METHOD

. Excluding some special cases, computing the critical inverse-temperature β c of a mixed p -spin spin glass model is a diﬃcult task. The only known method to calculate its value for a general model requires the full power of the Parisi formula. On the other hand, for suﬃciently small inverse-temperatures β , an extremely easy application of the second moment method to the partition function Z N,β shows that 1 N E log Z N,β ≈ 1 N log E Z N,β and thus β ≤ β c . The second moment method works up to an explicit threshold β m ≤ β c . Interestingly, in the important case of the Sherrington-Kirkpatrick model β m = β c . In this work we consider the multi-species spherical mixed p -spin models without external ﬁeld, and characterize by a simple condition the models for which the second moment method works in the whole replica symmetric phase, namely, models such that β m = β c . In particular, for those models we obtain the value of β c . Our proof is independent of the Parisi formula.


Introduction
The high-temperature phase of a mixed p-spin spin glass model (with no external field) consists of inverse-temperatures β ≤ β c such that, for large N , (1.1) Computing the mean of the partition function Z N,β is trivial, and thus provided the above, one has the mean of the free energy F N,β in the large N limit.Moreover, in the high-temperature phase, no 'replica symmetry breaking' occurs.Namely, independent samples from the Gibbs measure are typically orthogonal to each other.Despite the simple behavior of the system in the high-temperature phase, computing the critical value β c is generally a difficult problem.For general mixed p-spin models, the only known method to achieve its value heavily relies on the Parisi formula [20,21,32], one of the deepest, most complicated results in mean-field spin glass theory.See the works of Chen [11] and Talagrand [31] where a characterization for sub-critical inverse-temperatures β is derived from the optimality criterion for the Parisi distribution, for (single-species) models with Ising and spherical spins respectively.
In sharp contrast to the usage of the Parisi formula in its extreme simplicity, an application of the second moment method to the partition function Z N,β very easily allows one to lower bound the critical inverse-temperature β c .Interestingly, for the important SK model [22], the bound actually gives the correct critical value β c .In this paper we focus on the question: when does the second moment work up to the critical β c ?We answer this question for the multi-species spherical mixed p-spin models.
1.1.Definition of the model.Consider a finite set of species S , which will be fixed throughout the paper.For each N ≥ 1, we will suppose that {1, . . ., N } = s∈S I s , for some disjoint I s .Denoting Z + := {0, 1, . ..} and |p| := s∈S p(s) for p ∈ Z S + , let Given some nonnegative numbers (∆ p ) p∈P , define the mixture polynomial in x = (x(s) x(s) p(s) .
We will assume that ξ(1 + ǫ) < ∞ for some ǫ > 0, where for a ∈ R we write ξ(a) for the evaluation of ξ at the constant function x ≡ a.
The multi-species mixed p-spin Hamiltonian H N : S N → R corresponding to the mixture ξ is given by where J i1,...,i k are i.i.d.standard normal variables and if #{j ≤ k : i j ∈ I s } = p(s) for any s ∈ S , then ∆ i1,...,i k = ∆ i1,...,i k (N ) is defined by By a straightforward calculation, the covariance function of H N (σ) is given by where we define the overlap vector Identifying S N with the product space s∈S S(N s ), let µ be the product of the uniform measures on each of the spheres S(N s ).The partition function and free energy at inverse-temperature β ≥ 0 are, respectively, defined by The Gibbs measure is the random measure on S N with density σ) .
If |S | = 1, all the definitions above coincide with the usual (single-species) spherical mixed p-spin model.By Jensen's inequality, It is not difficult to check (also using Jensen's inequality, see Lemma 13 below) that for any β ′ < β, Hence, there exists a critical inverse-temperature β c such that 1.2.The second moment method.By the Paley-Zygmund inequality, if we are able to show for some β that (1.9) lim then for any θ ∈ (0, 1), Z N,β ≥ θEZ N,β with probability not exponentially small in N .Combined with the well-known concentration of the free energy, (see e.g., [17,Theorem 1.2]) this easily implies that β ≤ β c .By Fubini's theorem and symmetry we have that where σ 0 ∈ S N is some arbitrary point.Using the coarea formula, one can then check that where ω d denotes the volume of the unit sphere in R d .Therefore, obviously, (1.10) lim where If we define the threshold inverse-temperature then (1.9) holds if and only if β ≤ β m , from which we have that β m ≤ β c .The second moment method similarly works for models with Ising spins up to a threshold β m as above, if one appropriately modifies the logarithmic entropy term in the definition of f β (r).For the SK model, Talagrand used the method in [29, Section 2] to show that β m = 1/ √ 2 (the fact that β c ≥ 1/ √ 2 was first proved in [1]).Exploiting special properties of the SK model, Comets proved in [14] that β c ≤ 1/ √ 2 and therefore β c = β m .For the p-spin generalization of the SK models, Talagrand [30] used a truncated second moment argument to prove a lower bound for the critical β c .Bolthausen [9] applied the second moment method conditional on an event related to the TAP equations to compute the free energy of the SK model with an external field at high-temperature.
