Some Observations for Mean-Field Spin Glass Models
aa r X i v : . [ m a t h - ph ] D ec SOME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS
SHANNON STARR AND BRIGITTA VERMESI
Abstract.
We obtain bounds to show that the pressure of a two-body, mean-field spin glass is aLipschitz function of the underlying distribution of the random coupling constants, with respect toa particular semi-norm. This allows us to re-derive a result of Carmona and Hu, on the universalityof the SK model, by a different proof, and to generalize this result to the Viana-Bray model. Wealso prove another bound, suitable when the coupling constants are not independent, which is whatis necessary if one wants to consider “canonical” instead of “grand canonical” versions of the SKand Viana-Bray models. Finally, we review Viana-Bray type models, using the language of L´evyprocesses, which is natural in this context. Continuity of pressure with respect to the coupling distribution
Let us consider a mean-field spin glass Hamiltonian of the form − H N ( σ ) = N X i,j =1 J N ( i, j ) σ i σ j + h N X i =1 σ i , where the coefficients J N ( i, j ) are i.i.d. random variables, chosen from some distribution represented bya cumulative distribution function, F N , and h ∈ R is nonrandom. The spins themselves are assumedto be Ising type spins, so that σ = ( σ , . . . , σ N ) lives in Ω N = { +1 , − } N . This general definitionencompasses the SK model, where F N is the c.d.f., for a N (0 , β / N ) random variable. That is, allthe J N ( i, j )’s are normal (Gaussian) random variables with mean 0 and variance 1 / N . This definitionalso includes a nice version of a Viana-Bray type model, wherein F N is the c.d.f. for a PoissonizedGaussian random variable. More specifically, for each i and j , we have J ( i, j ) = K ( i,j ) X k =1 g k ( i, j ) , where K ( i, j ) is a Poisson random variable, with mean α/N , and all the random variables g k ( i, j ), for k = 1 , , . . . , are i.i.d. N (0 , β ) random variables.Note that we write J N ( i, j ) and H N ( σ ) using the sans serif font to indicate that they are randomvariables. As usual, we also define the random partition function, Z N = X σ ∈ Ω N e − H N ( σ ) , and the “pressure”, p N = 1 N ln( Z N ) . This random variable is not actually the pressure, which is only defined in the thermodynamic limit.But we will call it the “random pressure”, anyway. Let us write p N ( F N ) = E [ p N ] , which is the definition of the “quenched pressure”. Note that at this point we explicitly denote theunderlying distribution F N for the coupling constants ( J N ( i, j ) : 1 ≤ i, j ≤ N ). This is because p N ( F N ) is not random, but a function of F N .A very basic, but important bound is the following. Date : November 8, 2018.
Lemma 1.1.
Let w , w , . . . , w K ≥ be given, and suppose w + · · · + w K ≥ . Also, suppose s , . . . , s K ∈ [ − , are given. Define a function f : R → R by f ( x ) = ln K X k =1 w k e xs k ! . Then f is globally Lipschitz, with Lipschitz constant 1. Proof:
Note that f ′ ( x ) = P Kk =1 w k e xs k s k P Kk =1 w k e xs k = K X k =1 θ k s k , where θ k = w k e xs k P Kℓ =1 w ℓ e xs ℓ for k = 1 , . . . , K . So f ′ ( x ) is in the convex hull of { s , . . . , s K } , which is a subset of [ − ,
1] by our assumption. So | f ′ ( x ) | ≤ x , which proves the claim. (cid:3) By the lemma, we see that p N is a jointly globally Lipschitz function of the random couplingconstants ( J N ( i, j ) : 1 ≤ i, j ≤ N ). Let F denote the set of all c.d.f.’s F which have finite firstmoment. Then, from the above we see that, if F N ∈ F , then p N also has a finite first moment.Therefore, p N : F → R is a well-defined function. Part of our goal is to understand the continuityproperties of p N . Another goal is to derive useful inequalities for the study of “real” mean-field spinglass models, such as the SK and VB models.1.1. Continuity of p N . Given F ∈ F , let us define a ( F ) = Z ∞−∞ ln(cosh( x )) dF ( x ) , and for k = 1 , , . . . , define a k ( F ) = 1 k Z ∞−∞ tanh k ( x ) dF ( x ) . Proposition 1.2.
