Annals of Mathematics The Parisi formula By Michel Talagrand Annals of Mathematics, 163 (2006), 221–263 The Parisi formula By Michel Talagrand* Dedicated to Francesco Guerra Abstract Using Guerra’s interpolation scheme, we compute the free energy of the Sherrington-Kirkpatrick model for spin glasses at any temperature, confirming a celebrated prediction of G. Parisi. 1. Introduction The Hamiltonian of the Sherrington-Kirkpatrick (SK) model for spin glasses [10] is given at inverse temperature β by H N (σ)=− β √ N  i<j g ij σ i σ j .(1.1) Here σ =(σ 1 , ,σ N ) ∈ Σ N = {−1, 1} N , and (g ij ) i<j are independent and identically distributed (i.i.d.) standard Gaussian random variable (r.v.). It is unexpected that the simple, basic formula (1.1) should give rise to a very intricate structure. This was discovered over 20 years ago by G. Parisi [8]. The predictions of Parisi became the starting point of a whole theory, the breadth and the ambitions of which can be measured in the books [6] and [9]. Literally hundreds of papers of theoretical physics have been inspired by these ideas. The SK model is a purely mathematical object, but the methods by which it has been studied by Parisi and followers are not likely to be recognized as legitimate by most mathematicians. The present paper will correct this dis- crepancy and will make one of the central predictions of Parisi, the computa- tion of the “free energy” of the SK model appear as a consequence of a general mathematical principle. This general principle will also apply for even p to the “p-spin” generalization of (1.1), where the Hamiltonian is given at inverse *Work partially supported by an NSF grant. 222 MICHEL TALAGRAND temperature β by H N (σ)=−β  p! 2N p−1  1/2  i 1 < <i p g i 1 , ,i p σ i 1 ···σ i p .(1.2) We consider for each N a Gaussian Hamiltonian H N on Σ N , that is a jointly Gaussian family of r.v. indexed by Σ N . (Here, as everywhere in the paper, by Gaussian r.v., we mean that the variable is centered.) We assume that for a certain sequence c(N) → 0 and a certain function ξ : R → R,we have ∀σ 1 , σ 2 ∈ Σ N ,    1 N EH N (σ 1 )H N (σ 2 ) −ξ(R 1,2 )    ≤ c(N),(1.3) where R 1,2 = R 1,2 (σ 1 , σ 2 )= 1 N  i≤N σ 1 i σ 2 i (1.4) is called the overlap of the configurations σ 1 and σ 2 . A simple computation shows that for the Hamiltonian (1.2), we have (1.3) for ξ(x)=β 2 x p /2 and c(N) ≤ K(p)/N , where K(p) depends on p only. When ξ is three times continuously differentiable, and satisfies ξ(0) = 0,ξ(x)=ξ(−x),ξ  (x) > 0ifx>0 ,(1.5) we will compute the asymptotic free energy of Hamiltonians satisfying (1.3). We fix once and for all a number h (that represents the strength of an “external field”). Consider an integer k ≥ 1 and numbers 0=m 0 ≤ m 1 ≤···≤m k−1 ≤ m k =1(1.6) and 0=q 0 ≤ q 1 ≤···≤q k+1 =1.(1.7) It helps to think of m  as being a parameter attached to the interval [q  ,q +1 [. To lighten notation, we write m =(m 0 , ,m k−1 ,m k ); q =(q 0 , ,q k ,q k+1 ).(1.8) Consider independent Gaussian r.v. (z p ) 0≤p≤k with Ez 2 p = ξ  (q p+1 ) −ξ  (q p ).(1.9) We define the r.v. X k+1 = log ch  h +  0≤p≤k z p  and recursively, for  ≥ 0 X  = 1 m  log E  exp m  X +1 ,(1.10) THE PARISI FORMULA 223 where E  denotes expectation in the r.v. z p ,p ≥ . When m  = 0 this means X  = E  X +1 .ThusX 0 = E 0 X 1 is a number. We set P k (m, q) = log 2 + X 0 − 1 2  1≤≤k m   θ(q +1 ) −θ(q  )  (1.11) where θ(q)=qξ  (q) −ξ(q).(1.12) We define P(ξ, h) = inf P k (m, q),(1.13) where the infimum is over all choices of k and all choices of the sequences m and q as above. One might notice that giving sequences m and q as in (1.8) is the same as giving a probability measure µ on [0, 1] that charges at most k points (the points q  for 1 ≤  ≤ k, the mass of q  being m  − m −1 ). One can then write P(µ) rather than P k (m, q). Moreover Guerra [3] proves that this definition can be extended by a continuity argument to any probability measure µ on [0, 1], and the distribution function of such a probability is the “functional order parameter” of the theoretical physicists. We do not adopt this point of view since an essential ingredient of our approach is that we need only consider discrete objects rather than continuous ones. We refer the reader to [18] for further results in this direction. Theorem 1.1 (The Parisi formula). We have lim N→∞ 1 N E log  σ exp  H N (σ)+h  i≤N σ i  = P(ξ, h).