In the context of the spherical models but in a different direction than the above, the second moment method was used in the study of critical points.In [4,8,15,23,28] it was applied to the complexity of critical points to prove its concentration around the mean [2,3,16].1.3.Main results.Our main result is the following theorem.We will make the following assumption, (A) x ∈ [0, 1] S \ {0} =⇒ ξ(x) > 0.
In the single-species case |S | = 1, which is also covered by our results, the assumption always holds.Since d dr(s) f β (0) s∈S = 0, the matrix in (1.11) determines the local behavior of f βm (r) around 0. Roughly speaking, the theorem says that the second moment method works up to the critical inverse-temperature if for β slightly above β m the condition that max r∈[0,1) S f β (r) = 0 is broken around r = 0.
The proof that β m < β c if the matrix in (1.11) is regular is the easy part of the theorem.It will follow by showing that in this case β m < βm ≤ β c for some other threshold βm , which will arise from applying the second moment method to a random variable different from Z N,β .The argument essentially uses the same idea as in the truncated second moment method used by Talagrand in [30].
The main part of the theorem, concerning the case that the matrix (1.11) is singular, is a consequence of the following proposition.For the spherical single-species mixed p-spin models, the Parisi formula for the limit of the free energy was proved by Talagrand [31] for models with even interactions, and later generalized to arbitrary mixtures by Chen [10].For the multi-species spherical mixed p-spin models, the Parisi formula was recently proved by Bates and Sohn [7,6], assuming that the mixture polynomial ξ(x) is convex on [0, 1] S .For the proof of the Parisi formula for models with Ising spins, see the works of Talagrand and Panchenko for the single-species case [18,32] and the work of Panchenko [19] for the multi-species SK model, where it was also assumed that ξ(x) is convex.
For the single-species spherical models, Talagrand also proved in [31] the following characterization of sub-critical inverse-temperatures, using the Parisi formula: β ≤ β c if and only if (1.13) ∀r ∈ [0, 1) : Using that log(1 − r) = − n≥1 r n /n, we have that in the single-species case Reassuringly, this is consistent with Theorem 1. Possibly, an analogue of the characterization of the high-temperature phase from [31] for multi-species models can be proved using the results of [7,6], assuming the convexity of ξ(x).
For single-species spherical models, one can deduce Theorem 1 from the characterization (1.13) and (1.14).Still, even in this setting, it is interesting to understand from basic principles rather than the Parisi formula when does a basic tool like the second moment method fails or succeeds.For the multi-species models, our main result allows one to compute β c for a certain class of models.This result is new, as no analogue for the criterion from [31] is known in this case.In particular, this class includes models which do not saisfy the assumption that ξ(x) is convex1 as in the proof of the Parisi formula in the multi-species setting [5,7,19].Finally, one of the main motivations for this work is that our results are crucial to [27] where we compute the free energy for pure multi-species spherical models using the TAP representation developed in [26].
It is well-known (see Section 3) that for any β < β c and ǫ > 0, with probability going to 1 as N → ∞, where we define the subset Moreover, with high probability, For a point σ ∈ S N , an overlap vector r ∈ [−1, 1] S and width δ > 0, define the subset We will show using (1.15) that for most points σ in L β (ǫ), the free energy on B(σ, r, δ), namely is close to its conditional expectation given H N (σ) (see Lemma 5).By estimating the conditional expectation on B(σ, r, δ) given H N (σ), we will prove the following in Section 3.
Proposition 3. Assume (A) and let β < β c .Then, for any r ∈ [0, 1) S and t > 0, for some c > 0, if δ, ǫ > 0 are sufficiently small, (1.17) lim where A(t) ⊂ S N is the set of points σ such that and κ(x) is a deterministic function such that κ(x) → 0 as x → 0, which only depends on ξ.