Suppose F N , ˜ F N ∈ F . Then | p N ( F N ) − p N ( ˜ F ) N | ≤ N ∞ X k =0 | a k ( F N ) − a k ( ˜ F N ) | . Proof:
Enumerate all N pairs ( i, j ) in any way, as ( i , j ) , . . . , ( i N , j N ). Then the two Hamiltoniansassociated to F N and ˜ F N are − H N ( σ ) = N X n =1 J N ( i n , j n ) σ i n σ j n + h N X i =1 σ i and − ˜ H N ( σ ) = N X n =1 ˜ J N ( i n , j n ) σ i n σ j n + h N X i =1 σ i . For n = 1 , . . . , N define w n ( σ ) = exp n − X k =1 ˜ J N ( i k , j k ) σ i k σ j k + N X k = n +1 J N ( i k , j k ) σ i k σ j k + h N X i =1 σ i . Then, by a telescoping sum, p N − ˜ p N = 1 N N X n =1 " ln X σ ∈ Ω N w n ( σ ) e J N ( i n ,j n ) σ in σ jn ! − ln X σ ∈ Ω N w n ( σ ) e ˜ J N ( i n ,j n ) σ in σ jn ! =: 1 N N X n =1 d n . (1) OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 3
Let us define ˆ w n ( σ ) = w n ( σ ) / P σ ∈ Ω N w n ( σ ). Then we can rewrite d n = ln X σ ∈ Ω N ˆ w n ( σ ) e J N ( i n ,j n ) σ in σ jn ! − ln X σ ∈ Ω N ˆ w n ( σ ) e ˜ J N ( i n ,j n ) σ in σ jn ! , since the normalizing multipliers cancel in the difference of the logarithms. Also, let us define h σ i n σ j n i n = X σ ∈ Ω N ˆ w n ( σ ) σ i n σ j n . Then, since e xσ = cosh( x ) + σ sinh( x ) for ± σ , e J N ( i n ,j n ) σ in σ jn = cosh( J N ( i n , j n )) [1 + σ i n σ j n tanh( J N ( i n , j n ))] ⇒ ln X σ ∈ Ω N ˆ w n ( σ ) e J N ( i n ,j n ) σ in σ jn ! = ln[cosh( J N ( i n , j n ))] + ln [1 + tanh( J N ( i n , j n )) h σ i n σ j n i n ] . Since | tanh( J N ( i n , j n )) h σ i n σ j n i n | <
1, we can use the Taylor expansion of ln(1 + x ):ln(1 + x ) = − ∞ X k =1 ( − k x k k . Therefore, defining φ ( x ) = ln(cosh( x )) and φ k ( x ) = tanh k ( x ) for k = 1 , , . . . , we have d n = φ ( J N ( i n , j n )) − φ (˜ J N ( i n , j n )) − ∞ X k =1 ( − k ( h σ i n σ j n i n ) k k h φ k ( J N ( i n , j n )) − φ k (˜ J N ( i n , j n )) i . Importantly, by our assumption, J N ( i n , j n ) and ˜ J N ( i n , j n ) are independent of all J N ( i k , j k ) aand˜ J N ( i k , j k ) for k = n . Therefore, J N ( i n , j n ) and ˜ J N ( i n , j n ) are both independent of h σ i n σ j n i n . Also,obviously, E [ φ k ( J )] = a k ( F ) for an F -distributed random variable J . E [ d n ] = a ( F ) − a ( ˜ F ) − ∞ X k =1 ( − k E h ( h σ i n σ j n i n ) k i [ a k ( F ) − a k ( ˜ F )] . Since, |h σ i n σ j n i n | < | E [ d n ] | ≤ ∞ X k =0 | a k ( F N ) − a k ( ˜ F N ) | . This is true for all 1 ≤ n ≤ N . Plugging into (1), we obtain the result. (cid:3) Re-derivation of a result of Carmona and Hu.
In an important paper, Carmona and Huproved universality for the Sherrington-Kirkpatrick model [2]. Using the proposition, we can re-derivepart of their results, in fact with slightly weaker hypotheses. Suppose F is a cumulative distributionfunction such that(2) µ ( F ) := Z ∞−∞ x dF ( x ) = β / , and(3) µ ( F ) := Z ∞−∞ x dF ( x ) = 0 . In other words, letting J be distributed according to F , we have E [ J ] = 0 and Var( J ) = β /
2. Thec.d.f. of J / √ N , then, is F N ( x ) = F ( √ N x ). Let us write p N ( F N ) just to make explicit the dependenceof p N on the distribution of all the random coupling constants ( J N ( i, j ) : 1 ≤ i, j ≤ N ). OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 4
Let F ∗ β be the c.d.f. for an N (0 , β /
2) random variable and F ∗ N,β ( x ) = F ∗ β ( √ N x ). Then, defining p SK N ( β ) := p N ( F ∗ N,β ), this is the quenched pressure for the SK model at inverse temperature β . By animportant result of Guerra and Toninelli, the thermodynamic limit, p SK ( β ) := lim N →∞ p SK N ( β ), exists.Among other important results, Carmona and Hu proved the following result, however with slightlydifferent hypotheses: Corollary 1.3.
Suppose F satisfies (2) and (3), and F N ( x ) = F ( √ N x ) for all N > . Then p N ( F N ) → p SK ( β ) , as N → ∞ , almost surely, and in probability. This is a non-quantitative version of Carmona and Hu’s Theorem 1. They obtained explicit boundson | p N ( F N ) − p SK N ( β ) | . On the other hand, to do so, they assumed that F also has a finite thirdmoment, which is slightly stronger than our hypotheses. (But without such an assumption, it isimpossible to get quantitative bounds.) We will re-prove Corollary 1.3 in order to demonstrate thatit can be derived from Proposition 1.2, rather than from Carmona and Hu’s alternative, and veryinteresting, approach of “approximate Gaussian integration by parts”. The reason we do this is thatlater we will use Proposition 1.2 to prove an analogous result for the Viana-Bray model. (Incidentally,Carmona and Hu’s result was extended to quantum spin glasses by Crawford in [3]. We have notconsidered whether a similar generalization is possible for Proposition 1.2.)We can prove convergence in mean, easily, using the proposition. But then convergence in probabilityfollows trivially using an important and well-known result of Pastur and Shcherbina. In an importantpaper, they proved that the spin-spin overlap order parameter of the SK model is not self-averagingin a certain regime [7]. They also proved the much simpler, but even more widely cited, result in anappendix, that the random pressure is self-averaging. It is that second result which we now state, andfor completeness prove, in the context of general mean-field spin glasses. Lemma 1.4.
