2D Ising Model: Near-Critical Scaling Limit and Magnetization - PowerPoint PPT Presentation

2D Ising Model: Near-Critical Scaling Limit and Magnetization Critical Exponent ∗ Charles M. Newman Courant Institute of Mathematical Sciences newman @ courant.nyu.edu ∗ Based on joint work with Federico Camia and Christophe Garban.

Ising Model on Z 2 S x = +1 or −1 �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� Probability ∝ exp ( β � { x,y } S x S y + h � x S x ) Spins: S x , S y = ± 1 Edges: e = { x, y } ( || x − y || = 1) Continuum scaling limit: replace Z 2 by a Z 2 and let a → 0. 1

Ising Model in a Finite Domain 1 P β,h e − β E L ( S )+ h M L ( S ) L ( S ) := Z L,β,h Λ L := [ − L, L ] 2 ∩ Z 2  domain        E L ( S ) := − � interaction energy  { x,y } S x S y     M L ( S ) := � x ∈ Λ L S x total magnetization in Λ L          S e − β E L ( S )+ h M L ( S )  Z L,β,h := � partition function   2

h = 0 Case: The Three Regimes Evidence of a phase transition. 3

√ The Critical Point : β = β c = 1 2 log(1 + 2) 4

Thermodynamic Limit h > 0 or β ≤ β c ⇒ P β,h has a unique infinite volume limit as L L → ∞ : L →∞ P β,h → P β,h − L �·� β,h denotes expectation with respect to P β,h 5

The Magnetization Exponent (F. Camia, C. Garban, C.M.N.; arXiv:1205.6612) Theorem. Consider the Ising model on Z 2 at β c with a positive external magnetic field h > 0, then ∗ 1 15 . � S 0 � β c ,h ≍ h ∗ f ( a ) ≍ g ( a ) as a ց 0 means that f ( a ) /g ( a ) is bounded away from 0 and ∞ . 6

Critical Exponents  C ( T ) ∼ | T − T c | − α Heat capacity:        M ( T ) ∼ | T − T c | b  Order parameter:     χ ( T ) ∼ | T − T c | − γ Susceptibility:          M ( h ) ∼ h 1 /δ  Equation of state ( T = T c ) :   7

2D Ising Critical Exponents Onsager’s solution shows that • susceptibility has logarithmic divergence ⇒ α = 0 • b = 1 / 8 (Yang) Scaling theory predicts • correlation length at T c : ξ ( h ) ∼ h − 8 / 15 8

Scaling Laws  Rushbrooke: α + 2 b + γ = 2   Widom: γ = b ( δ − 1)   ⇓ δ = 2 − α − b 2D Ising = 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 �·� denote expectation with respect to P β,h ( h ≥ 0). Then, for any vertices L x, y, z ∈ Λ L , � � � S x S y S z �− � S x � � S y S z � + � S y � � S x S z � + � S z � � S x S y � +2 � S x �� S y �� S z � ≤ 0 . Corollary. The GHS inequality implies that ∂ 3 h log( Z L,β,h ) ≤ 0 . 11

Magnetization 1 1 � S 0 � β c ,h = | Λ L |� M L � β c ,h ≤ | Λ L |� M L � β c ,h, + (+ b.c. on Λ L ) ∂ � M L � β c ,h, + = � M L e hM L � β c , 0 , + ∂h � e hM L � β c , 0 , + = � e hM L � β c , 0 , + � e hM L � β c , 0 , + 12

Consequences of GHS �� � ∂ 3 S e − β c E L ( S )+ hM L ( S ) GHS ⇒ ∂h 3 log ≤ 0 �� e − βcEL + hML � ∂ 3 ∂h 3 log ≤ 0 ⇔ � e − βcEL � ∂ ∂h � e hML � βc, 0 , + � ∂ 2 = ∂ 2 ⇔ ∂h 2 � M L � β c ,h, + ≤ 0 ∂h 2 � e hML � βc, 0 , + ∂h � e hML � βc, 0 , + ∂ Let F ( h ) ≡ F L ( h ) := = � M L � β c ,h, + , then � e hML � βc, 0 , + F (0) + h F ′ (0) F ( h ) ≤ � � M 2 L � β c , 0 , + − � M L � 2 � = � M L � β c , 0 , + + h β c , 0 , + 13

