Overlap distribution in the Spherical Sherrington-Kirkpatrick model - PowerPoint PPT Presentation

Overlap distribution in the Spherical Sherrington-Kirkpatrick model with V.-L. Nguyen, Benjamin Landon Cornell University and Massachussetts Institute of Technology May 10th, 2019

The model 2-spin Spherical Sherrington-Kirkpatrick model with zero magnetic field 1 � S N − 1 e − βH N ( σ ) d ω N ( σ ) , Z N ( β ) = | S N − 1 | Spins: √ σ ∈ S N − 1 := { σ ∈ R N , | σ | = N } . Hamiltonian: 1 � √ H N ( σ ) = − g ij σ i σ j . 2 N 1 ≤ i � = j ≤ N Disorder: g ij , 1 ≤ i � = j ≤ N i.i.d. Gaussian.

Interpretation: “soft maximization” “Low temperature” limit β → ∞ of free energy density is the top eigenvalue of GOE minus diagonal: 1 βN log Z N ( β ) → max | σ | =1 � σ, Mσ � , M ij = g ij , M ii = 0 . SSK model partition function is a positive temperature version of the top eigenvalue.

Model features 1 � H N ( σ ) = − √ g ij σ i σ j . 2 N 1 ≤ i � = j ≤ N Mean field model: all spins interact, no geometry, like the Curie-Weiss model (easier than lattice) Disordered model: interactions are random, with no sign (much harder than ferromagnetic models)

History Introduced by Kosterlitz, Thouless and Jones (1976) after the original spin glass model with Ising spins σ i ∈ {± 1 } by Sherrington and Kirkpatrick (1975). More explicit computations are possible than for Sherrington-Kirkpatrick (SK) model. KTJ compute the limit of the partition function lim N →∞ 1 N log Z N ( β ). The corresponding computation for SK is much harder. Sherrington-Kirkpatrick (1975) for high-temperature phase ( β small). For all β Parisi (2003), Guerra, Talagrand (2008), Panchenko (2011).

p -spin models Both SK ( σ ∈ {± 1 } ) and SSK ( σ ∈ S N − 1 ) can be generalized to p -spin models: 1 � H N ( σ ) = g i 1 ,...,i p σ i 1 · · · σ i p . √ qN ( p − 1) / 2 1 ≤ i 1 <...<i p ≤ N Crisanti-Sommers (1992) give a variational formula for the free energy in this case. Proved rigorously by Talagrand (2006) and Chen (2013).

Connection to spiked models Spiked Wigner model: � λ N x ∗ ( x ∗ ) t . Y = M + M : traceless GOE x ∗ : N × 1 vector from a distribution P (d x ). Can the spike be detected? Density of Y : � � − 1 λ � P λ (d Y ) = Z − 1 � N x i x j ) 2 � exp ( Y ij − P (d x ) 2 i<j

Connection to spiked models Likelihood ratio: �� L ( λ ) = P λ � λ � � x 2 i x 2 � = exp Y ij x i x j − λ P (d x ) . j P 0 N i<j i<j If x is uniform on the sphere, this is the partition function of the SSK model. Likelihood ratio test: compare L ( λ ) to threshold and reject λ = 0 if L ( λ ) > c . SSK phase transition corresponds to detectability of spike: if signal is too small (high temperature), likelihood ratio test fails. (Onatski, Moreira, Hamlin (2014), Johnstone, Onatski (2015)).

SSK Kosterlitz-Thouless-Jones: � β 2 1 β < 1 2 lim N log Z N ( β ) := F ( β ) = . (1) 2( β 2 − 3 2 − log 2) N →∞ β > 1 . Phase transition at β = 1. For σ ∈ {± 1 } , limit is given by an infinite dimensional variational formula (Parisi functional).

Spiked model For the spiked model with i.i.d. prior x ∗ , Lelarge and Miolane (2016) prove: 1 N E λ [log L ( λ )] → φ ( λ ) = sup F ( λ, q ) , q ≥ 0 where F ( λ, q ) = ψ ( λq ) − λq 2 4 , exp( √ rzx + rxx ∗ − rxx ∗ − r � 2 x 2 ) d P (d x )] . ψ ( r ) = E [log Hre z ∼ N (0 , 1) and x ∗ ∼ P (d x ).

Fluctuations: high temperature Aizenman-Lebowitz-Ruelle (1987): central limit theorem for the partition function for Ising spins σ ∈ ± 1 in the low temperature phase: log Z N − N (log 2+ β 2 (1 4(log(1 − β 2 )+ β 2 ) , − 1 � 2(log(1 − β 2 )+ β 2 ) � 2 ) → N . Stein’s method approach: Talagrand (2011). May be applied to spherical spin glasses. For general spherial p -spin model, Chen and Sen prove that the fluctuations of the partition function are Gaussian in the high temperature phase.

Baik-Lee results Baik and Lee (2015) find the fluctuations of the spherical model for any (off-critical) temperature: Theorem For the SSK model, when β < 1 (1 4(log(1 − β 2 ) + β 2 ) , − 1 2(log(1 − β 2 ) + β 2 ) � � log Z N − NF ( β ) → N When β > 1 , N 2 / 3 β − 1( 1 N log Z N − F ( β )) → TW GOE .

In contrast with this, for σ ∈ {± 1 } (Ising spins), we have almost no information about the fluctuations in the low temperature phase. Chatterjee (2009) shows Var(log Z N ) ≤ C ( β ) N log N for any temperature for the SK model with Ising spins. For p -spin models with p ≥ 3, Subag (2016) shows that the free energy fluctuations are O (1) (tight) in the low temperature phase.

