Large Deviations of the Free-Energy in Diluted Mean-Field Spin-Glass
aa r X i v : . [ c ond - m a t . d i s - nn ] O c t Large Deviations of the Free-Energy in Diluted Mean-Field Spin-Glass
Giorgio Parisi , and Tommaso Rizzo Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, P.le Aldo Moro 2, 00185 Roma, Italy Statistical Mechanics and Complexity Center (SMC) - INFM - CNR, Italy
Sample-to-sample free energy fluctuations in spin-glasses display a markedly different behaviourin finite-dimensional and fully-connected models, namely Gaussian vs. non-Gaussian. Spin-glassmodels defined on various types of random graphs are in an intermediate situation between thesetwo classes of models and we investigate whether the nature of their free-energy fluctuations isGaussian or not. It has been argued that Gaussian behaviour is present whenever the interactionsare locally non-homogeneous, i.e. in most cases with the notable exception of models with fixedconnectivity and random couplings J ij = ± ˜ J . We confirm these expectation by means of variousanalytical results. In particular we unveil the connection between the spatial fluctuations of thepopulations of populations of fields defined at different sites of the lattice and the Gaussian natureof the free-energy fluctuations. On the contrary on locally homogeneous lattices the populationsdo not fluctuate over the sites and as a consequence the small-deviations of the free energy arenon-Gaussian and scales as in the Sherrington-Kirkpatrick model. I. INTRODUCTION
The problem of sample-to-sample free energy fluctuations in spin-glasses has attracted large interest in recent yearsboth from the theoretical and numerical point of view [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].In general at fixed system size N the free energy per spin of a given sample is a random variable with mean f N and variance σ N . In the thermodynamic limit f N approaches a definite value f typ while σ N approaches zero: thefree energy is self-averaging and does not depend on the sample. It is expected that in the thermodynamic limit,the rescaled variable x = ( f − f N ) /σ N has a limiting probability distribution that describes the small deviations ofthe free energy, i.e. those that occur with finite probability and whose scale decreases with N . On the other handlarge deviations of the free energy, i.e. those that remains finite in the large N limit have a probability that vanishesexponentially with the size of the system and are described by a large deviation function [13, 14, 15, 16].In finite-dimensional models one expects the small deviations of the free energy to have a Gaussian distributionwith variance proportional to the volume of the system i.e. σ N ∝ N − / [7, 12, 17]. In mean-field models however thestandard arguments leading to this expectation no longer hold and more exotic situations can be observed. Indeed thescaling of σ N with N in the Sherrington-Kirkpatrick(SK) model [18] has been largely studied numerically in recentyears [1, 2, 3, 4, 5, 6, 10] and there is growing consensus that σ N ∝ N − / . This scaling had been conjectured earlyin [16] using a large deviations result by Kondor [19] under the assumption that there is a matching between largeand small deviations (for a recent discussion see [14]). The large deviation result of Kondor had been questioned in[7, 8] but was later proved to be correct [20]. The distribution of negative large deviations was computed down to zerotemperature and excellent agreement with numerical results was found [13, 14]. Currently there is no analytical toolto compute directly σ N nevertheless the recent computation of positive large deviations [15] adds further support tothe 5 / σ N and the shape of probability distribution of the ground state energy of a spin-glass definedon random graphs with fixed connectivity depends on the distribution of the couplings. In particular on a randomgraph with fixed connectivity and Gaussian distributed couplings J ij it turned out that σ N ∝ N − / (as in finitedimensional models) and that the skewness of the small-deviation distribution tends to zero at large N , consistentlywith the assumption that it is a Gaussian. However in the case of fixed connectivity and bimodal distribution of thecouplings ( J ij = ± ˜ J ) they observed a scaling of σ N definitively different from N − / , (possible values being N − / or N − / ) . Furthermore it turned out that the skewness of the small deviation distribution instead of vanishing atlarge N tends to go to a finite value consistent with that of the SK model.In this paper we compute the large deviations of the models considered by [2], i.e. spin-glass models defined ongraphs with fixed connectivity. We consider the following functional [13, 14, 16]:Φ N ( n, β ) = − βnN ln Z J ( β ) n , (1)where different systems (or samples) are labeled by J , Z J ( β ) is the partition function of a given sample and thebar denotes the average over different disordered samples. The above functional is the generating function of thecumulants of the sample-to-sample fluctuations of the free-energy. In order to determine the moments of the smalldeviation distribution we should first compute the derivatives with respect to n of Φ N ( n ) at n = 0, and then takethe limit N → ∞ . This apparently simple step is a complex and open problem also in the SK model [13, 14, 15];instead the opposite case in which one takes first the N → ∞ limit and then the n → n, β ) = lim N →∞ Φ N ( n, β ) (2)It is well known that the probability of large deviations is related to the function Φ( n, β ). Indeedexp( − βnN Φ( n, β )) = Z J ( β ) n = exp( − nN βf J ( β )) , (3)where f J is the system-dependent free energy per spin . The region of positive