(1.14) The summation is of course over all values of σ ∈ Σ N . To lighten the exposition, we do not follow the convention of physics to put a minus sign in front of the Hamiltonian. We learned the present formulation in Guerra’s work [3], to which we refer for further discussion of its connections with Parisi’s original formulation. In this truly remarkable paper Guerra proves that the left-hand side of (1.14) is bounded by the right-hand side, using an interpolation scheme that is the back- bone of the present work. Guerra and Toninelli [5] had previously established the existence of the limit in (1.14). Even in concrete cases, the computation of the quantity P(ξ, h) is certainly a nontrivial issue. In fact, it is possibly a difficult problem. This problem however is of a different nature, and we will not investigate it. It should be pointed out that one of the reasons that make our proof of Theorem 1.1 possible is that we have succeeded in separating the proof of this theorem from the issue of computing P(ξ, h). 224 MICHEL TALAGRAND When the infimum in (1.13) is a minimum, and if k ≥ 1 is the smallest integer for which P(ξ, h)=P k (m, q) for a certain choice of m and q, one says in physics that the system exhibits “k −1 steps of replica symmetry breaking”. Only the case k = 1 (“high temperature behavior”) and k = 2 (as in the p-spin interaction model for p ≥ 3 at suitable temperatures) have been described in the physics literature but it is possible (elaborating on the ideas of [14]) to show that suitable choices of ξ can produce situations where k is any integer. The most interesting situation is however when the infimum is not attained in (1.13), which is expected to be the case for the SK model (where ξ(x)=βx 2 /2) when β is large enough. The Parisi formula can be seen as a theorem of mathematical analysis. The proof we present is self-contained, and requires no knowledge whatso- ever of physics. It could however be of some interest to briefly discuss some of the results and of the ideas that led to this proof. This discussion, that occupies the rest of the present paragraph, assumes that the reader is some- what familiar with the area and its recent history, and understands it is in no way a prerequisite to read the rest of the paper. We will discuss only the history of the SK model (where ξ(x)=βx 2 /2). In that case, at given h, for β small enough, the infimum in (1.13) is obtained for k = 1, and the corresponding value is known as the “replica-symmetric solution”. The re- gion of parameters β,h where this occurs is known as the “high-temperature region”. For sufficiently small β, (say, β ≤ 1/10), and any value of h, the author [21] first proved in 1996 the validity of (1.14) using the so-called “cav- ity method” (which is developed at length in his book [16]). Soon after, and independently, M. Shcherbina [11] produced a proof using somewhat different ideas, valid in a larger region of parameters and, in particular, for all h and all β ≤ 1. It became soon apparent however that the cavity method is powerless to obtain (1.14) in the entire high-temperature region. One of the key ideas of our approach is the observation (to be detailed later) that, in order to prove lower bounds for the left-hand side of (1.14), it is sufficient to prove upper bounds on similar quantities that involve two copies of the system (what is called real replicas in physics). The author observed this in 1998 while writing the paper [13]. This observation was not very useful at that time, since there was no method to prove upper bounds. In 2000, F. Guerra [2] invented an interpolation method (which he later improved in his marvelous paper [3] that plays