Spherical Sherrington-Kirkpatrick model and random matrix Jinho Baik - PowerPoint PPT Presentation

Spherical Sherrington-Kirkpatrick model and random matrix Jinho Baik University of Michigan 2019 April at CIRM Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Ji Oon Lee (KAIST, Korea) Hao Wu (grad student, U.Michigan) Pierre le Doussal (ENS) Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

The largest eigenvalue of a symmetric matrix satisfies λ 1 = max � x � =1 � x , Mx � Consider its finite temperature version �� � 1 β = 1 e β � x , Mx � d Ω( x ) β log , T � x � =1 This is known as the free energy of the Spherical Sherrington-Kirkpatrick (SSK) model Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Spherical spin glass In general, a spherical spin glass model is defined by a random symmetric polynomial H ( σ ). Its random Gibbs measure is defined by √ p ( σ ) = 1 for σ ∈ R N with � σ � = Z N e β H ( σ ) N and its free energy by �� � 1 1 e β H ( σ ) d Ω( σ ) F N = N β log Z N = N β log √ � σ � = N One may consider models in other manifold or graph. The case with {− 1 , 1 } N is especially important and this is the usual spin glass model. Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Goal We consider the spherical spin glass model as N → ∞ for H ( σ ) = 1 2 σ T M σ (1) SSK H ( σ ) = 1 2 σ T M σ + m 2 N σ T σ (2) SSK + CW H ( σ ) = 1 2 σ T M σ + h σ T g (3) SSK + external field We use random matrix theory to study fluctuations of the free energy and the spin distribution. Assume that the semicircle law has support [ − 2 , 2]. For (1), RMT tells us that ≃ 1 + TW 1 D at T = 0 F N 2 N 2 / 3 For T > 0? For (2) and (3)? Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Outline (1) SSK model H ( σ ) = 1 2 σ T M σ (i) Fluctuation results (ii) History (iii) Random single integral formula (iv) Linear statistics vs largest eigenvalue (2) SSK+CW (3) SSK+external field Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Fluctuations of free energy Theorem [Baik and Lee 2016] For T < 1, � � 1 − 3 T 4 + T log T + 1 − T D ≃ F N 2 N 2 / 3 TW 1 2 For T > 1, 4 T + T 1 D F N ≃ 2 N N ( − α, 4 α ) where α = − 1 2 log(1 − T − 2 ) T = 1 is an open problem Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

History: limiting free energy SSK: Kosterlitz, Thouless, Jones (1976), Guionnet and Ma¨ ıda (2005), Panchenko and Talagrand (2007) General spin glass: Parisi formula (1980), Crisanti and Sommers formula (1992) Guerra (2003), Talagrand (2006), Panchenko (2014) Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

History: fluctuations SK for high temperature , T > 1: Gaussian, N − 1 [Aizenmann, Lebowitz, Ruelle 1987], [ Fr¨ ohlich and Zegarli´ nski 1987], [Comets and Neveu 1995] pure p -spin spin glass high temperature : Gaussian, N − p / 2 [Bovier, Kurkova, and L¨ owe 2002] pure p -spin Spherical spin glass with p ≥ 3 zero temperature , T = 0: Gumbel N − 1 [Subag and Zeitouni 2017] (spherical) spin glass with external field for all temperatures , T > 0: Gaussian N − 1 / 2 [Chen, Dey, Parchenko 2017] No phase transition! Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Random single integral formula Lemma [Kosterlitz, Thouless, Jones 1976] � γ + i ∞ N G ( z ) = β z − 1 N 2 G ( z ) dz , � log( z − λ k ) Z N = C N e with γ > λ 1 N γ − i ∞ k =1 N e βσ T M σ d Ω( σ ) = i λ i u 2 N e β � i d Ω( u ) Proof: By definition, Z N = � � √ √ � σ � = � u � = i λ i u 2 Let f ( r ) = r N / 2 − 1 � � u � =1 e r � i d ω ( u ) R N e − z � y 2 i + � λ i y 2 � ∞ e − zr f ( r ) d r = i d N y � Laplace transform L ( z ) = 0 � By Gaussian integral, L ( z ) = � N π i =1 z − λ i 1 � e rz L ( z ) d z Inverse Laplace transform f ( r ) = 2 π i Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Does the method of steepest-descent apply to random integrals? Yes, thanks to Rigidity of eigenvalues [Erd¨ os, Yau and Yin (2012)] | λ k − γ k | ≤ ˆ k − 1 / 3 N − 2 / 3+ ǫ uniformly for 1 ≤ k ≤ N with high probability where ˆ k = min { k , N + 1 − k } and γ k is the classical location (i.e. quantile of √ � 2 4 − x 2 d x = k the semicircle law), γ k 2 π N Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Critical point of the random function G ( z ) N G ′ ( z ) = β − 1 1 � z − λ k , Re( z ) > λ 1 N k =1 � d σ sc ( x ) ≃ 0 implies that z c ≃ β + 1 For β < 1, β − z c − x β For β > 1, z c = λ 1 + O ( N − 1+ ǫ ) with high probability Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