n corresponds to fluctuations wherethe free energy is smaller than the typical one and the region on negative n corresponds to fluctuations where the freeenergy is larger than the typical one.We define the large deviation function for the free energy, L ( f ), (that we will call in the following the samplecomplexity because it is related to the number of samples with free energy equal to f ) as the logarithm divided by N of the probability density of samples with free energy per spin f in the thermodynamic limit: L ( f ) ≡ lim N →∞ log( P N ( f )) N . (4)For large N the majority of the samples has free energy per spin equal to f typ , and all other values have exponentiallysmall probability. Consistently L ( f ) is less or equal than zero, the equality holding f = f typ , i.e. L ( f typ ) = 0. Forsome values of f it is possible that L ( f ) = −∞ , meaning that the probability of large deviations goes to zero fasterthan exponentially with N . In the thermodynamic limit the function Φ( n, β ) defined in eq. (1) yields the Legendretransform of L ( f ) [16], indeed we have: − βn Φ( n ) = − βnf + L ( f ) (5)where f is determined by the condition: βn = ∂L∂f (6)and equivalently we have: L ( f ) = βnf − βn Φ( n ) (7)where βn is determined by the condition: f = ∂n Φ ∂n . (8)Note that while numerical methods are best suited to study small deviations (precisely because they are typical)present day theoretical methods deal mainly with large deviations. This is because at the theoretical level largedeviations requires essentially treating a replica field theory at the mean-field level, while small deviations requires thecomputation of the loop corrections. Furthermore the problem is made even more complicated by the fact that thesecorrections are singular [15]. Nevertheless it is usually assumed that information on small deviations can be extractedfrom large deviations [14, 16]. In particular if the expansion of n Φ( n ) around n = 0 reads n Φ( n ) = nf typ + c n one expects that σ N ≈ ( − c /β ) / N − / and that the small deviation function is a Gaussian. In the SK modelinstead n Φ( n ) = nf typ + c n for positive n and n Φ( n ) = nf typ for negative n [24] and this has led to the 5 / σ N [16]. Correspondingly the small deviation distribution is not expected to be a Gaussian,as confirmed by the numerics [2]. In this case however the constant c is not related to the sixth moment of thesmall deviation distribution but rather to its right tail [14]. Note that the fact that the function Φ( n ) is constant fornegative n leads to L ( f ) = −∞ for f > f typ , indeed in this region the probability vanishes as exp[ − O ( N )] [15]. Thisinterpretation scheme allows us to make contact between our results and those of [2]. Indeed we find that in randomgraphs with fixed connectivity the presence of an n term in the expansion of n Φ( n ) depends on the nature of thecoupling distribution. For a generic distribution ( e.g. Gaussian) of the coupling the term is present and the smalldeviations are expected to be Gaussian. Nevertheless if the couplings have a bimodal distribution J ij = ± ˜ J the n term is absent and Φ( n ) has the same qualitative properties of the SK model, namely n Φ( n ) = nf typ + c n (with adifferent c ) for positive n and n Φ( n ) = nf typ for negative n . Thus in the bimodal case we expect that σ N ∝ N − / and that the small deviation distribution is not a Gaussian, possibly the same of the SK model [15]. With respect tothe data of [2] we think that the expected 5 / n ) are thus in agreement with the findings of [2]. Interestingly enough the authors of [2] suggestedthat the peculiar behavior of free energy fluctuations for bimodal distribution of the couplings is caused by the factthat the Hamiltonian is locally homogeneous. Indeed around a given site the negative couplings can be transformedin positive coupling through a Gauge transformation. The process must stop when the one of the sites alreadyencountered in the process is reached again due to a loop of the graph, however since loops are typically large thedisordered nature of the model appears “at infinity”. As we will see in the following our results put this intuition ona firm ground. Indeed we will show that are precisely the spatial inhomogeneities of the interactions that generate a n term in n Φ( n ). On the contrary in the bimodal case, the local homogeneity allows to obtain a solution that doesnot fluctuate over the sites (in a sense to be specified below) and this guarantees that Φ( n ) = nf eq for positive andnegative n both in the Replica-Symmetric (RS) case and for n < n ) for thespin-glass on a random lattice with fixed connectivity (the Bethe lattice). In sections III and IV we will considerrespectively the RS and RSB solution. In both cases we will show that local inhomogeneities lead to the presence of a O ( n ) term whose coefficient can be expressed in terms of the spatial fluctuations of the local fields. On the contrarywhen the interactions have a bimodal distribution the resulting local homogeneity allows to obtain a solution thatdoes not fluctuate over the sites and leads to the vanishing of the n term. This also implies that Φ( n ) = nf eq forpositive and negative n in the RS case and for n < n we resort to an expansion of Φ( n ) in powers of theorder parameter. This is presented in section V. We will confirm that in the bimodal case the O ( n ) term vanish andthat the first non-trivial term in the expansion of Φ( n ) is O ( n ) much as in the SK model. At the end we will giveour conclusions and discuss some interesting consequences of our results. II. THE FUNCTIONAL Φ( n ) OF THE BETHE LATTICE SPIN-GLASS
