σx +
a15/8σx + . . . = ξ1 Areaa(C1) + ξ2 Areaa(C2) + . . . where (ξi)i≥1 are i.i.d coin flips.
Claim
Uniformly in a → 0, E
ξi Areaa(Ci) 2 ≤ C ǫ7/4
Magnetization field of the Ising model 13 / 25
Magnetization field of the Ising model 14 / 25
a Z2
Magnetization field of the Ising model 14 / 25
a Z2 ǫ
Magnetization field of the Ising model 14 / 25
a Z2 ǫ #points =
Xi ∼
Yi K ǫ a 15/
Magnetization field of the Ising model 14 / 25
a Z2 #points =
Xi ∼
Yi K ǫ a 15/
E
a)15/8Y 2 =
E
a)15/8Yi Xj − K( ǫ a)15/8Yj
)
Magnetization field of the Ising model 14 / 25
Theorem (Dubédat or Chelkak, Hongler, Izyurov)
a Z2
Theorem (Dubédat or Chelkak, Hongler, Izyurov)
a Z2
There exist n-point correlation functions z1, . . . , zn → z1, . . . , zn+
Ω ,
Theorem (Dubédat or Chelkak, Hongler, Izyurov)
a Z2
There exist n-point correlation functions z1, . . . , zn → z1, . . . , zn+
Ω ,
so that as the mesh a → 0, and if the points z1, . . . , zn remain “macroscopically far apart”, a− n
8 E +
Ω
1 . . . σza n
Ω
Theorem (Dubédat or Chelkak, Hongler, Izyurov)
a Z2
There exist n-point correlation functions z1, . . . , zn → z1, . . . , zn+
Ω ,
so that as the mesh a → 0, and if the points z1, . . . , zn remain “macroscopically far apart”, a− n
8 E +
Ω
1 . . . σza n
Ω
This suggests that one should have for each n ≥ 1: E +
Ω
a
→
a→0
Ω dz1 . . . dzn
In order to show that Φa converges to a limiting random distribution, two things thus remain to be proved:
In order to show that Φa converges to a limiting random distribution, two things thus remain to be proved:
Ω
a
µn :=
Ω dz1 . . . dzn
as a → 0.
In order to show that Φa converges to a limiting random distribution, two things thus remain to be proved:
Ω
a
µn :=
Ω dz1 . . . dzn
as a → 0.
probability law m (moment problem).
In order to show that Φa converges to a limiting random distribution, two things thus remain to be proved:
Ω
a
µn :=
Ω dz1 . . . dzn
as a → 0.
probability law m (moment problem). The second property will follow from the following proposition:
Proposition (Camia, G., Newman)
For all fixed t > 0, one has sup
a>0
E +
Ω
< ∞
Magnetization field of the Ising model 17 / 25
Theorem
(i) m[0,λ]2
(d)
≡ λ15/8 m[0,1]2
Magnetization field of the Ising model 18 / 25
Theorem
(i) m[0,λ]2
(d)
≡ λ15/8 m[0,1]2
Magnetization field of the Ising model 18 / 25
Theorem
(i) m[0,λ]2
(d)
≡ λ15/8 m[0,1]2
(ii) f∗Φ(ω) and Φ(˜ ω) are a.c. Furthermore, ∀ φ ∈ C ∞
c (Ω)
˜ φ = φ ◦ f −1 ∈ C ∞
c (˜
Ω) , one has
ω), ˜ φ
Magnetization field of the Ising model 18 / 25
Let’s try to understand P
Magnetization field of the Ising model 19 / 25
Let’s try to understand P
Fact
There is a universal constant C > 0 s.t. P
Magnetization field of the Ising model 19 / 25
Let’s try to understand P
Fact
There is a universal constant C > 0 s.t. P
This suggests
Magnetization field of the Ising model 19 / 25
Theorem
There exists a universal constant c > 0 such that for any boundary conditions ξ around [0, 1]2, as x → ∞: log P
Theorem
There exists a universal constant c > 0 such that for any boundary conditions ξ around [0, 1]2, as x → ∞: log P
Proof: Goes through the study of the moment generating function of mξ:
Theorem
There exists a universal constant c > 0 such that for any boundary conditions ξ around [0, 1]2, as x → ∞: log P
Proof: Goes through the study of the moment generating function of mξ:
Proposition
There exists a universal constant b > 0 such that for any boundary conditions ξ around [0, 1]2, one has as t → ∞: log E
∼ −b t
16 15
Theorem
There exists a universal constant c > 0 such that for any boundary conditions ξ around [0, 1]2, as x → ∞: log P
(by a Tauberian Theorem) Proof: Goes through the study of the moment generating function of mξ:
Proposition
There exists a universal constant b > 0 such that for any boundary conditions ξ around [0, 1]2, one has as t → ∞: log E
∼ −b t
16 15
COROLLARY: mξ is indeed not Gaussian.