Let F N satisfy µ ( F N ) < ∞ . Let p N denote the random pressure associated to thedistribution of the coupling constants with c.d.f. F N . Then Var( p N ) ≤ σ ( F N ) = µ ( F N ) − µ ( F N ) . Proof of Lemma 1.4:
The proof follows the standard martingale method. Again, enumerate all N pairs ( i, j ) as ( i , j ) , . . . , ( i N , j N ). Let F n = σ ( J ( i k , j k ) : k ≤ n ) be the σ -algebra generated by J ( i k , j k ) for k ≤ n . Let F be the trivial σ -algebra. For n = 0 , , . . . , N , define M n = E [ p N | F n ].Note that p N = M N while E [ p N ] = M . Therefore, by a telescoping sum,(4) Var( p N ) = N X n =1 E [ M n − M n − ] . In addition to the coupling constants J N ( i, j ) distributed according to F N , let us define additionalindependent copies of these random variables, called J N ( i, j ) and J N ( i, j ). Define, for α, β ∈ { , } , − H α,βN,n ( σ ) = n − X k =1 J N ( i k , j k ) σ i k σ j k + J αN ( i n , j n ) σ i n σ j n + N X k = n +1 J βN ( i k , j k ) σ i k σ j k + h N X i =1 σ i . Also define p α,βN,n = 1 N ln X σ ∈ Ω N exp( − H α,βN,n ( σ )) ! . Then E [ M n − M n − ] = 12 E [( p , N,n − p , N,n )( p , N,n − p , N,n )] . Finally, we want to bound this. So, define − H ∅ ,βN,n ( σ ) = n − X k =1 J N ( i k , j k ) σ i k σ j k + N X k = n +1 J βN ( i k , j k ) σ i k σ j k + h N X i =1 σ i . OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 5
Then, for β = 1 , | p ,βN,n − p ,βN,n | = 1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln X σ ∈ Ω N e − H ∅ ,βN,n ( σ ) e J N ( i n ,j n ) σ in σ jn ! − ln X σ ∈ Ω N e − H ∅ ,βN,n ( σ ) e J N ( i n ,j n ) σ in σ jn !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . By Lemma 1.1, the right hand side is bounded above by N − | J N ( i n , j n ) − J N ( i n , j n ) | . Therefore, E [ M n − M n − ] ≤ N E [ | J N ( i n , j n ) − J N ( i n , j n ) | ] = σ ( F N ) N . Using this bound for all 1 ≤ n ≤ N , and plugging in to (4) gives the result. (cid:3) Proof of Corollary 1.3:
By the triangle inequality and Proposition 1.2, we can bound | p N ( F N ) − p SK N ( β ) | ≤ δ N ( F N ) + δ N ( F ∗ N,β ) , where δ N ( F N ) = (cid:12)(cid:12)(cid:12)(cid:12) N a ( F N ) − β (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) N a ( F N ) − β (cid:12)(cid:12)(cid:12)(cid:12) + N ∞ X k =1 k =2 | a k ( F N ) | . We will show that δ N ( F N ) →
0, as N → ∞ . Since F ∗ β also satisfies (2) and (3), this also implies δ N ( F ∗ N,β ) → N → ∞ . Note that β − N a ( F N ) = Z ∞−∞ x dF ( x ) − Z ∞−∞ N ln(cosh( x )) dF N ( x )= Z ∞−∞ (cid:18) x − N ln (cid:18) cosh (cid:18) x √ N (cid:19)(cid:19)(cid:19) dF ( x ) . Note that, since ln(cosh( x )) ≤ x /
2, the integrand is nonnegative. Similarly, β − N a ( F N ) = Z ∞−∞ (cid:18) x − N (cid:18) x √ N (cid:19)(cid:19) dF ( x ) , and the integrand is nonnegative because tanh ( x ) ≤ x . Also, notice that a k ( F N ) ≥ k = 0 , , . . . , because tanh k ( x ) ≥
0, and also ∞ X k =2 tanh k ( x )2 k = −
12 ln(1 − tanh ( x )) − tanh ( x ) = ln(cosh( x )) − tanh ( x )2 . Therefore, (cid:12)(cid:12)(cid:12)(cid:12)
N a ( F N ) − β (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) N a ( F N ) − β (cid:12)(cid:12)(cid:12)(cid:12) + N ∞ X k =2 | a k ( F N ) | = Z ∞−∞ (cid:16) x − N tanh (cid:16) x/ √ N (cid:17)(cid:17) dF ( x ) . By the dominated convergence theorem, the last quantity approaches 0 as N → ∞ . Note that, since µ ( F ) = 0, − N a ( F N ) = N ( µ ( F N ) − a ( F N )) = N Z ∞−∞ ( x − tanh( x )) dF N ( x )= N Z ∞−∞ (cid:18)Z x tanh ( y ) dy (cid:19) dF N ( x )= N Z ∞ tanh ( y ) Z | x |≥√ N y sgn( x ) dF ( x ) dy . So, N | a ( F N ) | ≤ Z ∞ tanh ( y ) y Z | x |≥√ N y x dF ( x ) dy . OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 6
Since µ ( F ) < ∞ , this quantity also approaches 0 as N → ∞ , by the dominated convergence theorem.Finally, ∞ X k =1 | a k +1 ( F N ) | = ∞ X k =1 k + 1 (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ tanh k +1 ( x ) dF N ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∞−∞ ∞ X k =1 | tanh k +1 ( x ) | k + 1 dF N ( x )= Z ∞−∞ (cid:18)
12 ln (cid:18) | tanh( x ) | − | tanh( x ) | (cid:19) − | tanh( x ) | (cid:19) dF N ( x )= Z ∞−∞ ( | x | − | tanh( x ) | ) dF N ( x ) . Therefore, the same argument as the one just above shows thatlim N →∞ N ∞ X k =1 | a k +1 ( F N ) | = 0 . This completes the proof that lim N →∞ δ N ( F N ) = 0. Therefore, it also shows that lim N →∞ | p N ( F N ) − p SK N ( β ) | = 0. Since lim N →∞ p SK N ( β ) = p SK ( β ), in order to complete the proof, all we need to show isthat p N ( F N ) − p N ( F N ) →
0, as N → ∞ , in probability. But, by Pastur and Shcherbina’s bound, E [( p N ( F N ) − p N ( F N )) ] ≤ µ ( F N ) = µ ( F ) N .
So, we have the even stronger result: p N ( F N ) − p N ( F N ) → L . (cid:3) An application to the Viana-Bray model.
Now, let us now reconsider Proposition 1.2 in thecontext of the Viana-Bray model. Define p VB N ( α, β ) = p N ( F ∗ N,α,β ), where F ∗ N,α,β = e − α/N ∞ X k =0 ( α/N ) k k ! ( F ∗ ,β ) ⋆k , in which ⋆ is the convolution product. Hence, defining K to be a Poisson-( α/N ) random variable, anddefining g , g , . . . to be i.i.d., N (0 ,
1) random variables, F ∗ N,α,β is the c.d.f. for the random variable J = K X k =1 g k . This is the “Poissonized Gaussian” coupling used in one version of the Viana-Bray model.
Corollary 1.5.