In this section we discuss the potential Φ( n ) of the spin-glass defined on the Bethe lattice with fixed connectivity M + 1. Following [25] we express Φ( n ) as a variational functional of the order parameter ρ ( σ ) that is a functiondefined on n Ising spins σ . The variational expression of the free energy reads: nβ Φ( n ) = M ln Tr { σ } ρ M +1 ( σ ) − M + 12 ln Z Tr { σ } Tr { τ } ρ M ( σ ) ρ M ( τ ) h exp βJ X α σ α τ α i (9)Where the square brackets mean average with respect to the distribution of J . The above expression has to beextremized with respect to ρ ( σ ). We note that it is invariant under a rescaling of ρ ( σ ) so that we can choose anynormalization for it. If we normalize ρ ( σ ) to the corresponding variational equation in terms of ρ ( σ ) reads: ρ ( σ ) = (cid:10) Tr τ ρ M ( τ ) exp Jστ (cid:11) h Tr τ,σ ρ M ( τ ) exp βJστ i . (10)where στ ≡ P α σ α τ α .In the next two sections we discuss the RS and RSB ansatz of ρ ( σ ) that are characterized respectively by fields anddistributions of fields. In the RS case we will find that a crucial condition in order not to have O ( N − / ) Gaussianfluctuations is that the fields do not fluctuate over the sites (which is possible in the low temperature phase only ifthe interactions are locally homogeneous). In the RSB case this condition becomes a condition of homogeneity of thepopulations of fields, meaning that the populations do not fluctuate. Note that at one-step RSB level this correspondsto the fact that we have to consider the so-called factorized solution and the fact that fluctuating solutions are actuallyobtained at this level [26] confirms that the true solution is full-RSB.In the RS case the homogeneity condition guarantees that the same solution valid at n = 0 can be used at n differentfrom zero yielding Φ( n ) = nf eq exactly. Much as in the SK model we expect that this statement holds in the RSBcase only for negative n , because the full-RSB solution at positive n cannot be the same at n = 0 if we require that x min ≥ n where x min is the first breaking point of the q ( x ). Indeed the expansion in the order parameter of section Vshows that the model is mapped in the SK model with different coefficients and an explicit computation shows Φ( n )for positive n has on O ( n ) behaviour as SK. III. THE REPLICA-SYMMETRIC SOLUTION
In this section we study the replica symmetric ansatz on ρ ( σ ). We normalize ρ ( σ ) to one, following [26]. In the RScase ρ ( σ ) is function of P a σ a and is parameterized by a function R ( u ) as: ρ ( σ ) = Z duR ( u ) exp βu P a σ a (2 cosh βu ) n (11)where R ( u ) must satisfy R duR ( u ) = 1 because of the normalization of ρ ( σ ). Accordingly we have:ln Tr { σ } ρ M +1 ( σ ) = ln Z β P M +1 i u i Q M +1 i βu i ! n M +1 Y i =1 R ( u i ) du i (12)We are interested in evaluating what is the dependence on n of the previous quantity for fixed R ( u ). Expanding inpowers of n we get: ln Tr { σ } ρ M +1 ( σ ) = n [ hh A ii ] + n (cid:2) ( hh A ii − hh A ii ) (cid:3) + O ( n ) (13)where we have used: hh A p ii ≡ Z M +1 Y i =1 R ( u i ) du i ln 2 cosh β P M +1 i u i Q M +1 i βu i ! p (14)thus we see that i.f.f. R ( u ) = δ ( u − u ) ( i.e. R ( u ) is concentrated on some value u ) there is no O ( n ) term. On theother hand it is easily seen that in this case there are no higher terms as well, and the following relationship is validat all orders in n : ln Tr { σ } ρ M +1 ( σ ) = n (ln 2 cosh β ( M + 1) u − ( M + 1) ln 2 cosh βu ) (15)So the crucial condition in order not to have a O ( n ) term is that R ( u ) is a delta function, and a sufficient conditionfor this is that the interactions are locally homogeneous. The other term entering the free energy can be expressedas: ln Tr { σ } Tr { τ } ρ M ( σ ) ρ M ( τ ) h exp( β X a σ a τ a ) i == ln Z M Y i =1 R ( u i ) du i M Y i =1 R ( v i ) dv i * X σ,τ exp[ β P Mi u i σ + β P Mi v i τ + βJστ ] Q Mi βu i cosh βv i ! n + (16)where the square brackets mean average with respect to the distribution of J . As above we can expand in powers of n and obtain:ln Tr { σ } Tr { τ } ρ M ( σ ) ρ M ( τ ) h exp( β X a σ a τ a ) i = n [ hh B ii ] + n (cid:2) ( hh B ii − hh B ii ) (cid:3) + O ( n ) (17)where we have used: hh B p ii ≡ Z M Y i =1 [ R ( u i ) du i R ( v i ) dv i ] * ln X σ,τ exp[ β P Mi u i σ + β P Mi v i τ + βJστ ] Q Mi βu i cosh βv i ! p + (18)thus we see that the O ( n ) is absent if R ( u i ) = δ ( u i ) and ρ ( J ) = ± ˜ J , according to the criterion of local homogeneityof the interactions. On the other hand the O ( n ) term is present also if R ( u i ) = δ ( u i ) but ρ ( J ) is not bimodal, e.g. in the high temperature phase of the corresponding model. A. Fluctuations of the Free Energy in the Replica-Symmetric Solution
The presence of a O ( n ) term in the large deviation function leads naturally to assume that the small deviationsof the free energy are Gaussian. A straightforward computation shows that the variance of the small-deviation isproportional to the coefficient c of the O ( n ) in n Φ( n ) according to: h ∆ F i − h ∆ F i = − c β N (19) Around n = 0 the coefficient c can be computed noticing that since the Φ( n ) is stationary with respect to R ( u ) thederivative with respect to n of Φ( n ) is given by its partial derivative with respect to n .Using the definition of Φ( n ), eq. (9), and the expansions eq. (13) and eq. (17) we get: βn Φ( n ) = n (cid:20) M hh A ii − M + 12 hh B ii (cid:21) + n (cid:20) M ( hh A ii − hh A ii ) − M + 12 ( hh B ii − hh B ii ) (cid:21) + O ( n ) (20)where we have used definitions (14) and (18). The previous expression has to be evaluated using the variational R ( u )obtained at n = 0.The RS solution with R ( u ) = δ ( u ) is correct above the critical temperature specified by the condition h tanh β c J i =1 /M . Therefore we conclude