2D Ising Model: Near-Critical Scaling Limit and Magnetization Critical Exponent ∗

Home , Ising critical exponents

Charles M. Newman

Courant Institute of Mathematical Sciences

newman @ courant.nyu.edu

∗Based on joint work with Federico Camia and Christophe Garban. 2 Ising Model on Z

Sx = +1 or −1

P P Probability ∝ exp (β {x,y} SxSy + h x Sx)

Spins: Sx,Sy = ±1 Edges: e = {x, y} (||x − y|| = 1)

2 2 Continuum scaling limit: replace Z by a Z and let a → 0. 1 Ising Model in a Finite Domain

1 β,h −β EL(S)+h ML(S) PL (S) := e ZL,β,h

 Λ := [−L, L]2 ∩ 2 domain  L Z    P  EL(S) := − SxSy interaction energy  {x,y}  P  ML(S) := x∈Λ Sx total magnetization in ΛL  L    P −β E (S)+h M (S)  ZL,β,h := S e L L partition function

2 h = 0 Case: The Three Regimes

Evidence of a phase transition.

3 1 √ The Critical Point: β = βc = 2 log(1 + 2)

4 Thermodynamic Limit

β,h h > 0 or β ≤ βc ⇒ PL has a unique inﬁnite volume limit as L → ∞: β,h L→∞ β,h PL −→ P

β,h h·iβ,h denotes expectation with respect to P

5 The Magnetization Exponent (F. Camia, C. Garban, C.M.N.; arXiv:1205.6612)

2 Theorem. Consider the Ising model on Z at βc with a positive external magnetic ﬁeld h > 0, then ∗ 1 15 hS0iβc,h h .

∗f(a) g(a) as a & 0 means that f(a)/g(a) is bounded away from 0 and ∞.

6 Critical Exponents

 Heat capacity: ( ) −α  C T ∼ |T − Tc|     Order parameter: M(T ) ∼ |T − T |b  c

 Susceptibility: χ(T ) ∼ |T − T |−γ  c    1/δ  Equation of state (T = Tc): M(h) ∼ h

7 2D Ising Critical Exponents

Onsager’s solution shows that

• susceptibility has logarithmic divergence ⇒ α = 0

• b = 1/8 (Yang)

Scaling theory predicts

−8/15 • correlation length at Tc: ξ(h) ∼ h

8 Scaling Laws

 Rushbrooke: α + 2b + γ = 2   Widom: γ = b(δ − 1)

⇓

2 − α − b 2D Ising δ = = 15 b

9 Proof of the Exponent Theorem

Lower bound: Use Ising ghost spin representation + standard percolation arguments; tools: FKG + RSW for FK percolation.

Upper bound: Combine GHS inequality with ﬁrst and second moment bounds for the magnetization; tools: GHS + FKG + RSW for FK percolation.

RSW for Ising-FK proved by Duminil-Copin, Hongler, Nolin (2011).

10 GHS Inequality

Theorem [Griﬃth, Hurst, Sherman, 1970]. Let h·i denote β,h expectation with respect to PL (h ≥ 0). Then, for any vertices x, y, z ∈ ΛL, hSxSySzi− hSxi hSySzi+hSyi hSxSzi+hSzi hSxSyi +2hSxihSyihSzi ≤ 0 .

Corollary. The GHS inequality implies that

3 ∂h log(ZL,β,h) ≤ 0 .

11 Magnetization

1 1 hS0iβc,h = hMLiβc,h ≤ hMLiβc,h,+ (+ b.c. on ΛL) |ΛL| |ΛL|

hML ∂ hML hML e iβc,0,+ ∂hhe iβc,0,+ hMLiβ ,h,+ = = c hML hML he iβc,0,+ he iβc,0,+

12 Consequences of GHS

3 GHS ⇒ ∂ log P e−βcEL(S)+hML(S) ≤ 0 ∂h3 S 3 P −βcEL+hML ⇔ ∂ log e ≤ 0 ∂h3 P e−βcEL 2 ∂ hehMLi 2 ⇔ ∂ ∂h βc,0,+ = ∂ hM i ≤ 0 ∂h2 hML ∂h2 L βc,h,+ he iβc,0,+

∂ hehMLi Let F (h) ≡ F (h) := ∂h βc,0,+ = hM i , then L hML L βc,h,+ he iβc,0,+ F (h) ≤ F (0) + h F 0(0) = + 2 2 hMLiβc,0,+ h hMLiβc,0,+ − hMLiβc,0,+

13 Magnetization Bounds

Theorem [T.T. Wu, 1966]. There exists an explicit constant c > 0 such that as n → ∞ −1/4 ρ(n) := hS(0,0)S(n,n)iβc,0 ∼ c n .