Suppose that, for each
N > , the sequence ( f N, , f N, , . . . ) is a probability massfunction, such that lim N →∞ N f N, = α and lim N →∞ N ∞ X k =2 kf N,k = 0 . For each
N > , define F N = ∞ X k =0 f N,k ( F ∗ ,β ) ⋆k . Then lim N →∞ | p N ( F N ) − p VB N ( α, β ) | = 0 . As a particular application, we could take f N,k = (1 − αN ) δ k, + αN δ k, . I.e., instead of taking K to bePoisson, with mean α/N , we could take it to be Bernoulli with the same mean. OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 7
Proof:
Note that ∞ X k =0 | a k ( F N ) − f N, a k ( F ∗ ,β ) | = Z ∞−∞ x )) [ dF N ( x ) − f N, dF ∗ ,β ( x )] ≤ Z ∞−∞ x ∞ X k =2 f N,k d ( F ∗ ,β ) ⋆k ( x )= β ∞ X k =2 k f N,k . Defining f ∗ N,k ( α ) = e − α/N ( α/N ) k /k !, we see that exactly the same property is true of it. But also, byassumption and calculation,lim N →∞ N ∞ X k =2 k f ∗ N,k ( α ) = lim N →∞ α (1 − e − α/N ) = 0 = lim N →∞ N ∞ X k =2 k f N,k . Therefore, lim sup N →∞ N ∞ X k =0 | a k ( F N ) − a k ( F ∗ N,α,β ) | = lim sup N →∞ N | f N, − f ∗ N, ( α ) | ∞ X k =0 a k ( F ∗ ,β ) ≤ β N →∞ N | f N, − f ∗ N, ( α ) | . But, of course, by assumption and calculationlim N →∞ N f ∗ N, ( α ) = lim N →∞ e − α/N α = α = lim N →∞ N f N, . Since a k ( F N ) = a k ( F ∗ N,α,β ) = 0, for odd k , because the Gaussian is symmetric, we then see thatlim N →∞ N ∞ X k =0 | a k ( F N ) − a k ( F ∗ N,α,β ) | = 0 , and we can apply Proposition 1.2. (cid:3) Also note that, again, Pastur and Shcherbina’s self-averaging bounds prove that p N − p N ( F N ) →
0, in L , as N → ∞ . Indeed, σ ( F N ) = ∞ X k =0 f N,k σ (( F ∗ ,β ) ⋆k ) = β ∞ X k =0 kf N,k , and the right-hand-side goes to 0, as N → ∞ , by hypothesis.1.3.1. Poisson thinning.
Let us briefly address one possible point of confusion. The Hamiltonian wewrote for the Viana-Bray model was − H N ( σ ) = N X i,j =1 K ( i,j ) X k =1 g k ( i, j ) σ i σ j + h N X i =1 σ i , where, for each ( i, j ) ∈ { , . . . , N } × { , . . . , N } the random variable K ( i, j ) is a Poisson randomvariable, with mean α/N , such that all the variables { K ( i, j ) : 1 ≤ i, j ≤ N } are independent, andall the random variables g k ( i, j ), for k = 1 , , . . . , are i.i.d. N (0 , β ) random variables, all of whichare independent, and independent of the K ( i, j )’s. This is not literally the model that is written down OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 8 in some references on the Viana-Bray model. Let us call the model we wrote above the
PoissonizedViana Bray model . The original Viana-Bray model considered by many authors is − ˜ H N ( σ ) = K X k =1 g k σ i k σ j k + h N X i =1 σ i , where K is a Poisson random variable, with mean αN , and i , i , . . . and j , j , . . . are i.i.d., randomvariables, uniformly distributed on the N sites { , . . . , N } . Also, g , g , . . . are i.i.d., N (0 , β ) randomvariables, independent of everything else.These two versions of the model are statistically equivalent. So all expectations of all functionsof the Hamiltonians are equal, including the quenched pressure. To see this, we use a well-knownproperty of Poisson random variables, which is commonly called “Poisson thinning”. We can constructa direct correspondence between the random variables of the first Hamiltonian and the second one.For instance, given K , i , i , . . . , j , j , . . . and g , g , . . . , do the following: First, let ˜ K ( i, j ) = { k : k ≤ K , i k = i , j k = j } for each 1 ≤ i, j ≤ N . Note that E exp N X i,j =1 λ i,j ˜ K ( i, j ) = E " exp K X k =1 λ i k , j k ! = E N − N X i,j =1 e λ i,j K = exp αN N − N X i =1 N X j =1 e λ i,j − = N Y i,j =1 exp (cid:16) αN [ e λ i,j − (cid:17) , which is exactly the joint moment generating function for i.i.d., Poisson random variables with mean α/N . Therefore, the random variables ˜ K ( i, j ) have identical joint distribution to K ( i, j ). Similarly,there is a way to construct ˜ g k ( i, j )’s from the g k ’s, by merely letting ˜ g k ( i, j ) equal g n k ( i,j ) where n k ( i, j ) is the k th smallest integer n such that ( i n , j n ) = ( i, j ). Using independence, it is trivial tosee that the collection of { ˜ g k ( i, j ) : 1 ≤ i, j ≤ N , k = 1 , , . . . } is equivalent to the collection of { g k ( i, j ) : 1 ≤ i, j ≤ N , k = 1 , , . . . } .2. Continuity for non-independent couplings
A key ingredient in the proof of Proposition 1.2 was the assumption that all the coefficients wereindependent. But for some purposes, one wants to drop that assumption, instead assuming that allthe coefficients J ( i, j ) and ˜ J ( i, j ) are defined on a common probability space, and are close in somesense. Let us state a bound that works in that case. Proposition 2.1.
Suppose the following random variables are defined on one probability space: arandom integer N ≥ ; random spin sites i , . . . , i N , j , . . . , j N ∈ { , . . . , N } ; and random couplings J , . . . , J N and ˜ J , . . . , ˜ J N , which may be dependent and possibly not identically distributed. Define therandom Hamiltonians − H N ( σ ) = N X n =1 J n σ i n σ j n + h N X i =1 σ i and − ˜ H N ( σ ) = N X n =1 ˜ J n σ i n σ j n + h N X i =1 σ i . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N E " ln X σ ∈ Ω N e − H N ( σ ) ! − N E " ln X σ ∈ Ω N e − ˜ H N ( σ ) ! ≤ N E " N X n =1 | J n − ˜ J n | . We thank an anonymous referee for suggesting this name, as well as for raising the issue of demonstrating the factthat the two versions of the model are statistically equivalent.
OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 9
Proof:
Consider the linear interpolation H N,t = t H N + (1 − t ) ˜ H N . Then ddt E " ln X σ ∈ Ω N e − H N,t ( σ ) ! = − E P σ ∈ Ω N e − H N,t ( σ ) (cid:16) H N ( σ ) − ˜ H N ( σ ) (cid:17)P σ ∈ Ω N e − H N,t ( σ ) , and integrating over (0 ,
1) it readily follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N E " ln X σ ∈ Ω N e − H N ( σ ) ! − N E " ln X σ ∈ Ω N e − ˜ H N ( σ ) ! ≤ N E (cid:20) max σ ∈ Ω N (cid:12)(cid:12)(cid:12) H N ( σ ) − ˜ H N ( σ ) (cid:12)(cid:12)(cid:12)(cid:21) . Now conditioned on N , for all σ ∈ Ω N , (cid:12)(cid:12)(cid:12) H N ( σ ) − ˜ H N ( σ ) (cid:12)(cid:12)(cid:12) ≤ N X n =1 | J n − ˜ J n | . Taking expectations gives the desired result. (cid:3)
A canonical, versus grand canonical, version of the Viana-Bray model.
An equivalentdefinition of the Hamiltonian for the Viana-Bray model [9] is − H N ( σ ) = K X n =1 J n σ i n σ j n + h N X i =1 σ i , where: K is a Poisson random variable, with mean αN ; the J n ’s are i.i.d., N (0 , β ) Gaussian randomvariables; and i , i , . . . , j , j , . . . are i.i.d., integer-valued random variables, uniformly distributed onthe set { , , . . . , N } . Then, p VB N ( α, β ) = 1 N E " ln X σ ∈ Ω N e − H N ( σ ) ! . Note that the number of edges present is a random variable K .In statistical mechanics, we often consider the canonical ensemble to be a model of a gas where thenumber of particles (but not the energy) is held fixed. In the grand canonical ensemble, the numberof particles is random, although usually highly concentrated around its mean value. Since the numberof edges in the Viana-Bray model is random (but highly concentrated around its average), this is likea grand canonical ensemble for the number of edges. Consider an alternative “canonical ensemble”Hamiltonian − H can N ( σ ) = ⌊ αN ⌋ X n =1 J n σ i n σ j n + h N X i =1 σ i , where all the J n ’s are i.i.d., N (0 , β ) Gaussian random variables, and i , i , . . . , j , j , . . . are as before.Now the number of edges is nonrandom: it is ⌊ αN ⌋ is nonrandom, the greatest integer ≤ αN . Define p can N ( α, β ) = 1 N E " ln X σ ∈ Ω N e − ˜ H N ( σ ) ! . Corollary 2.2.
With the definitions above, p can N ( α, β ) − p VB N ( α, β ) = O (cid:18) √ N (cid:19) . Remark 2.1.
A qualitative version of this result was stated without proof in the papers of Franz andLeone [5] and Guerra and Toninelli [6] . OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 10
Proof:
Define g , g , . . . to be i.i.d. N (0 ,
1) random variables. Let K be a P ( αN ) random variable,and let i , i , . . . , j , j , . . . be i.i.d., and uniform on { , . . . , N } , as before. For n ≤ K , let J n = β g n .For n > K , define J n = 0. For n ≤ ⌊ αN ⌋ define ˜ J n = β g n . For n > ⌊ αN ⌋ define ˜ J n = 0. Let N = max( K , ⌊ αN ⌋ ). It is easy to see that H N ( σ ) and ˜ H N ( σ ), defined as in Prop 2.1, have the correctdistributions for the Viana-Bray and the “canonical ensemble” models, respectively. Therefore, | p can N ( α, β ) − p VB N ( α, β ) | ≤ N E " N X n =1 | J n − ˜ J n | = β N E [ | K − ⌊ αN ⌋| ] E [ | g N | ] ≤ βN (1 + Var( K ) / ) Var( g N ) / = βN (1 + √ αN ) . (cid:3) The canonical, versus grand canonical, version of the SK model.
An analogous resultholds for the SK model. A definition of the SK model is as follows: Let X N be a χ N random variable.Let V N = ( V ( i, j ) : 1 ≤ i, j ≤ N ) be a uniform random point on the unit sphere S N − = n V = ( V ( i, j ) : 1 ≤ i, j ≤ N ) (cid:12)(cid:12)(cid:12) X Ni,j =1 ( V ( i, j )) = 1 o . Then, defining J N ( i, j ) = β V N ( i, j ) X N / √ N , p SK N ( β ) = 1 N E ln X σ ∈ Ω N exp N X i,j =1 J ( i, j ) σ i σ j + h N X i =1 σ i . Again, note that the norm of the coupling constant vector, ~ J N = ( J N ( i, j ) : 1 ≤ i, j ≤ N ), equals X N / √ N , which is random itself. One could consider this to be a grand canonical ensemble. In the“canonical ensemble” the only thing random about the coupling constant vector would be the direction.Therefore, define ˜ J N ( i, j ) = β V ( i, j ) p N/
2, and define p can N ( β ) = 1 N E ln X σ ∈ Ω N exp N X i,j =1 ˜ J ( i, j ) σ i σ j + h N X i =1 σ i . This is the analogue of the SK model, but where the couplings constant vector, (˜ J N ( i, j ) : 1 ≤ i, j ≤ N ), is constrained to lie on a sphere with radius R N and satisfying R N = β N/ E [ k ~ J N k ]. Corollary 2.3.