that above the critical temperature Φ( n ) in general has a term O ( n ) different from zeroand its coefficient c is given by: c = − M + 14 (cid:0) h (ln cosh βJ ) i − h ln cosh βJ i (cid:1) (21)Clearly this coefficient vanishes in the case of a bimodal distribution. Above the critical temperature the solution is R ( u ) = δ ( u ) also for n = 0 and Φ( n ) reads:Φ( n ) = − ln 2 β − M + 12 βn ln h cosh n βJ i (22)again we see that in the case of a bimodal distribution Φ( n ) does not depend on n . In the low temperature phase weknow that the ρ ( σ ) is no longer a constant. The correct parameterization in the low-temperature spin-glass phase isfull-RSB. In the following section we will describe the RSB ansatz and show that in general the expansion of n Φ( n )has a O ( n ) term. Nevertheless we will see that in the case of a bimodal distribution the O ( n ) term vanishes andthat Φ( n ) is constant for n < n ) for n > O ( n ) much as in the SK model. IV. THE REPLICA-SYMMETRY-BREAKING SOLUTION
The replica-symmetry breaking (RSB) parametrization of ρ ( σ ) in terms of distributions of fields was presented in[26] and we refer to that paper for an explanation of the main ideas underlying it. In particular we will work in theReplica framework rather than using the cavity method. The resulting equations are the same but while the formerallow a quicker derivation the latter unveils the physical meaning of the populations and the appearance of free energyshifts.We introduce the field u that parameterizes a distribution over the values of an Ising spin σ according to the formula P ( σ ) = exp( βuσ ) / βu . We define a probability distribution (population) P (0) ( u ) of such fields a 0-distribution,correspondingly a 1-distribution is a probability distribution on probability distributions (population of populations)and so on. In the following a k -distribution will be written as P ( k ) , and it defines a measure P ( k ) dP ( k − over thespace of k − ρ ( σ ) with K steps of RSB we need: • a K -distribution P ( K ) ; • K integers 1 ≤ x , . . . , x K ≤ n (as usual for n < ≤ x , . . . , x K ≤ n fixed and consider just the dependency on thedistributions. The construction is iterative and requires a set of functions ρ P ( k ) ( σ ) of x k +1 spins with k = 1 , . . . , K + 1(we define x k +1 ≡ n and x ≡ ρ P ( k ) is crucial, we choose to normalize all of them to 1. Wedefine ρ P ( k ) ( σ ) starting from ρ P ( k − ( σ ), we first divide the x k +1 spins in x k +1 /x k groups { σ C } of x k spins labelledby an index C = 1 , . . . , x k +1 /x k . Then we have: ρ P ( k ) ( σ ) = Z P ( k ) dP ( k − x k +1 /x k Y C =1 ρ P ( k − ( { σ } C ) (23)Thus ρ ( σ ) ≡ ρ P ( K ) ( σ ) is defined iteratively starting from the Replica-Symmetric case corresponding to k = 0: ρ P (0) ( σ ) = Z P (0) ( u ) du x Y i =1 exp βuσ i βu (24)With the above definitions it is possible to express the variational free energy (9) in terms of the populations ofpopulations in the same way as we derived eq. (12) and eq. (16). In order to do that we need to introduce twofunctions: ∆ F ( k )1 [ P ( k )1 , . . . , P ( k ) M +1 ] is a function of M + 1 k -populations and ∆ F ( k )12 [ P ( k )1 , . . . , P ( k )2 M , J ] is a function of2 M k -populations and a coupling constant J . Their definition is iterative, i.e. the function at level k is defined interm of the function at level k − F ( k )1 ( P ( k )1 , . . . , P ( k ) M +1 ) ≡ − βx k +1 ln Z " M +1 Y i =1 P ( k ) i dP ( k − i e − βx k +1 ∆ F ( k − ( P ( k − ,...,P ( k − M +1 ) (25)∆ F ( k )12 ( P ( k )1 , . . . , P ( k )2 M , J ) ≡ − βx k +1 ln Z " M Y i =1 P ( k ) i dP ( k − i e − βx k +1 ∆ F ( k − ( P ( k − ,...,P ( k − M ,J ) (26)The above definitions have to be supplemented with the definitions for k = 0 that read:∆ F (0)1 ( P (0)1 , . . . , P (0) M +1 ) ≡ − βx ln Z " M +1 Y i =1 P (0) i du i β P M +1 i =1 u i Q M +1 i =1 βu i ! x . (27)∆ F (0)12 ( P (0)1 , . . . , P (0)2 M , J ) ≡ − βx ln Z " M Y i =1 P (0) i du i σ,τ exp[ β P Mi =1 u i σ + β P Mi = M +1 u i τ + βJστ ] Q Mi =1 βu i ! x . (28) A. The Functional Φ( n ) in the Replica-Symmetry-Breaking solution The variational free energy expressed in term of the K -population P ( K ) that parameterizes ρ ( σ ) reads: nβ Φ( n ) = nM β ∆ F ( K )1 ( P ( K ) , . . . , P ( K ) ) + M + 12 ln h e − βn ∆ F ( K )12 ( P ( K ) ,...,P ( K ) ,J ) i (29)where the square brackets mean average over the coupling constant J and we have used the functions defined above.The proof that the above expression is equivalent to (9) is not very complicated and we will not report it. We justmention that it can be obtained in an iterative way much as we will do in the appendix for the variational equation.In order to determine the K -population that extremizes (29) we need to solve the corresponding variational equationsobtained differentiating it with respect to P ( K ) . An equivalent way to obtain P ( K ) is to consider the variationalequation (10) and rewrite it in terms of P ( K ) , we will show how to do this in the appendix. For practical purposesthis second method is to be preferred because the corresponding equations can be solved by means of a populationdynamics algorithm [26], however in order to study the small- n behaviour of Φ( n ) is more useful to consider thatvariational expression (29).In general the K -population that extremizes (29) depends on the value of n . In order to determine the expressionof Φ( n ) at small values of n we can expand expression (29) at the second order in n around n = 0. This expressionis variational in P ( K ) and therefore the total second derivative of nφ ( n ) with respect to n is equal to the partialsecond derivative. However we must keep in mind that ∆ F ( K )1 and ∆ F ( K )12 both