Obviously, the free energy (1.16) on B(σ, r, δ) lower bounds the total free energy F N,β .Hence, for β < β c , the proposition in particular gives us a lower bound for the free energy F N,β by using only one point from the set L β (ǫ) ∩ A(t), with high probability.To prove Proposition 2, assuming (1.12) we will show that if there were some inverse-temperature β < β ′ < β c , then this lower bound would imply that EF N,β ′ > 1 2 β ′2 ξ(1) in contradiction to (1.7).In Section 2 we will prove Proposition 2 and Theorem 1.Those proofs will assume Proposition 3, which will be proved in Section 3 that will occupy the rest of the paper.

Proof of the main results
In this section we prove Theorem 1 and Proposition 2, assuming Proposition 3 which we will prove in Section 3.

Proof of Proposition 2. Write f
Note that for any r and real z, where as usual (zr)(s) = zr(s).
Let β be some inverse-temperature and assume that the matrix in (1.12) has some non-negative eigenvalue.There exists some r such that (2.1) From (1.2), it is easy to see that we may assume that this r belongs to [0, 1) S .Assume towards contradiction that β < β c and let β ′ ∈ (β, β c ).With the same r as in (2.1), Hence, we may choose some small enough z and t > 0 such that , where κ(x) is as in Proposition 3. By the latter proposition and the simple Lemma 4 below, with probability going to 1 as N → ∞, there exists a point σ ∈ S N such that Combined with the well-known concentration of the free energy (see [17,Theorem 1.2]), this contradicts (1.7).We therefore conclude that β c ≤ β.

Proof of Theorem 1. Recall that
is singular, then by Proposition 2, β c ≤ β m and thus Henceforth, assume that the matrix above is regular.We will prove that in this case β m < β c .As mentioned in the introduction, the argument we will use is essentially equivalent to the second moment method with truncation used in [30].
Note that β m ≤ βm , since Recall that we assume that the Hessian matrix (2.2) is regular.By the definition of β m , since for some constant α > 0, all the eigenvalues of the Hessian (2.2) of f βm (r) at r = 0 are less than −α.From continuity, the eigenvalues of the Hessian of f β (r) at r = 0 are less than −α/2 for all β in some right neighborhood of β m .Combined with (2.4), for such β, this implies that for some open neighborhood A of r = 0. Obviously, for such β and some small δ > 0, Recall the assumption (A).Since [0, 1 − δ] S \ A is closed, from (2.3) and the continuity of f β (r) and fβ (r) in r and β, for some small c > 0 and Combining the above, we have that β ≤ βm for any β in some small right neighborhood of β m .Therefore, β m < βm ≤ β c , which completes the proof.

Proof of Proposition 3
To prove Proposition 3, we will need the three auxiliary results below.The first is a simple well known computation of the volume of approximate level sets, or entropy for sub-critical β.Lemma 4. If β < β c , then for any ǫ, t > 0, there exists some c > 0 such that, for large N , Define the random fields and the random set Proposition 3 concerns the volume of points in L β (ǫ) such that φ N,β (σ, r, δ) is close to a certain value.The next lemma shows that φ N,β (σ, r, δ) and ϕ N,β (σ, r, δ) are close to each other, up to a subset of small volume.It will allow us to work with the conditional expectation ϕ N,β (σ, r, δ), which only depends on the value of the Hamiltonian at σ. Lemma 5. Suppose that β < β c .For any r ∈ (−1, 1) S and positive δ, ǫ and t, for large N , 1) .
The main ingredient in the proof of Proposition 3 is the following estimate on the conditional expectation ϕ N,β (σ, r, δ).Proposition 6. Assume (A) and let β < β c .Then, for any r ∈ [0, 1) S , almost surely, , where c ξ is some constant and κ(x) is a deterministic function such that κ(x) → 0 as x → 0, and both only depend on ξ.
Next we will prove Proposition 3, assuming the three results above.They will be proved in the following subsections.Let t, ǫ, δ > 0 and r ∈ [0, 1) . By Lemmas 4 and 5, for some a > 0 and large enough N , with probability at least 1 − e −N a , 1 By Proposition 6, a.s., lim sup for c ξ and κ(x) as in the proposition.Hence, a.s., lim sup and thus, for small enough ǫ and δ, lim This completes the proof of Proposition 3. It remains to prove the three results above.
By Markov's inequality, Choose some β 0 ∈ (β, β c ) and δ, x > 0 such that (βξ(1), βξ(1) and To complete the proof it will be enough to show that for some c > 0, for large N , (3.4) where we define since the probability in (3.4) is an upper bound for From the well-known concentration of the free energy (see e.g.[17, Theorem 1.2]), for any x > 0 and N , .