With the definitions above, p can N ( β ) − p SK N ( β ) = O (cid:18) √ N (cid:19) . Proof:
By direct application of Prop 2.1, | p can N ( β ) − p SK N ( β ) | ≤ N E N X i,j =1 | J N ( i, j ) − ˜ J N ( i, j ) | = β √ N N X i,j =1 E [ | V N ( i, j ) | · | X N − N | ] ≤ β √ N N X i,j =1 ( E [ V N ( i, j ) ]) / (cid:0) E [( X N − N ) ] (cid:1) / = β √ N (cid:0) E [ X N ] + N − N E [ X N ] (cid:1) / . OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 11
But, as X N is χ N , that means E [ X N ] = N , while E [ X N ] = √ N + 1) / N /
2) = N − O (cid:18) N (cid:19) . Therefore, E [ X N ] + N − N E [ X N ] = O (1) . (cid:3) Mean-field spin glass models with infinitely divisible couplings
All of the results so far were motivated by the beautiful results for the Viana-Bray models obtainedby Franz and Leone [5], Guerra and Toninelli [6], and De Sanctis [4]. Additionally, we were motivatedby Carmona and Hu’s universality result for the SK model [2]. The Viana-Bray model, as studied inthe papers listed above, basically relies upon one property of the coupling distribution. That is that F N is infinitely divisible. Let us digress briefly, to discuss infinitely divisible distributions.We will specialize our attention to symmetric distributions. Suppose that Λ is a nonnegative measureon (0 , ∞ ) (not including 0) satisfying Z ∞ min( y ,
1) Λ( dy ) < ∞ . (We write R ∞ in place of R (0 , ∞ ) .) Also suppose v is a nonnegative number. Then one can define afunction Ψ (Λ ,v ) ( k ) = vk Z ∞ (1 − cos( ky )) Λ( dy ) . This function is conditionally negative semidefinite, and 0 at 0. In other words (c.f. Schoenberg’stheorem), exp (cid:2) − Ψ (Λ ,v ) ( k ) (cid:3) is a positive semidefinite function of k ∈ R , and equals 1 at k = 0.Therefore, by Bochner’s theorem, it is the characteristic function of a unique c.d.f. We define F (Λ ,v ) to be this c.d.f. Thus, Z ∞−∞ e ikx dF (Λ ,v ) ( x ) = e − Ψ (Λ ,v ) ( k ) . It is a basic fact that F (Λ ,v ) ⋆ F (Λ ,v ) = F (Λ +Λ ,v + v ) . Therefore, F (Λ ,v ) is infinitely divisible: infact F (Λ ,v ) = F ⋆n (Λ /n,v/n ) . By the L´evy-Khinchine formula, specialized to symmetric distributions, everysymmetric, infinitely divisible c.d.f. is of this form for a unique pair (Λ , v ).Let us suppose that we have a spin glass Hamiltonian, defined as previously − H N ( σ ) = N X i,j =1 J N ( i, j ) σ i σ j + h N X i =1 σ i . But, now we suppose that the random couplings, ( J N ( i, j ) : 1 ≤ i, j ≤ N ) are i.i.d., and distributedaccording to F N = F (Λ /N, v/N ). We may denote F := F = F (Λ , v ). For a pure Gaussian, with v = β /
2, we have F N = F (0 , β / N ) = F ∗ N,β , as before. For the Poissonized Gaussian, we have F ∗ N,α,β = F N = F ( α Λ ∗ β /N, ∗ β ( dy ) = 2 e − y /β p πβ (0 , ∞ ) ( y ) dy . Therefore, this does, indeed, generalize the two cases we considered before, of the SK model and oneversion of the Viana-Bray model. Then we write p ∗ N (Λ , v ) = 1 N E " ln X σ ∈ Ω N e − H N ( σ ) ! . OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 12
We will not necessarily introduce a new symbol for the Hamiltonian, when the underlying distributionfor the couplings, ( J N ( i, j ) : 1 ≤ i, j ≤ N ), changes. Rather, we will endeavor to write the distributionexplicitly, when we take expectations, as in E F [ · ]. Let us write E (Λ ,v ) [ · ] instead of E F (Λ ,v ) [ · ].Let us introduce the fundamental definitions of Franz and Leone, Guerra and Toninelli, and DeSanctis. First of all, given a function of n spin configurations, u : (Ω N ) n → R , let us write h u i = h u ( σ (1) , . . . , σ ( n ) ) i := ( Z n ) − X σ (1) ,...,σ ( n ) ∈ Ω N e − [ H N ( σ (1) )+ ··· + H N ( σ ( n ) )] u ( σ (1) , . . . , σ ( n ) ) . Note that, in our choice of convention, this is still a random variable depending on the underlyingcoupling constants. But E (Λ ,v ) [ h u ( σ (1) , . . . , σ ( n ) ) i ] has had the expectation taken (with respect to thei.i.d. product of F (Λ ,v ) distributions). Let us also define the degree- n multi-overlap function R N,n :(Ω N ) n → R , as R N,n ( σ (1) , . . . , σ ( n ) ) = 1 N N X i =1 σ (1) i · · · σ ( n ) i . Then a fundamental result of Franz and Leone and Guerra and Toninelli, suitably generalized to thepresent context, gives an integral formula for p ∗ N (Λ , v ) − p ∗ N (Λ , v ) in terms of the expectations ofthese multi-overlaps. Let us generalize the definition of a k ( F ) as follows: a ∗ (Λ , v ) := v Z ∞ ln(cosh( y )) Λ( dy ) ; a ∗ (Λ , v ) := v Z ∞ tanh ( y ) Λ( dy ) ; and a ∗ k (Λ , v ) = a ∗ k (Λ) := 12 k Z ∞ tanh k ( y ) Λ( dy ) , for k = 2 , , . . . . (In some sense, when we write (Λ , v ) this really means the distribution Λ + δ ′′ , acting on smooth testfunctions φ satisfying φ (0) = φ ′ (0) = 0.) Then the result is as follows. Proposition 3.1.