Magnetization Bounds Theorem [T.T. Wu, 1966]. There exists an explicit constant c > 0 such that as n → ∞ ρ ( n ) := � S (0 , 0) S ( n,n ) � β c , 0 ∼ c n − 1 / 4 . Proposition [F. Camia, C. Garban, C.M.N.]. There is a universal constant C > 0 such that for L sufficiently large, one has (i) � M L � β c , 0 , + ≤ C L 2 ρ ( L ) 1 / 2 , L � β c , 0 , + ≤ C L 4 ρ ( L ). (ii) � M 2 14

Upper Bound � S 0 � β c ,h ≤ 1 L 15 / 8 + h L 15 / 4 � /L 2 � L 2 � M L � β c ,h, + ≤ C (optimize in L = L ( h )) ⇔ (choose L ( h ) ≍ h − 8 / 15 ∼ ξ ( h )) 1 L ( h ) 2 L ( h ) 15 / 8 = O (1) L ( h ) − 1 / 8 � S 0 � β c ,h ≤ O (1) O (1) h 1 / 15 ≤ 15

Scaling Limit : Z 2 replaced by a Z 2 ; a → 0 � � Spin Approach 1 : Boundaries of clusters as conformal loop FK ensembles in plane related to Schramm-Loewner Evolution ( SLE κ ) � � 3 with κ = (Schramm, Smirnov) 16 / 3 Approach 2 ( today ): Random (Euclidean) field, Φ a ( z ) = Θ a � S x δ x x ∈ a Z 2 16

Heuristics for Magnetization Field β,h ≡ Cov β,h ( S x , S y ) ∼ e −|| x − y || /ξ ( β,h ) as || x − y || → ∞ � S x S y � T 1. β < β c fixed ( h = 0) with ξ ( β ) < ∞ : Φ 0 trivial (i.e., Gaussian white noise) by some CLT. 2. β = β c ( h = 0), ξ ( β c ) = ∞ : Φ 0 massless; 3. β = β ( a ) ↑ β c ( h = 0) or β = β c , h ( a ) ↓ 0 s. t. a ξ ( a ) → 1 /m ∈ (0 , ∞ ) as a → 0: “near-critical” Φ 0 is massive. 17

Continuum Scaling Limits for the Magnetization ( h = 0 ) 1 a → 0 � High temperature: S x − → M ∼ Normal dist. � 1 /a 2 x ∈ square Critical temperature: classical CLT does not hold. 18

Scaling Limit at β c and h = 0 In the scaling limit ( a → 0) one hopes that � a → 0 a 15 / 8 � [0 , 1] 2 Φ( z ) dz S x − → x ∈ square for some magnetization field Φ = Φ 0 . 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) Φ a := a 15 / 8 � S x δ x x ∈ a Z 2 Critical scaling limit: β = β c , h = 0, a → 0 Φ a → random generalized function Φ 0 : massless field (power-law decay of correlations). The limiting magnetization field is not Gaussian: log P (Φ 0 ([0 , 1] 2 ) > x ) x →∞ − c x 16 ∼ 20

Conformal Covariance (F. Camia, C. Garban, C.M.N.; arXiv:1205.6610) The magnetization field Φ = Φ 0 exists as a random generalized function and is conformally covariant: If f is a conformal map, − 1 / 8 Φ( z )” . “Φ( f ( z )) dist. � � � f ′ ( z ) = � � � E.g., for a scale transformation f ( z ) = αz ( α > 0), � [ − αL,αL ] Φ( z ) dz dist. � = α 15 / 8 [ − L,L ] Φ( z ) dz . 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: β = β c , a → 0, h → 0, ha − 15 / 8 → λ ∈ (0 , ∞ ). Heuristics: choose h = λa 15 / 8 and note that ξ ( h ) = ξ ( λa 15 / 8 ) ∼ ( a 15 / 8 ) − 8 / 15 = 1 . Limit yields one-parameter ( λ ) family of fields [in progress: mas- sive; i.e., exponential decay of correlations]. R 2 Φ 0 ( z ) dz );” Heuristics: multiply zero-field measure by “exp( λ � exponential decay based on FK percolation props. of critical Φ 0 . 22