+ o(1), by Markov's inequality Thus, for any 0 < x < δ 2 2ξ(1) , for some c = c(x) > 0, On the event in (3.5), 3.2.Proof of Lemma 5. Let σ ∈ S N be an arbitrary point.Conditional on H N (σ) the value at σ, the Hamiltonian is a Gaussian field whose variance is bounded by the variance before conditioning N ξ (1).Hence, from the well-known concentration of the free energy, for any x ∈ R, 1) .

3.3.
Proof of Proposition 6.The proof will be based on the three lemmas below which will be proved in the next subsections.Define Note that we may identify B(σ, r) with the product spheres, one for each s ∈ S , of codimension 1 in S(N s ).Endow each of those spheres with the uniform probability measure and let ν = ν σ,r denote the product measure on B(σ, r).Similarly to ϕ N,β (σ, r, δ), (see (3.1)) define Lemma 7. Let β ≥ 0 and r ∈ [0, 1) S .For large enough N , almost surely, where c ξ > 0 is a constant that only depends on ξ.
Remark 8.In the Introduction we defined the multi-species mixtures (1.2) with coefficients for p with |p| ≥ 2 and their corresponding Hamiltonians in (1.3).Of course, one may consider mixtures with non-zero coefficients also for p with |p| = 1, for which the summation in the definition of the corresponding Hamiltonian in (1.3) starts from k = 1.
Note that ξr (x) is a mixture as in the remark above and let Hr N (σ) be the corresponding Hamiltonian.We remark that the same mixture has been considered in several previous works in the study of the Gibbs measure [8,24] and in the context of the TAP approach [12,13,25,26].Lemma 9. Let β ≥ 0 and r ∈ [0, 1) S .Then, almost surely, (3.9) lim The last lemma we need approximates the free energy of Hr N (σ), for r close to 0. Lemma 10.Assume (A) and let β < β c .Then, for r ∈ [0, 1) S , (3.10) lim sup where κ(x) is a deterministic function such that κ(x) → 0 as x → 0, which only depends on ξ.
Suppose that ξ(x) satisfies the assumption in (A) and let β < β c .Combining the three lemmas above, we have that for r ∈ [0, 1) S , almost surely, lim sup where c ξ and κ(x) are as in the lemmas.By [26,Lemma 25], for some constant C ξ > 0 that depends only on ξ, By the Borell-TIS inequality and the Borel-Cantelli lemma, almost surely, lim sup Note that from (3.7), using the fact that ∆ p > 0 only if |p| ≥ 2, ξr (1) = ξ(1) + κ( r 2 2 ), where κ(x) is a deterministic function such that κ(x) → 0 as x → 0, which only depends on ξ.Proposition 6 follows by combining the above.It remains to prove the three lemmas above.This will be done in Subsections 3.4-3.6below.
3.4.Proof of Lemma 7. Fix some σ ⋆ ∈ S N .Since the Hamiltonian H N (σ) is a Gaussian process, it can be decomposed as where and ĤN (σ) is a centered Gaussian process, independent of H N (σ ⋆ ), with covariance function 1) .
Note that, since η(σ) = ξ(r) ξ( 1) on B(σ ⋆ , r), For any σ ∈ B(σ ⋆ , r, δ), and therefore Hence, to prove the lemma it will be enough to show that for some c > 0 depending only on ξ, for large N , Since the variance of ĤN (σ) is bounded uniformly in σ by N ξ( 1), the variance of the unconditional Hamiltonian, from the concentration of the free energies around their mean (see [17,Theorem 1.2]) it will be enough to show that with probability that goes to 1 as N → ∞, for c as above.
By [26,Lemma 25], for any C > 0, for some L > 0 that depends only on ξ, with probability at least 1 − e −N C , for all σ, σ ′ ∈ S N , 1 For any σ, σ ′ ∈ S N , Hence, from (3.12) and the fact that H N (σ ⋆ ) is a Gaussian variable with zero mean and variance N ξ (1), with the probability going to 1 as On this event, (3.14) holds since by the co-area formula, where ω d denotes the volume of the unit sphere in R d , ρ denotes the volume measure corresponding to the Riemannian metric on T (σ, δ) induced by the Euclidean structure in R N , and where P s (σ) is the orthogonal projection of (σ i ) i∈Is to the orthogonal space to (σ ⋆,i ) i∈Is .