Overlaps Phase transition occurs also in the geometry of spins. Consider σ (1) , σ (2) sampled from Gibbs measure µ β ( σ ) = e − βH ( σ ) Z N ( β ) . σ ( i ) are “near minimizers” of H ( σ ), called replicas. Overlap: inner product between replicas σ (1) , σ (2) . R 1 , 2 = 1 N σ (1) · σ (2) N = 1 � σ (1) σ (2) . i i N i =1

Overlap in spiked model The overlap R 1 , 2 in the spiked model corresponds to the overlap of a sample x from the posterior distribution P (d x | M ) , with the spike x ∗ (Nishimori identity). = 1 d N x · x ∗ . R 1 , 2

Overlap asymptotics Panchenko and Talagrand (2003) proved � 0 , β < 1 E [ � R 2 1 , 2 � ] → , q 2 , β > 1 for q = β − 1 . β They showed that overlaps concentrate around two values ± q : E [ �| R 2 12 − q 2 | ≥ ǫ � ] → 0 . �·� : Gibbs average.

Spiked model For the spiked model with i.i.d. prior x ∗ and x sampled from P (d x | M ), Lelarge-Miolane show that 1 N ( x ∗ x t ) 2 → q ∗ ( λ ) , where q ∗ is the maximizer in the limiting free energy density. “Reconstruction threshold” has the equivalent definitions: λ c = sup { λ > 0 : q ∗ ( λ ) = 0 } = sup { λ > 0 : φ ( λ ) = 0 } . Observation (El Alaoui, Krzkala, Jordan): λ c ( E [ x 2 ]) 2 ≤ 1 .

λ = 1 / ( E [ x 2 ]) 2 is the spectral threshold where λ 1 ( Y ) leaves the bulk and the first eigenvector is uniformative about x ∗ (P´ ech´ e, Capitaine, Feral, 200x, Benaych-Georges and Nadakuditi, 2011). For some distributions like sparse Rademacher, the inequality λ c < 1 /E [ x 2 ] 2 is strict. There is a conjectural “hardness” phase transition between the spectral threshold and λ c .

Results: high temperature We describe the fluctuations of the overlaps. Theorem (Nguyen-S., 2018) For β = β N = 1 − cN − 1 / 3+ τ , τ > 0 , √ N tR 12 � β,N = e t 2 + o (1) N (1 − β 2 � e with very high probability. The overlap remains Gaussian close to the critical temperature (but no closer than O ( N − 1 / 3+ ). � N (1 − β 2 N R 12 ) p � ] are Gaussian up to Talagrand: moments E [ � ( β N ≪ 1 − N − 1 / 3 , not Gaussian for β N = 1 − cN − 1 / 3 .

Results: low temperature With Benjamin Landon: describe the annealed fluctuations of R 1 , 2 for β > 1 fixed. We give expansions up to o ( N − 2 / 3 ), and identify the limiting distribution of | R 1 , 2 | . ( � R 12 � = 0 by symmetry of the Gibbs measure.)

Results: low temperature Theorem (Landon, S. 2019) Asymptotically with probability 1:   N � 2 � 1 − β + 2 β − 1  1 1 � � R 2 12 � = · λ j ( M ) − λ 1 ( M ) + 1  β β 2 N j =2   N 1  1 1 � −  Nβ 2 N ( λ j ( M ) − λ 1 ( M )) 2 j =2 2   N + 1  1 1 � λ j ( M ) − λ 1 ( M ) + 1  β 2 N j =2 + O ( N − 1+ ) .

Results: low temperature Theorem (Landon, S. 2019) Asymptotically with probability 1:   N �| R 12 |� = q + 1  1 1 � β · λ j ( M ) − λ 1 ( M ) + 1  N j =2 + O ( N − 1+ ) . Similar formulas were found using physics methods by Baik, Le Doussal and Wu. They also have precise predictions for the case with non-zero magnetic field.

Orders of magnitude By level repulsion λ 1 − λ 2 ≥ N − 2 / 3+ ǫ , and rigidity, we have N 1 1 � λ j ( M ) − λ 1 ( M ) + 1 = O ( N − 1 / 3+ ) , N j =2 and N 1 1 � ( λ j − λ 1 ) 2 = O ( N − 2 / 3+ ) . N 2 j =2 From the first term, we might expect tightness of N 1 / 3 ( � R 2 12 � − q 2 ), N 1 / 3 ( �| R 12 |� − q ).

“Renormalized” Airy series The term N 1 m N ( λ 1 ) = 1 1 � λ j ( M ) − λ 1 ( M ) − 1 ∼ N − 1 / 3 N ˜ N j =2 will be approximated by the Airy process. By rigidity: λ j − λ 1 ∼ ( j/N ) 2 / 3 . Let χ 1 ≥ χ 2 ≥ . . . be the points of an Airy GOE process. Define:   � ( 3 πn 2 ) 2 / 3 n 1 1 �  . π √ xdx Ξ := lim −  χ j − χ 1 n →∞ 0 j =2 Landon-S. (2019): The limit exists almost surely.

Distributional limits Theorem (Landon, S., 2019) The following limits hold in distribution: � β − 1 � N 1 / 3 � � R 2 12 � − q 2 � → 2 Ξ , β 2 and �| R 12 |� − 1 − β → 1 N 1 / 3 � � β Ξ . β Once the existence of Ξ is established, this means we need to show   N  1 1 � N 1 / 3 λ j ( M ) − λ 1 ( M ) − 1  N j =2 has the same limit.