have an implicit dependence from n ,therefore in order to derive with respect to n we make this dependence explicit by rewriting expression (29) as: nβ Φ( n ) = − M ln Z " M +1 Y i =1 P ( K ) dP ( K − i e − βn ∆ F ( K − ( P ( K − ,...,P ( K − M +1 ) ++ M + 12 ln Z " M Y i =1 P ( K ) dP ( K − i h e − βn ∆ F ( K − ( P ( K − ,...,P ( K − M ,J ) i (30)Expanding in powers of n we get: βn Φ( n ) = n (cid:20) M hh A ii − M + 12 hh A ii (cid:21) − n (cid:20) M ( hh A ii − hh A ii ) − M + 12 ( hh A ii − hh A ii ) (cid:21) + O ( n )where we have defined: hh A p ii ≡ Z M +1 Y i =1 P ( K ) dP ( K − i h ∆ F ( K − ( P ( K − , . . . , P ( K − M +1 ) i p hh A p ii ≡ Z M Y i =1 [ P ( K ) dP ( K − i ] Dh ∆ F ( K − ( P ( K − , . . . , P ( K − M , J ) i p E Thus we conclude that the coefficient c of the O ( n ) term in the small- n power series of n Φ( n ) is given by thecoefficient of the above O ( n ) term evaluated with the population P ( K ) corresponding to n = 0. Therefore in general c will be non-zero and we will expect Gaussian fluctuations with variance given by eq. (19).We note that the physical interpretation of the K -RSB ansatz is that on a given sample ( i.e. a graph with a givendisorder realization) the local fields on a given sites are described by a K − P ( K ) represents the distribution over the sites of these K − F ( K − and ∆ F ( K − areinterpreted as the free-energy variations (shifts) that are observed in the process of adding respectively a spin and abond to a given graph [26]. Thus the interpretation of the above equation is that the Gaussian fluctuations of the freeenergy are determined by the local fluctuations of the free energy shifts . B. The factorized solution
The results of the preceding subsection tell us that the O ( n ) term will be absent only if there are no spatialfluctuations in the distribution of the fields. This corresponds to the fact that P ( K ) is given by the so-called factorizedsolution P ( K ) = δ ( P ( K − − P ( K − ). Indeed it is easy to check that if P ( K ) is factorized the term ( hh A ii − hh A ii )vanishes. In order to have that also the term ( hh A ii − hh A ii ) vanishes we need also the condition J = ± ˜ J (in thehigh-temperature phase the O ( n ) term is determined solely by the fluctuations of J ).The variational equation for P ( K ) reported in the appendix shows that P ( K ) can be factorized only if the couplingshave a bimodal distribution J = ± ˜ J . We argue that in this case the correct distribution is indeed factorized. Notethat for K = 1 a non-factorized solution was found in [26] for the bimodal case. We believe that this is an artifact ofthe fact that the correct solution has an infinite number of RSB steps K = ∞ . This is similar to what happens forthe function q ( x ) of the SK model: in the 1RSB ansatz we find q (0) = 0 while q (0) = 0 in the full-RSB solution [18].In the bimodal case one can see that the factorized solution P ( K ) = δ ( P ( K − − P ( K − ) is such that P ( K − isindependent of n . As a consequence Φ( n ) is constant in n . Much as in the SK model we argue that Φ( n ) is constantonly for n <
0, while for n > n = 0 because of thecondition that the smallest RSB parameter x min must be larger than n . In order to study this effect and to determinethe first non-trivial term of Φ( n ) for positive n in the following section we will study an expansion of Φ( n ) in theorder parameter. V. EXPANSION OF Φ( n ) IN POWERS OF THE ORDER PARAMETER
In this section we report the expansion of the potential Φ( n ) of the spin-glass defined on the Bethe lattice with fixedconnectivity M + 1 in powers of the order parameter. Note that following [25] we will use a different normalizationof ρ ( σ ) with respect to the previous subsection, we write it as: ρ ( { σ } ) = n X k =0 b k X ( α ...α k ) q α ...α k σ α . . . σ α k (31)with b k ≡ h cosh n βJ tanh k βJ i and ˜ b k ≡ b k /b (32)The variational equation reads: ρ ( { σ } ) = Tr { τ } ρ M ( τ ) h exp βJ P α σ α τ α i Tr { τ } ρ M ( τ ) (33)expressed in terms of the q α ...α k reads: q α ...α k = Tr { σ } σ α . . . σ α k ρ M ( σ )Tr { σ } ρ M ( σ ) (34)We have expanded expression (9) in powers of the order parameter q ab at fourth order. The four-indexes order-parameter q abcd has been expressed in terms of q ab by means of its variational equation. In the appendix we give somedetails while here we report the results: nβ Φ = − M + 12 ln b − n ln 2 + F var . (35)Where F var ≡ − τ q − ω q − v q + y X abc q ab q ac − u X ab q ab + ˜ y q ) + O ( q ) (36)where the various coefficients depends on b k with k = 2 ,
4, see their explicit expressions below.In the high temperature region we have q = 0 and therefore F var = 0, thus the only relevant term is the firstterm in eq. 35. If J can take just two values J = ± ˜ J we have b = cosh n β ˜ J , therefore for T > T c Φ( n ) is just aconstant in n . From a high temperature expansion we can verify that in this case the fluctuations of the extensivefree energy scale as ∆ F = O (1). On the contrary if the distribution of J is not just peaked at ± ˜ J , Φ( n ) is linear in n and the fluctuations of the extensive free energy are Gaussian with ∆ F = O ( √ N ). Therefore if the graph is locallyhomogeneous the first term immediately yields small O (1) fluctuations above the critical temperature.Below the critical temperature we have to check whether n Φ( n ) is quadratic in n around n = 0. We note that thevarious coefficients { τ, ω, u, v, y } in eq. (36) depend on n and on the temperature, however we first study the modelwith fixed coefficients and variable number of replicas n ; we also reabsorb the coefficient ˜ y in y , which is possibleif we restrict the choice of the matrix q ab to those that verify the condition that ( q ) aa does not depend on a , thisproduce an additional dependence of y from n because y + n ˜ y → y but, like the other coefficients, we first considerit as independent of n .We have computed the free energy of this model, basically generalizing Kondor’s original computation to generalvalues of the coefficients and including all the fourth-order terms. We have found that much as in the SK model: i)the first non linear term in the expansion of n Φ( n ) for n ≥ O ( n ), due to non-trivial cancellations at order O ( n )and O ( n ) and ii) n Φ( n ) is just linear in n for n <