Extend the Hamiltonian HN from B(σ ⋆ , 0) to a centered Gaussian field on S N whose covariance is given by (3.17).Of course, (3.18) Since A(r) 2 := ξ(r 2 ) − ξ(r) 2 ξ(1) ≥ 0, we may write the Hamiltonian HN as HN (σ) = Hr where X is a standard Gaussian variable independent of the Hamiltonian Hr N (σ) with mixture ξr (x).Obviously, where we define Note that where Obviously, ηr (x) is a mixture of some multi-species model.From (3.21), so is η r (x) (see Remark 8).Let H N (σ).As we shall see in (3.26) below, the free energy of H (1) N (σ) is given by the replica symmetric solution.To estimate the free energy of Hr N (σ) we will define an interpolation from H (1) N (σ) to Hr N (σ) (see (3.22)) and control the derivative of the free energy along the interpolation.To the interpolating Hamiltonian we will add a perturbation Hamiltonian, whose role is ensure that the overlaps are positive.The definition of the perturbation is taken from [19] where Panchenko introduced a multi-species version of the Ghirlanda-Guerra identities, and showed that they are satisfied in the presence of the perturbation Hamiltonian.Provided that the identities are satisfied, the positivity of the overlaps follows from Talagrand's positivity principle [17,Theorem 2.16].The results of [19] concern the multi-species SK model, but they are general and also cover the multi-species spherical models.The proofs for the spherical case have been worked out in [7].
Let W be a countable dense subset of [0, 1] S .For any p ≥ 1 and vector w = (w s ) s∈S ∈ W define s i (w) = √ w s for i ∈ I s and s ∈ S , and consider the Hamiltonian where g w,p i1,...,ip are i.i.d.standard Gaussian variables, independent for all combinations of indices p ≥ 1 and 1 ≤ i 1 , . . ., i p ≤ N .
Consider some one-to-one function j : W → N. Let y = (y w,p ) w∈W ,p≥1 be i.i.d.random variables uniform in the interval [1,2] and independent of all other variables.Define the Hamiltonian h N (σ) = w∈W p≥1 2 −j(w)−p y w.p h N,w,p (σ).
Let γ be an arbitrary number in (0, 1/2) and set For any function f (σ 1 , . . ., σ n ) of n points from S N denote where (G t N,β ) ⊗n denotes the n-fold product measure of G t N,β with itself.Denote the free energy corresponding to X t N (σ) by Conditional on y = (y w,p ) w∈W ,p≥1 , h N (σ) is a Gaussian process with variance bounded by N C, for appropriate constant C > 0 (see e.g.[7, (3.6)]).Hence, using Jensen's inequality, we have that Since s 2 N /N → 0, as N → ∞ the right-hand side of both bounds above goes to 0. From Jensen's inequality, which gives one of the bounds in (3.10).The proof of Lemma 10 will be completed by the following two lemmas.
Lemma 11.At t = 0, the free energy is x(s) p(s)   are two mixtures such that ∆ 2 p ≥ ∆2 p for any p.Let F N,β and FN,β their corresponding free energies.Then lim Denote by H N (σ), HN (σ) and ĤN (σ) the Hamiltonians corresponding to ξ(x), ξ(x) and ξ(x), respectively.Define HN (σ) and ĤN (σ) on the same probability space such that they are independent.Note that in distribution (as processes) since the covariance functions and expectation of the Gaussian processes in both sides are equal.By Jensen's inequality, The matching upper bound follows from (1.7).
3.6.2.Proof of Lemma 12.Note that By Gaussian integration by parts [17, Lemma 1.1], (3.28) Combining this with (3.27), we obtain that To complete the proof of the lemma we need to show that (3.29) lim sup for some function κ(x) as in Lemma 10.
From a similar bound to (3.11) and the Borell-TIS inequality, for some constant C, (1) .
Using this bound, it is easy to check that (3.31) implies that for any ǫ > 0,

1
The subsets I s , of course, vary with N .Denoting N s := |I s |, we will assume that the proportion of each species converges limN →∞ N s N = λ(s) ∈ (0,1), for all s ∈ S .Let S(d) = {x ∈ R d : x = √ d} be the sphere of radius √ d in dimension d.The configuration space of the multi-species spherical mixed p-spin model is S N = (σ 1 , . . ., σ N ) ∈ R N : ∀s ∈ S , (σ i ) i∈Is ∈ S(N s ) .