an essential role in our approach) to prove such upper bounds, and soon after the author [15] attempted to combine Guerra’s method of proving upper bounds with his method to turn upper bounds into lower bounds to try to prove (1.14) in the entire high temperature region. The main difficulty is that when one tries to use Guerra’s method for two replicas, some terms due to the interaction between these replicas have the wrong sign. The device used by the author [15] in an attempt to overcome this THE PARISI FORMULA 225 difficulty unfortunately runs into intractable technical problems. The paper [15] inspired in turn a work by Guerra and Toninelli [4], with a more straight- forward approach, but that also fails to reach the entire high-temperature region. The author then improved in [16, Th. 2.9.10], the result of Guerra and Toninelli [4], and it was at this time that he made the simple, yet critical, observation that the difficulties occurring when one attempts to use Guerra’s scheme of [2] for two replicas largely disappear when, rather than considering the system consisting of two replicas, one considers instead the subsystem of the set of pairs of configurations with a given overlap. The region reached by this theorem still seems smaller than the high-temperature region. The au- thor obtained somewhat later, in spring 2003, the proof of (1.14) in the entire high-temperature region, and presented it in [16, Th. 2.11.16]. Even though our proof of Theorem 1.1 is self-contained, to penetrate the underlying ideas, the reader might find it useful to look first at this simpler use of our main techniques. The basic mechanism of the proof extracts crucial information from the fact that one cannot improve the bound obtained for k = 1 when one uses instead k = 2. This mechanism is simpler to describe in the case of the control of the high-temperature region than in the general case, which involves more details. It should be stressed however that the conventional wisdom, that asserted that the proof of (1.14) would be much easier in the high-temperature region than in general, turned out to be completely wrong. Rather surprisingly, the main ideas of our proof of the Parisi formula seem already required to prove it in the entire high temperature region. A crucial difficulty in the control of this region is that in some sense low temperature behavior seems to occur earlier when one considers two replicas rather than one. Even to control the high temperature region, our proof uses one idea of the type “symmetry breaking” (as inspired by Guerra [3]). Thus, unexpectedly, while it took many years to prove the Parisi formula in the entire high-temperature region, it took only a few weeks more to prove it for all values of the parameters. Interestingly, and despite Theorem 1.1, it is still not known exactly what is the high temperature region of the SK model. This is due to the difficulty of computing P(ξ, h). F. Guerra proved that for any values of β and h, if the r.v. z is standard Gaussian, the equation q = Eth 2 (βz √ q + h) has a unique solution, and F. Toninelli [22] deduced from Guerra’s upper bound of [3] that, if q is this unique solution, in the high temperature region one has β 2 E 1 ch 4 (βz √ q + h) ≤ 1.(1.15) It seems possible that the region where Condition (1.15) holds is exactly the high temperature region, but this has not been proved yet. (This question boils down to a nasty calculus problem, see [16, p. 154].) 