Magnetization field of the Ising model 21 / 25
Theorem (GHS inequality, Griffiths, Hurst, Sherman, 1970)
Zβ,h :=
σ e−βH(σ)+h P σx is such that
∂3
h
Magnetization field of the Ising model 21 / 25
Theorem (GHS inequality, Griffiths, Hurst, Sherman, 1970)
Zβ,h :=
σ e−βH(σ)+h P σx is such that
∂3
h
∂3
h
Magnetization field of the Ising model 21 / 25
Theorem (GHS inequality, Griffiths, Hurst, Sherman, 1970)
Zβ,h :=
σ e−βH(σ)+h P σx is such that
∂3
h
∂3
h
⇔ ∂3
h
e−βcH+h P σx e−βcH )
Magnetization field of the Ising model 21 / 25
Theorem (GHS inequality, Griffiths, Hurst, Sherman, 1970)
Zβ,h :=
σ e−βH(σ)+h P σx is such that
∂3
h
∂3
h
⇔ ∂3
h
e−βcH+h P σx e−βcH )
⇔ ∂3
h
≤ 0
Magnetization field of the Ising model 21 / 25
Theorem (GHS inequality, Griffiths, Hurst, Sherman, 1970)
Zβ,h :=
σ e−βH(σ)+h P σx is such that
∂3
h
∂3
h
⇔ ∂3
h
e−βcH+h P σx e−βcH )
⇔ ∂3
h
≤ 0 log E
≤ h
2 Var[
Magnetization field of the Ising model 21 / 25
log E
≤ h
2 Var[
With h := ta15/8, this gives sup
a>0
≤ sup
a>0
+ t2 2 Var[ma]
log E
≤ h
2 Var[
With h := ta15/8, this gives sup
a>0
≤ sup
a>0
+ t2 2 Var[ma]
Using the estimate with ˜ t > t > 0, we get
Fact
E
= lim
a→0 E
< ∞
Main tool: use scaling ! log E
= log E
Magnetization field of the Ising model 23 / 25
Main tool: use scaling ! log E
= log E
= log E
( with λt := t8/15)
Magnetization field of the Ising model 23 / 25
Main tool: use scaling ! log E
= log E
= log E
( with λt := t8/15) = λ2
t
1 λ2
t
log E
Magnetization field of the Ising model 23 / 25
Main tool: use scaling ! log E
= log E
= log E
( with λt := t8/15) = λ2
t
1 λ2
t
log E
∼ λ2
t
lim
L→∞
1 L2 log E
Magnetization field of the Ising model 23 / 25
Main tool: use scaling ! log E
= log E
= log E
( with λt := t8/15) = λ2
t
1 λ2
t
log E
∼ λ2
t
lim
L→∞
1 L2 log E
∼ t16/15C
Magnetization field of the Ising model 23 / 25
Magnetization field of the Ising model 24 / 25
Theorem
For all t ∈ R and any boundary condition ξ, one has |E ξ ei t m | ≤ e−c |t|
16 15 .
In particular, the density function f = fΩ is an entire function on the whole plane C !
Magnetization field of the Ising model 24 / 25
A first one:
Theorem
Consider Ising model on Z2 at βc with a positive external magnetic field h > 0, then σ0 ≍ h
1 15
Magnetization field of the Ising model 25 / 25
A first one:
Theorem
Consider Ising model on Z2 at βc with a positive external magnetic field h > 0, then σ0 ≍ h
1 15
A second one:
Theorem
Consider Ising model on aZ2 with external magnetic field h a15/8, then there is a scaling limit as a → 0 towards a massive near-critical model.
Magnetization field of the Ising model 25 / 25