0. More explicitly we have for n > F var = n (cid:18) τ ω + 2 u + 9 v + 6 y ω τ + O ( τ ) (cid:19) + − n (cid:18) ω u − ω u + 36 ω u + O ( τ ) (cid:19) + O ( n ) for n > ω = u = v = y = 1 and we recover Kondor’s value − / O ( n ) coefficient. Nowto study the actual n -dependence of the model we have to take into account the fact that the coefficients { τ, ω, u, v, y } depend on n , indeed the various coefficient reads: τ = 12 M (1 + M ) ˜ b (cid:16) − M ˜ b (cid:17) .ω = M (cid:0) − M (cid:1) ˜ b (cid:16) − M ˜ b (cid:17) .v = M (cid:0) − M (cid:1) ˜ b (cid:16) − M ) + M ( − M ) ˜ b + 2 M ˜ b (cid:16) − M + ( − M ) M ˜ b (cid:17)(cid:17) − M ˜ b .y = − M (cid:0) − M (cid:1) ˜ b (cid:16) − M + (8 − M ) M ˜ b + M ˜ b (cid:16) − M + M (1 + 3 M ) ˜ b (cid:17)(cid:17) − M ˜ b .u = − M (cid:0) − M (cid:1) ˜ b (cid:16) − − M ) + 3 (3 − M ) M ˜ b + 4 M ˜ b (cid:16) − M + M ( − M ) ˜ b (cid:17)(cid:17)(cid:16) − M ˜ b (cid:17) . ˜ y = M (1 + M ) ˜ b (cid:16) − M + (2 − M ) M ˜ b + M ˜ b (cid:16) M ˜ b − (cid:17) + 4 ( M − M ˜ b (cid:16) M ˜ b − (cid:17)(cid:17) (cid:16) M ˜ b − (cid:17) . (38)In order to recover the SK limit of the above coefficients, we have to rescale the couplings as J = ˜ J/M / where˜ J is a random variable with unit variance and take the limit M → ∞ . We obtain τ = 1 − T + O (1 − T ) and ω = u = v = y = 1, while ˜ y = 0 i.e. as it should.Each coefficient depends on β and n through ˜ b k and we have:˜ b k = h tanh k βJ i + n (cid:16) h ln cosh βJ tanh k βJ i − h ln cosh βJ ih tanh k βJ i (cid:17) + O ( n ) (39)Therefore the coefficients for a generic distribution of the J ′ s have a linear dependence on n that, included in expression(37), leads to a O ( n ) dependence of n Φ( n ). Note that the coefficients have a dependence from n that is regular around n = 0 and therefore the first term in eq. (37) is regular in n around n = 0, but there is still the O ( n ) term whichproduces a non regular dependence from n around n = 0. In other words in the general diluted model the function nφ ( n ) develops a regular O ( n ) dependence but there is still a singularity at n = 0 in the sixth derivative. Near thecritical temperature the leading O ( n ) term is given by the n dependence of τ in eq. (37) and therefore is O ( τ ),expanding eq. (38) in powers of n we get for the O ( n ) term of F var : n (cid:18) M M − ( M + 1) τ (cid:16) h ln cosh β c J tanh β c J i − h ln cosh β c J ih tanh β c J i (cid:17) + O ( τ ) (cid:19) (40)Where τ is given by eq. (38) computed at n = 0.We consider now the locally homogeneous case in which J = ± ˜ J with equal probability. We note first that in thiscase the coefficients ˜ b k do not depend on n anymore (see eq. 32) and as a consequence τ, ω, u and v do not depend on n neither. Thus the only dangerous coefficient is y in which we have reabsorbed the coefficient ˜ y through y + n ˜ y → y .The ˜ y coefficient turns out to be zero and therefore the behavior of the model is the same of the SK model, the firstnon linear term in nφ ( n ) being O ( n ). To be more precise we have checked that there are no O ( n ) terms in nφ ( n ) atthe first non-trivial order in τ , because ˜ y defined according to eq. (38) actually is zero only at the critical temperaturewhere ˜ b = 1 /M and ˜ b = 1 /M (because J = ± ˜ J ) but has small non-zero corrections O ( τ ) at higher orders. These O ( n ) corrections are likely cancelled by the O ( n ) higher order term Tr Q Tr Q not included in the computation.In the appendices we report a similar expansion for the variational equation and various quantities relevant for thecomputation. VI. CONCLUSION
We have investigated the large deviations free energy functional Φ( n ) in the case of the Bethe lattice spin-glassand we have confirmed that Gaussian behaviour of the free energy fluctuations has to be expected whenever there0is no local homogeneity of the interactions. In particular the only case in which we found a non-Gaussian SK-likebehaviour in when the random couplings can take only two possible opposite values J ij = ± ˜ J with equal probability.In general the quantity Φ( n ) can be expressed in terms of a distribution of fields. In the RS case we have a singledistribution corresponding to the possible values of the cavity fields at different sites of the lattice for a given disorderrealization. In the RSB phase we have a population of populations, i.e. on each site we have a population of fieldscorresponding to the presence of many states. We have found that if the system is locally homogeneous we can finda locally homogenates distribution of the fields and this leads to the vanishing of the O ( n ) term in Φ( n ). Thus weargue that the correct RSB solution in the bimodal case is the so-called factorized solution. Instead if the system isnot locally homogenates the O ( n ) terms in Φ( n ) can be evaluated using the n = 0 solution because of stationarity.We have also verified that the expansion in power of the order parameter near the critical temperature in the locallyhomogenates case is formally equivalent to that of the SK model and found that Φ( n ) has the same O ( n ) behaviourof SK for small positive n .We