Fluctuations of the free energy of spherical spin glass

Transcription

1 Fluctuations of the free energy of spherical spin glass Jinho Baik University of Michigan 2015 May, IMS, Singapore

2 Joint work with Ji Oon Lee (KAIST, Korea)

3 Random matrix theory Real N N Wigner matrix M with E[M 2 ij ] = 1 N and E[M2 ii ] = w 2 N. Gaussian orthogonal ensemble (GOE): M ij are Gaussian Edge fluctuations: largest eigenvalue Tracy-Widom distribution (λ 1 (N) 2)N 2/3 TW 1

4 DLPP (directed last passage percolation) Lattice Z 2. iid random variables w(i,j). Up/right path π from (1,1) to (M,N). L(M,N) = π E(π), As M = O(N), ( L(M,N) N E(π) = (i,j) π ) a bn 2/3 TW 2 w(i,j) Proved for a few special random variables (Johansson 2000). Conjectured to be universal except...

5 Directed polymers Finite temperature version of DLPP: [ ] L β (M,N) = log e βe(π) Same TW 2 fluctuations for all β > 0 (see Friday talks). Note: Intermediate disorder regime β = B N 1/4 (Alberts, Khanin, Quastel 2014) π

6 Random matrices: Finite temperature version of λ 1 (N)? Note that Define λ 1 = max x =1 x,mx F N (β) = 1 N log [ x =1e Nβ x,mx dω(x) ] Main result of this talk: TW 1 for all low temperature β > β c. For all high temperature β < β c, Gaussian fluctuations.

7 Spherical Sherrington-Kirkpatrick (SSK model) of spin glass Another connection: F N (β) is the free energy of SSK model (spherical mean-field spin glass model) 1 Quick introduction to spin glass 2 Result 3 Proof: uses results from random matrix theory

8 Spin glass: Edwards Anderson model Lattice L N in 2D or 3D. spin variable σ = (σ x ) x LN, σ x { 1,+1} Hamiltonian H N (σ) = (x,y)j xy σ x σ y : short-range interactions Gibbs measure p N (σ) = 1 Z N e βh N(σ) Free energy F N = 1 N logz N = 1 N log [ σ e β (x,y) Jxyσxσy ] Take J xy iid random variables. So, F N is a random variable.

9 Mean-field spin glass: Sherrington Kirkpatrick (SK) model Long range interactions spin variable σ = (σ 1,,σ N ) { 1,+1} N H N (σ) = i j J ij σ i σ j = σ,jσ, Z N = σ e β σ,jσ Let J be a symmetric random matrix: Diagonal entries are 0. Other entries are iid of mean 0 and variance 1 N. (Different conventions: N i,j=1 vs i j )

10 Some extensions: we don t consider them here external magnetic field H N (σ) = J ij σ i σ j +h i,j i p-spin H N (σ) = J i1,,i p σ i1 σ ip i 1,,i p dynamics; aging σ i

11 Some known results on SK model F N = 1 N log [ σ e β i j J ijσ i σ j ], E[J ij ] = 0, E[J 2 ij ] = 1 N Parisi s formula (1980) F = lim N F N Rigorous proof by Talagrand (2006) (see Auffinger and Chen 2014) Transition at β c = 1 2 quenched free energy vs annealed free energy: agree in high temperature 1 N E[logZ N] 1 N loge[z N] 0 disagree in low temperature 1 N E[logZ N] 1 N loge[z N] 0

12 Some known results on SK model, continued Fluctuations in the high temperature regime (β < 1/2): Aizenmann, Lebowitz, Ruelle (1987) (F N ( log2+β 2) )N f(β) N(0,c), c = 1 2 log(1 4β2 ) 2β 2 Fluctuations at low temperature?? open question

