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. 2 Ising Model on Z Sx = +1 or −1 P P Probability / exp (β fx;yg SxSy + h x Sx) Spins: Sx;Sy = ±1 Edges: e = fx; yg (jjx − yjj = 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 8 Λ := [−L; L]2 \ 2 domain > L Z > > > P > EL(S) := − SxSy interaction energy <> fx;yg > P > ML(S) := x2Λ 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 p 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 ! 1: β;h L!1 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 1. 6 Critical Exponents 8 Heat capacity: ( ) −α > C T ∼ jT − Tcj > > > > Order parameter: M(T ) ∼ jT − T jb <> c > Susceptibility: χ(T ) ∼ jT − T j−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 8 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 2 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) jΛLj jΛLj 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 −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 x2aZ2 16 Heuristics for Magnetization Field T −||x−yjj=ξ(β;h) hSxSyiβ;h ≡ Covβ;h(Sx;Sy) ∼ e as jjx − yjj ! 1 0 1. β < βc fixed (h = 0) with ξ(β) < 1:Φ trivial (i.e., Gaussian white noise) by some CLT. 0 2. β = βc (h = 0), ξ(βc) = 1:Φ massless; 3. β = β(a) " βc (h = 0) or β = βc, h(a) # 0 s. t. a ξ(a) ! 1=m 2 (0; 1) 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 x2square 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 x2square [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 x2aZ2 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!1 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 2 (0; 1). 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: mas- sive; 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.