note that the fact that in the bimodal case n Φ( n ) = nf typ for n < n ) is the Legendre transform of the large deviations function L ( f ) (see eqs. (5,6,7,8)) it follows that L ( f ) = −∞ for free energies per spin larger than the typical one f typ . This means that the probability of findinga sample with f > f typ is smaller than exp[ O ( N )]. Indeed for the SK model a recent computation [15] has shownthat P ( f ) ∝ exp[ O ( N )]. This scaling cannot hold for the Bethe lattice because while the total number of samples isactually exp[ O ( N )] in the SK model, the total number of samples on the Bethe lattice is exp[( M +1) N ln N ] at leadingorder. Thus we argue that in the Bethe lattice with bimodal distribution of the couplings P ( f ) ∝ exp[ O ( N ln N )]for f > f typ although the actual computation is beyond the scope of this work. For M = 1 detailed computationsare easy. Nevertheless we note that free energies larger than the typical one can only be observed on graphs withtopologies different from the typical one, ( e.g. a regular lattice). In other words the probability of observing a freeenergy (and in particular a ground state energy) larger than the typical one on a graph with typical topology is strictlyzero . Indeed suppose that by just changing the signs of the interactions of a typical graph ( i.e. without modifying theincidence matrix) we could raise the free energy per spin . Since the number of links on a graph is precisely M +12 N the probability of such a sample will be exp[ O ( N )] and this would lead to a non-constant Φ( n ) for n < APPENDIX A: THE VARIATIONAL EQUATIONS IN TERMS OF POPULATIONS
In this appendix we write the variational equations in terms of populations. These equations have been obtained atthe level of one-step RSB in [26] using the cavity method. In the following we write them down for a generic numberof RSB steps using the replica method. The variational equation that extremizes the free energy (9) reads: ρ ( σ ) = (cid:10) Tr τ ρ M ( τ ) exp Jστ (cid:11) h Tr τ,σ ρ M ( τ ) exp βJστ i . (A1)in terms of the K -population the above equation reads: P ( K ) ≡ h e − βn ∆ F ( K ) ( P ( K ) ,...,P ( K ) ,J ) i Z " M Y i =1 P ( K ) dP ( K − i ×× h δ ( P ( K − − ˜ P ( K − ) e − βn ∆ F ( K − ( P ( K − ,...,P ( K − M ,J ) i (A2)Where the square brackets mean average over the disorder. In the above equation we have used the followingfunctions of populations: i) a function ˜ P ( k ) [ P ( k )1 , . . . , P ( k ) M , J ] that yields a k -population from M other k -populationsand ii) a function ∆ F ( k ) [ P ( k )1 , . . . , P ( k ) M , J ] (also called the free-energy shift [26]) that yields a real number from Mk -populations. The definition is iterative: the function ˜ P ( k ) and ∆ F ( k ) at level k of RSB are defined starting fromthe functions ˜ P ( k − and ∆ F ( k − :˜ P ( k ) ( P ( k )1 , . . . , P ( k ) M , J ) ≡ e − βx k +1 ∆ F ( k ) ( P ( k )1 ,...,P ( k ) M ,J ) Z " M Y i =1 P ( k ) i dP ( k − i δ ( P ( k − − ˜ P ( k − ) ×× e − βx k +1 ∆ F ( k − ( P ( k − ,...,P ( k − M ,J ) (A3)and ∆ F ( k ) ( P ( k )1 , . . . , P ( k ) M , J ) = − βx k +1 ln Z " M Y i =1 P ( k ) i dP ( k − i e − βx k +1 ∆ F ( k − ( P ( k − ,...,P ( k − M ,J ) (A4)1The iterative definition has to be supplemented with the k = 0 case that reads:˜ P (0) ( P (0)1 , . . . , P (0) M , J ) ≡ e − βx ∆ F (0) ( P (0)1 ,...,P (0) M ,J ) Z " M Y i =1 P (0) i du i βJ cosh β P i u i Q Mi =1 βu i ! x ×× δ u − ˜ u X i u i , J !! (A5)and ∆ F (0) ( P (0)1 , . . . , P (0) M , J ) ≡ − βx ln Z " M Y i =1 P (0) i du i βJ cosh β P i u i Q Mi =1 βu i ! x (A6)where we used the definition [26]: ˜ u ( h, J ) = 1 β arctanh[tanh βJ tanh βh ] (A7)We recall also the relationship between the populations and ρ ( σ ): ρ P ( k ) ( σ ) = Z P ( k ) dP ( k − x k +1 /x k Y C =1 ρ P ( k − ( { σ } C ) (A8)In the following we will prove the equivalence between eq. (A1) and eq. (A2). We basic step is to prove that thefollowing fundamental equation holds at any level k : " Tr τ C M Y i =1 ρ P ( k ) i ( τ C ) ! exp βJσ C τ C = e − βx k +1 ∆ F ( k ) ( P ( k )1 ,...,P ( k ) M ,J ) ρ ˜ P ( k ) ( σ C ) (A9)In the above equations σ C and τ C are two sets of x k +1 spins and τ C σ C = P x k +1 a =1 σ a τ a . The proof is iterative: assumingthat the equation is satisfied at level k − k . In order to do that wedivide the x k +1 spins σ C in x k +1 /x k groups σ C ′ of x k spins and we use the definition (A8):Tr τ C M Y i =1 ρ P ( k ) i ( τ C ) ! exp βJσ C τ C = Tr τ C M Y i =1 Z P ( k ) i dP ( k − i x k +1 /x k Y C ′ =1 ρ P ( k − i ( τ C ′ ) exp βJσ C τ C == Z " M Y i =1 P ( k ) i dP ( k − i x k +1 /x k Y C ′ =1 Tr τ C′ " M Y i =1 ρ P ( k − i ( τ C ′ ) ! exp βJσ C ′ τ C ′ (A10)Now assuming that eq. (A9) holds true at level k − δ ( P ( k − − ˜ P ( k − ) we get:Tr τ C M Y i =1 ρ P ( k ) i ( τ C ) ! exp βJσ C τ C = (A11)= Z dP ( k − (" M Y i =1 P ( k ) i dP ( k − i δ ( P ( k − − ˜ P ( k − ) e − βx k +1 ∆ F ( k − ( P ( k − ,...,P ( k − M ,J ) ) x k +1 /x k Y C ′ =1 ρ P ( k − ( σ C ′ ) (A12)we see that the term in curly brackets corresponds to the one in the definition (A3), and using the definition (A8) weconclude that eq. (A9) holds true at level k .In order to complete the proof we need to show that eq. (A9) holds for k = 0. In this case σ C is a group of x spins, using eq. (A10) we have:Tr τ C M Y i =1 ρ P (0) i ( τ C ) ! exp βJσ C τ C = Z " M Y i =1 P (0) i du i x Y a =1 X τ a " M Y i =1 exp βu i τ a βu i ! exp βJσ a τ a now summing over each τ a and introducing a delta function δ ( u − ˜ u ( P Mi =1 u i , J )) and using the definitions (A5) and(A6) we can see that eq. (A9) holds true also at level k = 0. The equation (A9) can now be used to prove theequivalence between (A1) and (A2).2 APPENDIX B: THE ORDER-PARAMETER EQUATION