13 Spherical mean-field spin glass: Spherical Sherrington Kirkpatrick (SSK) model σ = σ σ2 N = N J = (J ij ) N i,j=1 is a symmetric random matrix: Diagonal entries are 0. Other entries are iid of mean 0 and variance 1 N [ ] free energy F N = 1 N log σ = e β σ,jσ dω(σ) N Kosterlitz, Thouless, Jones (1976) This talk: fluctuations of F N for SSK

14 SSK: Known results F = lim N F N Crisanti and Sommers (1992), Talagrand (2006) F is explicit: Panchenko and Talagrand (2007), Kosterlitz, Thouless, Jones (1976) F(β) is C 2 but not C 3 at β = 1 2. Fluctuations? F(β) = { β 2, β < 1/2, 2β 1 2 log(2β) 3 4, β > 1/2

15 SSK and random matrix theory [ ] F N = F N (β) = 1 N log σ = N e β σ,jσ dω(σ) zero-temperature case, β = : F N ( ) := min x =1 x,jx = λ min Random matrix theory: for real Wigner matrices, (F N ( ) 2)N 2/3 TW GOE True for all β > β c?

16 Theorem 1 for SSK: high temperature regime β < 1/2 Let J = M N with J ii = 0 and J ij = J ji. For i < j, assume E[M ij ] = 0, E[Mij 2] = 1, E[M4 ij ] = 3, and all moments are finite. (F N β 2 )N N(f,c) The variance c = 1 2 log(1 4β2 ) 2β 2 is same as SK (Aizenman, Lebowitz, Ruelle) The mean is f = 1 4 log(1 4β2 ) 2β 2

17 Theorem 2 for SSK: low temperature regime β > 1/2 Let J = M N with J ii = 0 and J ij = J ji. For i < j, assume E[M ij ] = 0, E[Mij 2 ] = 1 and all moments are finite. 1 (F β 1/2 N2/3 N (2β 12 log(2β) 34 ) ) TW 1 TW 1 is the limit law of the fluctuations of the largest eigenvalue of Gaussian orthogonal ensemble (GOE)

18 Order of fluctuations 1 β < 1/2: F N F(β)+ 1 N N(f,c) 2 β > 1/2: F N F(β)+ 1 N 2/3 (β 1/2)TW GOE Here c = 1 2 log(1 4β2 ) 2β 2

19 Intuitively, 1 Low temperature: λ 1 dominates. 2 High temperature: all eigenvalues contribute (linear statistic i f(λ i)) This is true even on the level of fluctuations; TW vs Gaussian We use recent advancements in random matrix theory to make this precise. The rigidity of the eigenvalues plays a key role.

20 Universality F N = 1 N log [ σ 2 =N e β N i,j=1 J ijσ i σ j dω(σ) Take symmetric Wigner matrix J with mean zero and E[J11] 2 = w 2 N, E[J2 12] = 1 N, E[J4 12] = W 4 N 2 Universal except that the mean and the variance of N in high temperature regime depend on w 2 and W 4. Can take J from orthogonal invariant ensemble and from real Wishart ensemble. Also can take Hermitian J and complex spin σ. ]

21 Integral representation of Z N for SSK Fix J. Diagonalize J = O T DO Z N = e βn x,( J)x dω(x) = Consider x 2 =1 Q(z) = R N e βn x 2 =1 i λ iy 2 i e z i y2 i dy e βn i λ ix 2 i dω(x) Evaluate this in two ways: (i) Gaussian integral, (ii) polar coordinates Q(z) = I(r)e zr2 r N/2 1 dr, I(r) = e r2 βn i λ ixi 2 dω(x) 0 Take inverse Laplace transform x 2 =1

22 Integral representation of Z N for SSK Z N = i ( ) π N/2 1 γ+i e N 2 G(z) dz, (γ > λ 1 ) S N 1 Nβ γ i where G(z) = 2βz 1 N N k=1 log(z λ k) 1 Method of steepest-descent. 2 Find the critical point G (z) = 0 in z > λ 1. 3 Key point: G(z) is random but the eigenvalues are rigid! Kosterlitz, Thouless, Jones (1976) Ben Arous, Dembo, Guionnet (2001)

23 Formally, approximate by semi-circle σ(x) = 1 2π 4 x 2 G(z) =2βz 1 N N log(z λ k ) k=1 G (z) = 2βz Steepest-descent analysis of 2 2 log(z x)σ(x)dx ZN = i ( ) π N/2 1 γ+i e N 2 G (z) dz S N 1 Nβ γ i Restriction on the contour: γ > 2.