226 MICHEL TALAGRAND It seems of interest to mention some of the developments that occurred during the rather lengthy interval that separated the submission of this work from its revision. The author [19] extended Theorem 1.1 to the case of the spherical model and obtained some information on the physical meaning of the parameters occurring in P(ξ, h) [18], [21]. Moreover, D. Panchenko [7] extended Theorem 1.1 to the case where the spins can take more general values than −1 and 1 . The Parisi conjecture (1.14) was probably the most widely known open problem about “spin glasses”, and it is certainly nice to have been able to prove it. The author would like however to stress that, when seen as part of the global area of spin glass models, this is a rather limited progress. It is not more than a very first step in a very rich area. Many of the most fundamental and fascinating predictions of the Parisi theory remain conjectures, even in the case of the SK model. This is in particular the case of ultrametricity and of the so-called chaos problem. These problems apparently cannot be solved using only the techniques of the present paper, or simple modifications of these. It is even conceivable that they will turn out to be very difficult. In fact, very little is presently known about the structure of the Gibbs measure. Moreover, the techniques of the present paper rely on rather specific arguments, namely using the convexity of ξ, to ensure that certain remainder terms are nonnegative. It is not known at this time how to use a similar approach for any of the important spin glass models other than the class described here (and variations of it). A detailed description in mathematical terms of some of the most blatant open problems on spin glasses can be found in [20]. Acknowledgment. I am grateful to WanSoo Rhee for having typed this manuscript and to Dmitry Panchenko for a careful reading. 2. Methodology To lighten notation, we will not indicate the dependence in N , so that our basic Hamiltonian is denoted by H. Central to our approach is the inter- polation scheme recently discovered by F. Guerra [3]. Consider an integer k and sequences m, q as above. Consider independent copies (z i,p ) 0≤p≤k of the sequence (z p ) 0≤p≤k of (1.9), that are independent of the randomness of H.We denote by E  expectation in the r.v. (z i,p ) i≤N,p≥ . We consider the Hamiltonian H t (σ)= √ tH(σ)+  i≤N σ i  h + √ 1 −t  0≤p≤k z i,p  .(2.1) We define F k+1,t = log  σ exp H t (σ),(2.2) THE PARISI FORMULA 227 and, for  ≥ 1, we define recursively F ,t = 1 m  log E  exp m  F +1,t .(2.3) When m  = 0 this means that F ,t = E  F +1,t . We set ϕ(t)= 1 N EF 1,t .(2.4) The expectation here is in both the randomness of H and the r.v. (z i,0 ) i≤N . We write, for 1 ≤  ≤ k, W  = exp m  (F +1,t − F ,t ).(2.5) (To lighten notation, the dependence in t is kept implicit.) We denote by Ξ  the σ-algebra generated by H and the variables (z i,p ) i≤N,p< so that F ,t is Ξ  -measurable, and W  is Ξ +1 -measurable.(2.6) Since E  (·)=E(·|Ξ  ), it follows from (2.3) that E  (W  )=1.(2.7) Using (2.6), and since E  = E  E +1 , we see inductively from (2.7) that E  (W  ···W k )=E  (W  )E +1 (W +1 ···W k )=1.(2.8) Let us denote by f t the average of a function f for the Gibbs measure with Hamiltonian H t , i.e. f t exp F k+1,t =  σ f(σ) exp H t (σ). We then see from (2.8) that the functional f → E   W  ···W k f t  is a probability γ  on Σ N . We denote by γ ⊗2  its product on Σ 2 N , and for a function f :Σ 2 N → R we set µ  (f)=E  W 1 ···W −1 γ ⊗2  (f)  .(2.9) Theorem 2.1 (Guerra’s identity [3]). For 0 <t<1 we have ϕ  (t)=− 1 2  1≤≤k m  (θ(q +1 ) −θ(q  ))(2.10) − 1 2  1≤≤k (m  − m −1 )µ   ξ(R 1,2 ) −R 1,2 ξ  (q  )+θ(q  )  + R where |R| ≤ c(N). 228 MICHEL TALAGRAND The convexity of ξ implies that ∀x, ξ(x) − xξ  (q)+θ(q) ≥ 0(2.11) so by (2.10) we have ϕ(1) ≤ ϕ(0) − 1 2  1≤≤k m   θ(q +1 ) −θ(q  )  + c(N).(2.12) One basic idea of (2.1) is that for t = 0, there is no interaction between the sites, so that ϕ(0) is easy to compute. In fact, if we denote by X i, the r.v. defined as in (1.10) but starting with the sequence (z i,p ) 0≤p≤k rather than with the sequence (z p ) 0≤p≤k , we see immediately by decreasing induction over  that F ,0 = N log 2 +  i≤N X i, (2.13) so that ϕ(0) = log 2 + X 0 (2.14) and (2.12) implies 1 N E log  σ exp  H N (σ)+h  i≤N σ i  ≤P k (m, q)+c(N),(2.15) which proves “half” of Theorem 1.1, the main result of [3]. Soon after the present work was submitted for publication, Aizenman, Sims and Starr [1] produced a generalization of Guerra’s interpolation scheme (nontrivial