In this appendix we report an order parameter expansion of the variational equation (34). Expanding equation (34)for q abcd in powers of the order parameters we get (see appendices C and D): q abcd = M ( M − − M ˜ b ˜ b ( q ab q cd + q ac q db + q ad q cb ) + O ( q ) (B1)Substituting this expression in eq. (34) for q ab we get at the third order in the order parameter q ab :0 = c q ab + c ( q ) ab + c , ( q ) ab + c , q ab (( q ) bb + ( q ) aa ) + c , q ab + c , q ab Tr Q (B2) c = M ˜ b − c = ˜ b ( M − M ) c , = − ˜ b ( M − M ( M ˜ b + M − M ˜ b − c , = ˜ b ( M − M ( M ˜ b + M − M ˜ b − c , = − b M − M ( M (2 M − b + M − M ˜ b − c , = − ˜ b M − M ( M ˜ b − M ˜ b − (cid:20) σ a σ b (cid:18) ρ ( σ ) − Tr τ ρ M ( τ ) h exp J P c σ c τ c i Tr ρ M ( τ ) (cid:19)(cid:21) (B3)while the equation one obtains by differentiating eq. (9) corresponds to:0 = Tr (cid:20) ρ M − ( { σ } ) σ a σ b (cid:18) ρ ( σ ) − Tr τ ρ M ( τ ) h exp J P c σ c τ c i Tr ρ M ( τ ) (cid:19)(cid:21) (B4)Thus the two expressions are equivalent in the sense that they have the same solution at the order at which they arevalid. It can be checked explicitly that the coefficient c , (as much as a , ) vanishes at zero-th order in the expansionin τ , noticing that at T = T c we have ˜ b = 1 /M and ˜ b = 1 /M (because J = ± ˜ J ).In the Sherrington-Kirkpatrick limit M → ∞ and J = ˜( J ) / √ M with J = 1 the coefficients of the order parameterequation go to the corresponding SK limit as can be also seen noticing that in this limit eq. (B3) reduces to thecorresponding SK equation: q ab = Tr σ a σ b exp[ β P a
18 (Tr q ) + 14 Tr q − X abc q ab q ac + 12 X ab q ab ! + O ( q ) (C4)12 n Tr ˜ g = ˜ b Tr q + 3˜ b ˜ b (cid:18) M ( M − − M ˜ b ˜ b (cid:19)
14 (Tr q ) + Tr q − X abc q ab q ac + 2 X ab q ab ! + O ( q ) (C5)12 n Tr ˜ g = ˜ b
34 (Tr q ) + 3Tr q − X abc q ab q ac + 4 X ab q ab ! + O ( q ) (C6)In order to sum over q abcd in Tr ˜ g we used the following identity valid for a general A abcd symmetric with respect topermutations of its indexes X a
In the following we report the values of traces over the spins. They have been computed using the following generalformula 12 n Tr σ a σ b σ c σ d . . . σ e σ f σ g σ h = X π δ ab δ cd . . . δ ef δ gh − X π δ abcd . . . δ ef δ gh ++ 16 X π δ abcdef . . . δ gh + 4 X π δ abcd δ efgh . . . + · · · (D1)The above expression represents the fact that each of the spins σ a , σ b , . . . must appear an even number of times inorder for the trace to be non zero. The first term describes the case in which each spin appears just two times inthe sum and the index π runs over all different permutations of the indexes that change δ ab δ cd . . . δ ef δ gh . The secondterm describes the case in which one spin appears four times and all the other appear two times. However if this isthe case the first term also give a non-zero contribution, for this reason the second term has the factor − π runs over all permutations of the indexex thatchange the summand. The third term corresponds to the case in which one spin appears six times in the sum, whilethe fourth corresponds to the case in which two diffent spins appears four times each in the sum. To give an example,in the case of four spins expression (D1) specializes to12 n Tr σ a σ b σ c σ d = δ ab δ cd + δ ac δ bd + δ ad δ cb − δ abcd (D2)Using these expression to couple the replica indexes we get:12 n Tr X ab q ab σ a σ b ! = 0 (D3)12 n Tr X ab q ab σ a σ b ! = 2Tr q (D4)12 n Tr X ab q ab σ a σ b ! = 8Tr q (D5)12 n Tr X ab q ab σ a σ b ! = 48Tr q − X abc q ab q ac + 64 X ab q ab + 12(Tr q ) (D6)Other traces necessary to the expansions are:12 n Tr X ab ( q ) ab σ a σ b ! X ab q ab σ a σ b ! = 4Tr q − X abc q ab q ac + 2(Tr q ) (D7)12 n Tr X mn q mn σ m σ n ! σ a σ b = 2 q ab (D8)12 n Tr X mn q mn σ m σ n ! σ a σ b = 48( q ) ab − q ab (( q ) aa + ( q ) bb ) + 32 q ab + 12 q ab Tr q (D9)12 n Tr X mn ( q ) mn σ m σ n ! X mn q mn σ m σ n ! σ a σ b = 8( q ) ab + 2 q ab Tr q − q ab (( q ) aa + ( q ) bb ) (D10) [1] S. Boettcher, Europhys. Lett. , 453 (2004)[2] J.-P. Bouchaud, F. Krzakala and O. C. Martin, Phys. Rev. B , 224404 (2003).[3] M. Palassini, cond-mat/0307713[4] S. Boettcher, Eur. Phys. J. B , 501 (2005)[5] H. G. Katzgraber, M. Korner, F. Liers, M. Junger and A. K. Hartmann, Phys. Rev. B , 094421 (2005)[6] K. F. Pal, Physica A , 261 (2006)[7] T. Aspelmeier and M.A. Moore, Phys. Rev. Lett. 90, 177201 (2003).[8] C. De Dominicis and P. Di Francesco, cond-mat/0301066.[9] T. Aspelmeier, M. A. Moore and A. P. Young, Phys. Rev. Lett. , 127202 (2003)[10] T. Aspelmeier, A. Billoire, E. Marinari, M.A. Moore Finite size corrections in the Sherrington-Kirkpatrick model , J. Phys.A. Math. Gen. in press.[11] A. Andreanov, F. Barbieri, O. C. Martin, Eur. Phys. J. B. 41 (3), 365 (2004).[12] T. Temesvari, to be published.[13] G. Parisi and T. Rizzo, Phys. Rev. Lett. , 117205. (2008)[14] G. Parisi and T. Rizzo, Phys. Rev. B 79, 134205 (2009).[15] G. Parisi and T. Rizzo, arXiv:0901.1100.[16] A. Crisanti, G. Paladin, H.-J. Sommers and A. Vulpiani, J. PHys. I France , 1325 (1992)[17] J. Wehr and M. Aizenman, J. Stat. Phys. 60, 287 (1990).[18] M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987)[19] I. Kondor, J. Phys. A L127 (1983) [20] M. Talagrand Large deviations, Guerra’s and A.S.S. Schemes, and the Parisi hypothesis , to appear in the proceedings ofthe conference Mathematical Physics of Spin-Glasses, Cortona (2005).[21] S. Boettcher, European Physics Journal B 31, 29-39 (2003).[22] S. Boettcher, Physical Review B 67, Rapid Communications 060403 (2003).[23] F. Liers, M. Palassini, A. K. Hartmann, M. Juenger, Phys. Rev. B 68, 094406 (2003).[24] Vik. Dotsenko, S. Franz and M. Mezard, J. Phys. A. Math. Gen.27