Suppose that (Λ , v ) and (Λ , v ) are parameters from the L´evy-Khinchine formula,satisfying the further requirement that a (Λ , v ) and a (Λ , v ) are finite. Then p ∗ N (Λ , v ) − p ∗ N (Λ , v ) = a ∗ (Λ , v ) − a ∗ (Λ , v ) − ∞ X k =1 [ a ∗ k (Λ , v ) − a ∗ k (Λ , v )] Z E ( t Λ +(1 − t )Λ ,tv +(1 − t ) v ) (cid:20)(cid:28)(cid:16) R N,n ( σ (1) , . . . , σ ( n ) ) (cid:17) (cid:29)(cid:21) dt . Remark 3.2.
One clearly sees the motivation for Proposition 1.2 in this formula.
An immediate corollary, along the lines of Proposition 1.2 can be deduced from this. Namely, | p ∗ N (Λ , v ) − p ∗ N (Λ , v ) | ≤ ∞ X k =0 | a ∗ k (Λ , v ) − a ∗ k (Λ , v ) | . This gives a bound which is uniform in N , and is often easier to calculate. Guerra and Toninelli werethe first to write this type of bound, when they used it to give a very simple proof thatlim α →∞ sup N ∈ Z > (cid:12)(cid:12)(cid:12)(cid:12) p VB N (cid:18) α, β √ α (cid:19) − p SK N ( β ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , for all β ≥
0. This is known as the “infinite connectivity limit”.The main advantage of Prop 3.1 is that it is an exact formula. For example, Franz and Leone useda close analogue of this formula to prove that, for the Viana-Bray model, the thermodynamic limitof the pressure exists. Guerra and Toninelli used similar methods to control the high-temperatureand low-connectivity regions of phase space, demonstrating replica symmetry in that domain. AndDe Sanctis used that method and other arguments to prove an extended variational principle, therebygeneralizing the results of Aizenman, Sims, and an author [1], from the SK to the Viana-Bray model.
OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 13
A specialized version of Prop 3.1, applicable to the standard Viana-Bray model, is contained im-plicitly or explicitly in each of the papers [5], [6] and [ ? ]. Accordingly, the reader may find the relevantproofs, there. However, a new issue arises in the generalized context. Namely, we should prove thatthe pressure function, p N , is still in the domain of the “generator”, despite the fact that it is not atypical test-function (because it does not vanish at infinity). Next, we will present the definition ofthe generator, as well as this technical result.3.1. The generator.
An important fact is that one can define a L´evy process associated to theinfinitely divisible distribution F (Λ ,v ) . This is a stochastic process ( X t : t ≥ s, t ≥
0, the increment ( X s + t − X s ) is independent of F s , where F s is the σ -algebra generated by ( X r : 0 ≤ r ≤ s ); and ( X s + t − X s ) has the c.d.f. F ( t Λ ,tv ) . In particular, theincrements are independent and stationary. (There are also continuity properties of the L´evy processwhich we will not need.) See [8], for example, for a reference. Let C ( R ) denote the set of function f ∈ R which are twice continuously differentiable, and such thatlim | x |→∞ f ( x ) = lim | x |→∞ f ′ ( x ) = lim | x |→∞ f ′′ ( x ) = 0 . Then the following is a specialization of Theorem 31.5 in [8]: If f ∈ C ( R ), then E [ f ( X t )] is differen-tiable, and(5) ddt E [ f ( X t )] = E [ G (Λ ,v ) f ( X t )] , where G (Λ ,v ) is the generator G (Λ ,v ) f ( x ) = v f ′′ ( x ) + Z ∞ (cid:18) f ( x + y ) − f ( x ) + 12 f ( x − y ) (cid:19) Λ( dy ) . In particular, if we instead consider X to be distributed by F (Λ ,v ) , and denote the expectation withrespect to F (Λ ,v ) as E (Λ ,v ) , then we have ddt E ( t Λ ,tv ) [ f ( X )] = E ( t Λ ,tv ) [ G (Λ ,v ) f ( X )] . The technical fact we want to prove now is the following:
Lemma 3.3.
Suppose (Λ , v ) and (Λ , v t ) satisfy a ∗ (Λ , v ) < ∞ and a ∗ (Λ , v ) < ∞ . Also supposethat f : R → R is a function in C ( R ) such that f ′ and f ′′ are in L ∞ ( R ) . (In particular, then, f isglobally Lipschitz.) Then G (Λ ,v ) f and G (Λ ,v ) f are both well-defined and (6) E (Λ ,v ) [ f ( X )] − E (Λ ,v ) [ f ( X )] = Z E ( t Λ +(1 − t )Λ ,tv +(1 − t ) v ) (cid:2)(cid:0) G (Λ ,v ) − G (Λ ,v ) (cid:1) f ( X ) (cid:3) dt . This lemma proves that the generator is applicable even to the function p N , which is a Lipschitzfunction of the random coupling constants J ( i, j ). The beautiful calculation of the generator for thisfunction can be read off of any of the references mentioned before, [5, 6, ? ]. We will spend the rest ofthis section proving Lemma 3.3.3.2. Proof.