High temperature regime β < 1 We have, with z c = β + 1 β , N G ( z c ) = β z c − 1 � log( z c − λ k ) N k =1 For T > 1, a linear statistic gives fluctuations: 4 T + T 1 � � log(1 − T − 2 ) − L N + O ( N − 2+ ǫ ) F N = 2 N with high probability where � 2 N � L N = g ( λ i ) − N g ( x ) d σ sc ( x ) − 2 i =1 T + T − 1 − x with g ( x ) = 1 � � 2 log Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Low temperature regime β > 1 / 2 The critical point is close to a branch point �� N � 2 G ( z ) dz ≃ N It still holds that log 2 G ( z c ) e Using z c = λ 1 + O ( N − 1+ ǫ ) and noting λ 1 = 2 + O ( N − 2 / 3+ ǫ ) N G ( z c ) = β z c − 1 log( z c − λ i ) − 1 � N log( z c − λ 1 ) N i =2 N � � ≃ βλ 1 − 1 1 � log(2 − λ i ) + 2 − λ i ( λ 1 − 2) N i =2 � 2 � 2 d σ sc ( s ) ≃ βλ 1 − log(2 − s ) d σ sc ( s ) − ( λ 1 − 2) 2 − s − 2 − 2 For T < 1, the largest eigenvalue gives the fluctuations: � 1 − 3 T 4 + T log T � + 1 − T ( λ 1 − 2) + O ( N − 1+ ǫ ) F N = 2 2 with high probability Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Outline (1) SSK model (2) SSK+CW (Curie-Weiss) N H ( σ ) = 1 2 σ T M σ + m σ i σ j = 1 M + m 2 σ T � N 11 T � � σ 2 N i , j =1 (3) SSK+external field Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

SSK+CW Random symmetric matrix with non-zero mean (spiked random matrix) Limiting free energy was obtained by [Kosterlitz-Thouless-Jones 1976] Fluctuations including Spin–Ferro ( m = 1 + aN − 2 / 3 ) [Baik-Lee 2017] Para–Ferro m = T + bN − 1 / 2 [Baik-Lee-Wu 2018] T linear statistics Paramagnetic N − 1 Gaussian 1 λ 1 λ 1 N − 1 / 2 Spin glass Ferromagnetic N − 2 / 3 Gaussian TW 1 m 0 1 Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

(1) SSK model (2) SSK + CW (3) SSK + external field (a) Free energy (b) Spin distribution Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

SSK + external field H ( σ ) = 1 2 σ T M σ + h σ T g g = ( g 1 , · · · , g N ) is a standard normal vector h is a coupling constant (strength of the external field) (Chen, Dey, Panchenko 2017) Gaussian N − 1 / 2 for all T > 0 if h > 0 On the other hand, if h = 0, there is a transition at T = 1 (Fyodorov and le Doussal 2014) For T = 0, the number of local max/min of H ( σ ) has a transition when h = O ( N − 1 / 6 ) Goal: Recover [CDP] result and study the free energy when h = HN − 1 / 6 Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Random integral formula Let u i be a unit eigenvector associated to λ i . We have the random integral formula with N N log( z − λ i ) + h 2 β n 2 G ( z ) = β z − 1 � � n i = u T i z − λ i , i g N N i =1 i =1 For h > 0 and every β > 0, � d σ sc ( x ) � d σ sc ( x ) G ′ ( z ) ≃ β − − h 2 β z − x ( z − x ) 2 has the unique root z c > 2 since G ′ (2) = −∞ and G ′ ( ∞ ) = β > 0. Insert this z c to G ( z ) and consider N N N n 2 n 2 1 i − 1 � � � i z c − λ i = z c − λ i + z c − λ i i =1 i =1 i =1 The first sum has fluctuations of O (1) from linear statistics . The second sum √ has fluctuations of O ( N ) by usual CLT . We recover [CDP] result. Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model

Conjecture [Baik, le Doussal, Wu 2019] For T < 1 and h = HN − 1 / 6 , + h 2 � � 1 − 3 T 4 + T log T F F N ≃ + N 2 / 3 2 2 with high probability Let { α i } be a GOE Airy point process ( α i ∼ − (3 π i / 2) 2 / 3 as i → ∞ ) and let { ν i } be independent standard normal random variables. Let s > 0 be the solution of the equation ∞ ν 2 1 − T � i = H 2 s + α 1 − α i i =1 Set (cf. [Landon and Sosoe 2019]) � ( 3 π n � n 2 ) 2 / 3 � ν 2 d x � i E ( s ) = lim s + α 1 − α i − √ x n →∞ 0 i =1 Then, = (1 − T )( s + α 1 ) + H 2 E ( s ) D F 2 Jinho Baik University of Michigan Spherical Sherrington–Kirkpatrick model