arguments are required to show that this scheme actually contains Guerra’s scheme). The main purpose of this scheme seems to have been to try to improve on Guerra’s bound (2.15). As Theorem 1.1 shows, this is not possible. However the scheme of [1] is still of interest, and is more transparent than Guerra’s scheme. It was used in particular by the author [17] to prove that Guerra’s bound (2.15) still holds if one relaxes condition (1.5) into assuming that ξ is convex on R + rather than on R as is assumed in [3]. It would be nice to be able to prove Theorem 1.1 under these weaker conditions on ξ. This would in particular cover the case of the p-spin interaction model for odd p. We will deduce the other half of Theorem 1.1 from the following, where we recall that ϕ depends implicitly on k,m and q. Theorem 2.2. Given t 0 < 1, there exists a number ε>0, depending only on t 0 ,ξ and h, with the following property. Assume that for some number k and for some sequences m and q as in (1.8), we have P k (m, q) ≤P(ξ,h)+ε,(2.16) THE PARISI FORMULA 229 P k (m, q) realizes the minimum over all choices of m and q.(2.17) Then, for t ≤ t 0 , we have lim N→∞ ϕ(t)=ψ(t):=ϕ(0) − t 2  1≤≤k m   θ(q +1 ) −θ(q  )  .(2.18) The existence of m and q satisfying (2.17) is obvious by a compactness argument. It is to permit this compactness argument that equality is allowed in (1.6) and (1.7). However, when m and q are as in (2.17), without loss of generality, we can assume (decreasing k if necessary) that 0=q 0 <q 1 < ···<q k <q k+1 =1, 0=m 0 <m 1 < ···<m k−1 <m k =1. (2.19) This is because if q  = q +1 then z  = 0, so that we can remove q +1 from the list q and m  from the list m without changing anything. If m  = m +1 we can “merge the intervals [q  ,q +1 [ and [q +1 ,q +2 [” and remove q +1 from q and m  from m. The central point of Theorem 2.2 is the fact that t 0 < 1 can be as close to 1 as one wishes. The expert about the cavity method should have already guessed that if instead of (2.17) we fix m and we assume that P k (m, q) realizes the minimum over all choices of q, then the conclusion of Theorem 2.2 holds for some t 0 > 0 (a result that is in the spirit of the fact that “the replica- symmetric solution is true at high enough temperature”). The key mechanism of the proof extracts information from the fact that P k (m, q) is also minimal over all choices of m to reach any value t 0 < 1 (a result that is in the spirit of “the control of the entire high-temperature region”). It might be useful to stress the considerable simplification that is brought by Theorem 2.2. One only has to consider structures with a “finite level on complexity” independent of N. It is of course much easier to bring out these structures in a large system than it would be to bring out the whole Parisi structure with “an infinite level of complexity”. One can surely expect that this idea of reducing to a “finite level of complexity” through interpolation to be useful in the study of other spin glass systems. When ξ  (0) > 0, one can actually take ε of order (1 −t 0 ) 6 in Theorem 2.2. We see no reason why this rate would be optimal. To prove Theorem 1.1, we see from Guerra’s identity that |ϕ  (t)|≤L + c(N), where, as everywhere in this paper, L denotes a number depending on ξ and h only, that need not be the same at each occurrence. Since ψ(1) = P k (m, q), we see from (2.18) that lim sup N→∞ |ϕ(1) −P k (m, q)|≤L(1 −t 0 ) [...]... η(1) with (3.26) Proof This relies on the same principles as the proof of Theorem 2.1 The main new feature is that new terms are created by the interaction between the two copies of the system we consider now These terms tend to have the wrong sign to make the argument of Theorem 2.1 work, but the device of restricting the summation to R1,2 = u in (3.23) makes these terms much easier to handle We write... of the others j For j = 1 or j = 2 the sequence (zp ) is as in (1.9); but (2.23) 1 2 1 2 zp = zp if p < r; zp and zp are independent if p ≥ r We consider the Hamiltonian (2.24) Ht (σ 1 , σ 2 ) = √ j