First, supposing a ∗ (Λ , v ) < ∞ , let us show that F ∗ (Λ ,v ) has finite first moment. Thisis equivalent to checking that ln(cosh( x )) is integrable, because ln(cosh( x )) ∼ | x | as | x | → ∞ . Butln(cosh( x )) − ǫ − ln(cosh( ǫx )) is a nonnegative function, for every 0 < ǫ <
1, and it is in C ( R ). Sothe generator applies to it. But it is easy to see that, defining u ǫ ( x ) = ǫ − ln(cosh( ǫx )), G (Λ ,v ) u ǫ ( x ) = ǫv ( ǫx ) + 12 ǫ Z ∞ ln (cid:18) ( ǫy )cosh ( ǫx ) (cid:19) Λ( dy ) . There are several things to note. First, the formula is well-defined. Second, an upper bound is obtainedby bounding cosh ( ǫx ) ≥ ǫ ) − ln(1 + sinh ( ǫy )) = ǫ − ln(cosh( ǫy )) ≤ ln(cosh( y )). Therefore, OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 14 G (Λ ,v ) u ǫ ( x ) ≤ a ∗ (Λ , v ), for all x . But finally, by the DCT, lim ǫ → G (Λ ,v ) u ǫ ( x ) →
0, for all x . So, byvarious trivial applications of the DCT, E (Λ ,v ) [ln(cosh( X ))] = lim ǫ → E (Λ ,v ) [( u − u ǫ )( X )] = lim ǫ → Z E ( t Λ ,tv ) (cid:2) G (Λ ,v ) ( u − u ǫ )( X t ) (cid:3) dt = Z E ( t Λ ,tv ) (cid:2) G (Λ ,v ) u ( X t ) (cid:3) dt ≤ a ∗ (Λ , v ) , A quantitative version shows that E (Λ ,v ) [ | X | ] ≤ (1 + a ∗ (Λ , v )) e . In particular, by assumption a ∗ (Λ , v )and a ∗ (Λ , v ) are finite, therefore, F ( t Λ +(1 − t )Λ ,tv +(1 − t ) v ) has a finite first moment for all t ∈ [0 , { a ∗ (Λ , v ) , a ∗ (Λ , v ) } ) e . Since f is globally Lipschitz, thismeans it is integrable against F ( t Λ +(1 − t )Λ ,tv +(1 − t ) v ) .Now we simply introduce a smooth cut-off, and perform basic estimates. Suppose that ψ ( x ) isany function which is twice continuously differentiable, compactly supported, and such that ψ (0) = 1.Then, defining, f ǫ ( x ) = f ( x ) ψ ( ǫx ) this is in C with compact support. So equation (6) holds with f replaced by f ǫ . Since f ǫ → f , pointwise, and since | f ǫ ( x ) | ≤ k ψ k ∞ | f ( x ) | , we have E (Λ ,v ) [ f ( X )] − E (Λ ,v ) [ f ( X )] = lim ǫ → (cid:0) E (Λ ,v ) [ f ǫ ( X )] − E (Λ ,v ) [ f ǫ ( X )] (cid:1) . To prove (6) for f (instead of f ǫ ), we just have to show that the right hand side of this equationconverges to the right hand side of (6) as ǫ →
0. Note that | f ǫ ( x ) − f ǫ ( y ) | ≤ ( k f ′ k ∞ + ǫ k ψ ′ k ∞ ) | x − y | . So, for some
A < ∞ , (cid:12)(cid:12)(cid:12)(cid:12) f ǫ ( x + y ) + 12 f ǫ ( x − y ) − f ǫ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ A | y | , which we use for | y | ≥
1. Also, k f ′′ ǫ ( x ) k ≤ k f ′′ k ∞ k ψ k ∞ + 2 ǫ k f ′ k ∞ k ψ ′ k ∞ + ǫ k ψ ′′ k ∞ | f ( x ) | . So, by Taylor’s theorem, for some
B, C < ∞ , (cid:12)(cid:12)(cid:12)(cid:12) f ǫ ( x + y ) + 12 f ǫ ( x − y ) − f ǫ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( B + ǫC | f ( x ) | ) y , which we use for | y | <
1. So, since k ln(cosh( y )) ≤ min( y , | y | ) ≤ K ln(cosh( y )), for some 0 < k < K < ∞ , we have (cid:12)(cid:12)(cid:12)(cid:12) f ǫ ( x + y ) + 12 f ǫ ( x − y ) − f ǫ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( ˜ B + ǫ ˜ C | f ( x ) | ) ln(cosh( y )) , for some different constants ˜ B, ˜ C < ∞ , and all y ∈ R . Since a ∗ (Λ , v ) and a ∗ (Λ , v ) are both finite,this upper bound is integrable against Λ and Λ . Therefore, for some other constants ˆ A, ˆ B < ∞ ,(7) | G (Λ i ,v i ) f ǫ ( x ) | ≤ ˆ A + ˆ B | f ( x ) | for i = 1 , . But also note that,lim ǫ → f ǫ ( x + y ) + 12 f ǫ ( x − y ) − f ǫ ( x ) = 12 f ( x + y ) + 12 f ( x − y ) − f ( x ) , for each x ∈ R and y >
0. Similarly, lim ǫ → f ′′ ǫ ( x ) = f ′′ ( x ), for each x ∈ R . So, by the DCT, we knowlim ǫ → G (Λ i ,v i ) f ǫ ( x ) = G (Λ i ,v i ) f ( x ) , for i = 1 , x ∈ R . But also, the upper bound of (7) is integrable against F ( t Λ +(1 − t )Λ ,tv +(1 − t ) v ) ,for all t , and is independent of ǫ . So, by the DCT again,lim ǫ → E ( t Λ +(1 − t )Λ ,tv +(1 − t ) v ) (cid:2)(cid:0) G (Λ ,v ) − G (Λ ,v ) (cid:1) f ǫ ( X ) (cid:3) = E ( t Λ +(1 − t )Λ ,tv +(1 − t ) v ) (cid:2)(cid:0) G (Λ ,v ) − G (Λ ,v ) (cid:1) f ( X ) (cid:3) . OME OBSERVATIONS FOR MEAN-FIELD SPIN GLASS MODELS 15
Of course, the integrated version of this is also true, once again by DCT.4.
Conclusion
We considered two different types of bounds for the difference of two pressures of two spin glasses.In Section 1, we considered a bound which demonstrates that the pressure is Lipschitz with respectto a seminorm, k F k = P ∞ k =0 | a k ( F ) | . This was strongly motivated by Carmona and Hu’s proof ofuniversality for the SK model, but generalized to also apply to the Viana-Bray model. In Section 2, weconsidered a bound which is useful if one does not assume that the coupling constants are independent.This proved a different type of universality, which was noted, but not proved, in papers of Franz andLeone, and Guerra and Toninelli. In Section 3, we briefly reviewed the theory of infinitely divisibledistributions, and applied it to the Viana-Bray model. All the results in this letter are simple. But wehope they add something to the growing wealth of knowledge for mean-field spin glass models. Acknowledgements
The research of S.S. was supported in part by a U.S. National Science Foundation grant, DMS-0706927.
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