Mathematical Statistical Physics, YITP Kyoto, July 30th, 2013 RIGOROUS RESULTS ON SHORT-RANGE FINITE-DIMENSIONAL SPIN GLASSES Pierluigi Contucci Department of Mathematics University of Bologna 1
Consider configurations of N Ising spins σ = { σ i } , τ = { τ i } , ... , introduce a centered Gaussian Hamiltonian H N ( σ ) de- fined by the covariance Av( H N ( σ ) H N ( τ )) = Nc N ( σ , τ ) . Examples of covariances: Sherrington-Kirkpatrick and Edwards-Anderson model 2 0 1 @ 1 1 X X σ i τ i , σ i σ j τ i τ j A N N i | i − j | =1 2
We are interested in the large volume properties of the random probability measure e − β H N ( σ ) p N ( σ ) = σ e − β H N ( σ ) P for all β > 0. Quantities of interest include: the pressure e − β H N ( σ ) X P N ( β ) = Av log σ the covariance moments 0 1 σ , τ c N ( σ , τ ) e − β [ H N ( σ )+ H N ( τ )] P A = Av @ σ , τ e − β [ H N ( σ )+ H N ( τ )] P 3
Z = < c > N = cp N ( c ) dc , Z c 12 c 23 p (12) , (23) < c 12 c 23 > N = ( c 12 , c 23 ) , N and especially the joint distribution (permutation invari- ant) p (12) , (23) ,..., ( kl ) ,... ( c 12 , c 23 , ..., c kl , ... ) N The joint distribution allows to compute internal en- ergy, specify heat, etc. What happens when N → ∞ ? 4
The mean-field theory (Replica Symmetry Breaking) is characterised by two properties: • p ( c ) has a non-trivial support • the joint distribution p (12) , (23) ,..., ( kl ) ,... ( c 12 , c 23 , ..., c kl , ... ) fulfils a factorisation property and can be recon- structed starting from p ( c ) through the ultrametric and replica equivalence rule. 5
Ultrametricity: p (12) , (23) , (31) ( c 12 , c 23 , c 31 ) = Z c 12 δ ( c 12 − c 23 ) δ ( c 23 − c 31 ) p ( c 12 ) p ( c ) dc 0 + θ ( c 12 − c 23 ) δ ( c 23 − c 31 ) p ( c 12 ) p ( c 23 ) + θ ( c 23 − c 31 ) δ ( c 31 − c 12 ) p ( c 23 ) p ( c 31 ) + θ ( c 31 − c 12 ) δ ( c 12 − c 23 ) p ( c 31 ) p ( c 12 ) no scalene triangles! 6
Replica Equivalence (Ghirlanda-Guerra) p (12) , (23) ( c 12 , c 23 ) = 1 2 p ( c 12 ) δ ( c 12 − c 23 ) + 1 2 p ( c 12 ) p ( c 23 ) p (12) , (34) ( c 12 , c 34 ) = 1 3 p ( c 12 ) δ ( c 12 − c 34 ) + 2 3 p ( c 12 ) p ( c 34 ) 7
Rigorous results on factorisation, mean-field models: • 1997, M. Aizenman, P.C. (stochastic stability), 1998 F.Guerra, S.Ghirlanda (based on Pastur-Scherbina) • 2005-2007, P.C. and C.Giardina (results for the first power moments, but hold also in finite-dimension short-range!), M. Talagrand (results in distribution, but only for the SK model) * 2011 D.Panchenko proved that Ghirlanda-Guerra identities in distribution imply ultrametricity (proof by contradiction, geometrical methods) 8
Result : Edwards-Anderson, and a wide class of finite-dimensional models, is ultrametric! P.C., E.Mingione, S.Starr [JSP, 2013] No claim on p ( c ) is made 9
More precisely : consider, in a box Λ ⊂ Z d , the model defined by the Hamiltonian X H Λ ( σ ) = J Λ ,X σ X X ⊆ Λ if (thermodynamic stability) Av ( H Λ ( σ ) H Λ ( σ )) ≤ cN then for all power p , any number of system copies n , any bounded measurable function f of the { c l,m } n l,m =1 in the thermodynamic limit it holds: n 1 1 < fc p < fc p n + 1 < f >< c p X n +1 ,n +1 > = k,n +1 > + 1 , 2 > , n + 1 k =1 and (by Panchenko result) is ultrametric. 10
Main idea : • Stochastic Stability and its extension (P.C., C.Giardina, C.Giberti, EPL 2011) Consider the quenched equilibrium state σ F ( σ ) e − β H N P ! < F > N = Av σ e − β H N P and the smooth deformation g of the Hamiltonian den- sity h : = < Fe − λ g ( h ) > < F > ( λ ) N < e − λ g ( h ) > 11
The spin glass quenched equilibrium state is stochasti- cally stable: < F > ( λ ) → < F > N (check that the perturbation doesn’t spoil thermody- namic stability). This implies bounds on thermal and disorder fluctuation Av( ω ( H 2 )) − Av( ω ( H ) 2 ) ≤ c 1 N Av( ω ( H ) 2 ) − Av( ω ( H )) 2 ≤ c 2 N 12
What’s next? • Overlap Equivalence: V N ( Q | q ) → 0 numerically seen in PRL 2006 by P.C., Cristian Gi- ardina, Claudio Giberti, Cecilia Vernia • . • ... • Triviality? Study P ( Q ) 13
Recommend
More recommend