24 Critical point z c of G (z) in Re(z c ) > 2 If β < 1/2, main contribution from z c = 2β + 1 2β > 2. If β > 1/2, G (z) has no critical point. Main contribution from z = 2 (branch point). Actually from z = λ 1. Kosterlitz, Thouless, Jones (1976)

25 To make it rigorous, we need (recent!) results from random matrix theory: 1 Rigidity of the eigenvalues, Erdös, Yau and Yin (2012) λ k γ k ˆk 1/3 N 2/3+ǫ, ˆk = min{k,n +1 k} 2 Tracy-Widom limit of the largest eigenvalue, Soshinkov (1999), Tao and Vu (2010), Erdös, Yau and Yin (2012) N 2/3 (λ 1 2) TW GOE 3 Gaussian fluctuations of linear statistics L f = N i=1 f(λ i), Johansson (1998), Bai and Yao (2005) (for f(z) = log(c z) where c > 2) For f analytic in a neighborhood of [ 2,2], L f E[L f ] N(0,σ 2 )

26 Critical point z c of G (z) vs z c of G(z) G (z) = 2β 1 N N 1 z λ k k=1 If β < 1/2, z c z c, O(1) away from the branch point z = 2. If β > 1/2, z c = λ 1 +O(N 1+ǫ )

27 Low temperature Lemma 1: z c = λ 1 +O(N 1+ǫ ) from rigidity Lemma 2: It is still true that γ+i γ i e N 2 G(z) dz Ke N 2 G(zc), N c K C

28 Low temperature β > 1/2 Lemma 3: Using z c = λ 1 +O(N 1+ǫ ) and λ 1 = 2+O(N 2/3+ǫ ), G(z c ) = 2βz c 1 N 2βλ 1 1 N 2βλ 1 1 N 2βλ 1 N log(z c λ i ) (λ 1 λ i = O(N 2/3+ǫ )) i=1 N log(λ 1 λ i ) i=2 N i=2 2 2 = (2β 1)λ [ log(2 λ i )+ 1 ] (λ 1 2) 2 λ i log(2 s)σ(s)ds 2 2 σ(s)ds 2 s (λ 1 2)

29 High temperature β < 1/2 Critical value z c of G(z) and the critical value z c of G (z): z c z c = 2β + 1 2β > 2 Method of steepest-descent: [ 2 γ+i ] N log e N 2 G(z) dz G(z c ) G(zc ) γ i G(z c ) = 2βz c 1 N N log(zc λ i ) i=1 Linear statistics of ϕ(x) = log(z c x), Johansson (1998), Bai-Yao (2005)

30 Extensions: work in progress (1) Near critical temperature: β = 1 2 ± C N δ. It appears that for δ < 1/3, we have same fluctuations as above (after some changes to the Gaussian fluctuations yet to be identified) Perhaps the critical window is β = 1 2 ±O(logN )? Standard N 1/3 deviations are of order log(1 2β) N and (β 1 2 ) N 2/3

31 Extensions: work in progress (2) Non-zero mean J ij +J 0 : rank-1 spiked random matrix Spin glass Hamiltonian plus ferromagnetic Hamiltonian H N (σ)+j 0 i j σ iσ j Scale J 0 = J 0 N T = 1 2β paramagnetic 1 spin glass ferromagnetic 1 J 0 Kosterlitz, Thouless, Jones (1976)