Proposition [F. Camia, C. Garban, C.M.N.]. There is a universal constant C > 0 such that for L suﬃciently large, one has

2 1/2 (i) hMLiβc,0,+ ≤ CL ρ(L) ,

(ii) 2 4 ( ). hMLiβc,0,+ ≤ CL ρ L

14 Upper Bound

1 15/8 15/4 2 hS0i ≤ hM i + ≤ C L + h L /L βc,h L2 L βc,h,

(optimize in L = L(h)) ⇔ (choose L(h) h−8/15 ∼ ξ(h))

1 15/8 −1/8 hS0i ≤ O(1) L(h) = O(1) L(h) βc,h L(h)2 ≤ O(1) h1/15

15 2 2 Scaling Limit: Z replaced by aZ ; a → 0

( Spin ) Approach 1: Boundaries of clusters as conformal loop FK ensembles in plane related to Schramm-Loewner Evolution (SLEκ)

( 3 ) with κ = (Schramm, Smirnov) 16/3

Approach 2 (today): Random (Euclidean) ﬁeld,

a X Φ (z) = Θa Sxδx x∈aZ2

16 Heuristics for Magnetization Field

T −||x−y||/ξ(β,h) hSxSyiβ,h ≡ Covβ,h(Sx,Sy) ∼ e as ||x − y|| → ∞

0 1. β < βc ﬁxed (h = 0) with ξ(β) < ∞:Φ trivial (i.e., Gaussian white noise) by some CLT.

0 2. β = βc (h = 0), ξ(βc) = ∞:Φ massless;

3. β = β(a) ↑ βc (h = 0) or β = βc, h(a) ↓ 0 s. t. a ξ(a) → 1/m ∈ (0, ∞) as a → 0: “near-critical” Φ0 is massive.

17 Continuum Scaling Limits for the Magnetization (h = 0)

1 X a→0 High temperature: q Sx −→ M ∼ Normal dist. 1/a2 x∈square

Critical temperature: classical CLT does not hold.

18 Scaling Limit at βc and h = 0

In the scaling limit (a → 0) one hopes that

a→0 Z a15/8 X S −→ Φ(z)dz x 2 x∈square [0,1] for some magnetization ﬁeld Φ = Φ0.

Φ should describe the ﬂuctuations of the magnetization around its mean (= 0).

However, Φ cannot be a function.

19 Critical Magnetization Field (F. Camia, C. Garban, C.M.N.; arXiv:1205.6610)

a 15/8 X Φ := a Sxδx x∈aZ2

Critical scaling limit: β = βc, h = 0, a → 0 Φa → random generalized function Φ0: massless ﬁeld (power-law decay of correlations).

The limiting magnetization ﬁeld is not Gaussian:

0 2 x→∞ 16 log P(Φ ([0, 1] ) > x) ∼ −c x

20 Conformal Covariance (F. Camia, C. Garban, C.M.N.; arXiv:1205.6610)

The magnetization ﬁeld Φ = Φ0 exists as a random generalized function and is conformally covariant:

If f is a conformal map,

dist. −1/8 0 “Φ(f(z)) = f (z) Φ(z)” . E.g., for a scale transformation f(z) = αz (α > 0),

Z dist. Z Φ(z) dz = α15/8 Φ(z) dz . [−αL,αL] [−L,L]

21 h → 0 Near-Critical Field

(F.C., C.G, C.M.N.; arXiv:1307.3926)

(Why?: Borthwick-Garibaldi, 2011; McCoy-Maillard, 2012)

Near-critical (oﬀ-critical) scaling limit: −15/8 β = βc, a → 0, h → 0, ha → λ ∈ (0, ∞).

Heuristics: choose h = λa15/8 and note that ξ(h) = ξ(λa15/8)∼(a15/8)−8/15 = 1 .

Limit yields one-parameter (λ) family of ﬁelds [in progress: massive; i.e., exponential decay of correlations].

Heuristics: multiply zero-ﬁeld measure by “exp(λ R Φ0(z)dz);” R2 exponential decay based on FK percolation props. of critical Φ0. 22