σi h + t H(σ 1 ) + H(σ 2 ) + j=1,2 i≤N √ j zi,p , 1−t 0≤p≤k 231 THE PARISI FORMULA 1 2 1 2 where (zi,p , zi,p )0≤p≤k are independent copies of the sequence (zp , zp )0≤p≤k , that are also independent of the. .. do not detail it 251 THE PARISI FORMULA 5 The main estimate The goal of this section is to prove Theorem 2.4 In the following statements, L1 , L2 , , denote specific quantities depending only upon ξ and h Proposition 5.1 If L1 ε1/6 ≤ 1 − t0 , then (5.1) qr−1 ≤ u ≤ qr , L1 (qr − u) ≤ 1 − t0 ⇒ Ψ(t, u) ≤ 2ψ(t) − (1 − t0 )2 (u − qr )2 L1 Proposition 5.2 If L1 ε1/6 ≤ 1 − t0 , then (5.2) qr ≤ u ≤ qr+1... summation of these formulas for 0 ≤ p ≤ k yields II(p) = − 0≤p≤k 1 ξ (1) + 2 (m 1≤ ≤k −1 − m )ξ (q )µ (R1,2 ) 237 THE PARISI FORMULA so that (3.17) 2ϕ (t) = ξ(1) − ξ (1) + (m −1 − m )µ ξ(R1,2 ) − R1,2 ξ (q ) + 2R 1≤ ≤k = −θ(1) − (m −1 − m )θ(q ) 1≤ ≤k (m + −1 − m )µ ξ(R1,2 ) − R1,2 ξ (q ) + θ(q ) + 2R 1≤ ≤k and the result follows using (3.3) for c = θ(q ) We now turn to the principle on which the paper... k, this is (2.34) For the induction from + 1 and that n ≤ n +1 , we see first that (2.37) J +1,u ≥J +1 + 1 log E n +1 V + 1 to , using (2.35) for +1 · · · Vk U 233 THE PARISI FORMULA and thus, using the definition of V in the second line, exp n J +1,u ≥ E +1 (V +1 · · · Vk U ) exp n =V E =E J +1 (V +1 · · · Vk U ) exp n +1 (V +1 J · · · Vk U ) exp n J j Since J does not depend on the r.v (zi,p ) for... Guerra’s bound and its extension We will first prove Theorem 2.1 Our approach to the computations is slightly simpler than Guerra’s [3] This simplification will be quite helpful when we will consider the more complicated situation of Theorem 3.1 The main tool of the proof is integration by parts Consider a jointly Gaussian family of r.v h = (hj )j∈J , J finite Then for a function F : RJ → R, of moderate growth,... L(1 − t0 ), N →∞ and this implies Theorem 1.1 since t0 < 1 is arbitrary We will deduce Theorem 2.2 from the following, where, for simplicity, we write µr (A) rather than µr (1A ) for a subset A of Σ2 N Proposition 2.3 Given t0 < 1, there exists ε > 0, depending only on t0 , ξ and h, with the following properties Assume that k, m, q are as in (2.16), (2.17) and (2.19) Then for any ε1 > 0, and any 1 ≤... the sum is taken only over all pairs (σ 1 , σ 2 ) for which R1,2 = u (We always assume that u is taken such that such pairs exist.) For ≥ 0, we define recursively (2.27) J ,t,u = 1 log E exp n J n +1,t,u j where E denotes expectation in the r.v zi,p for p ≥ , and we set (2.28) Ψ(t, u) = 1 EJ1,t,u , N j where the expectation is in the randomness of H and the r.v.zi,0 The a priori estimate on which the. .. and for N large enough the right-hand side is ≤ ε1 for all t ≤ t0 The proof of Proposition 2.5 has two parts The first part relies on a rather general principle, but the second will shed some light on the conditions (2.23) and (2.25) Keeping the dependence in t implicit, we define exp Ht (σ 1 , σ 2 ), Jk+1 = log (2.32) σ1 ,σ2 where the sum is now over all pairs of configurations, and we define recursively... We consider the set Su = {(σ 1 , σ 2 ) ∈ Σ2 ; R1,2 = u} N and, for a function f on Su , define f f v v by f (σ 1 , σ 2 ) exp Hv (σ 1 , σ 2 ) exp Fκ+1,v = (σ1 ,σ2 )∈Su We define a probability γ on Su by γ (f ) = E (V · · · Vκ f v ) 2 and for a function f on Su , we write µ (f ) = E V1 · · · V −1 γ ⊗2 (f ) As in the case of Theorem 2.1, we obtain (3.27) η (v) = I + II(p), 0≤p≤κ 239 THE PARISI FORMULA where . Annals of Mathematics The Parisi formula By Michel Talagrand Annals of Mathematics, 163 (2006), 221–263 The Parisi formula By Michel Talagrand* Dedicated. some of the developments that occurred during the rather lengthy interval that separated the submission of this work from its revision. The author [19] extended Theorem 1.1 to the case of the spherical. fascinating predictions of the Parisi theory remain conjectures, even in the case of the SK model. This is in particular the case of ultrametricity and of the so-called chaos problem. These problems apparently