The Aizenman-Sims-Starr and Guerra's schemes for the SK model with multidimensional spins

We prove upper and lower bounds on the free energy in the Sherrington-Kirkpatrick model with multidimensional (e.g., Heisenberg) spins in terms of the variational inequalities based on the corresponding Parisi functional. We employ the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the generalised random energy model-inspired processes and Ruelle's probability cascades. For this purpose an abstract quenched large deviations principle of the Gaertner-Ellis type is obtained. Using the properties of Ruelle's probability cascades and the Bolthausen-Sznitman coalescent, we derive Talagrand's representation of the Guerra remainder term for our model. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of the non-linear partial differential equations. Solving a problem posed by Talagrand, we show the strict convexity of the local Parisi functional. We prove the Parisi formula for the local free energy in the case of the multidimensional Gaussian a priori distribution of spins using Talagrand's methodology of the a priori estimates.


INTRODUCTION
The Sherrington-Kirkpatrick (SK) model of a mean-field spin-glass has long been one of the most enigmatic models of statistical mechanics. The recent rigorous proof of the celebrated Parisi formula for its free energy, due to Talagrand [30], based on the ingenious interpolation schemes of Guerra [17] and Aizenman, Sims, and Starr [1] constitutes one of the major recent achievements of probability theory. Recently, these results have been generalised to spherical SK-models [29] and to models with spins taking values in a bounded subset of Ê [21].
In this paper, we are mainly concerned with the question of the validity of the Parisi formula in the case where spins take values in a d-dimensional Riemannian manifold. We address the issue of extending the approach of Aizenman, Sims and Starr, and the one of Guerra to the multidimensional spins. We study the properties of the multidimensional Parisi functional. Motivated by a problem posed by [31], we show the strict convexity of the local Parisi functional in some cases.
We partially extend Talagrand's methodology of estimating the remainder term to the multidimensional setting. In the case of the multidimensional Gaussian a priori distribution of spins we prove the validity of the Parisi formula in the low temperature regime. Definition of the model. Let Σ ⊂ Ê d and denote Σ N ≡ Σ N . We define a family of Gaussian processes X ≡ {X(σ )} σ ∈Σ N as follows where the interaction matrix G ≡ {g i, j } N i, j=1 consists of i.i.d. standard normal random variables and, for x, y ∈ Ê d , x, y ≡ ∑ d u=1 x u y u is the standard Euclidean scalar product. In what follows all random variables and processes are assumed to be centred. We shall call H N (σ ) ≡ − √ NX N (σ ) a random Hamiltonian of our model.
Throughout the paper, we assume that we are given a large enough probability space (Ω, F , È) such that all random variables under consideration are defined on it. Without further notice, we shall assume that all Gaussian random variables (vectors and processes) are centred.
We shall be interested mainly in the free energy where β ≥ 0 is the inverse temperature and µ ∈ M f (Σ) is some arbitrary (not necessarily uniform or discrete) finite a priori measure. We assume that the a priori measure µ is such that (1.2) is finite. We shall be interested in proving bounds on the thermodynamic limits of these quantities, e.g., on Remark 1.1. Note that there is no need to include the additional external field terms into the Hamiltonian (1.1), since they could be absorbed into the a priori measure µ.
Mean-field spin-glass models (see, e.g., [7]) with multidimensional (Heisenberg) spins were considered in the theoretical physics literature, see, e.g., [25] and references therein. Rigorous results are, however, rather scarce. An early example is [15], where the authors get bounds on the free energy in the high temperature regime. Methods of stochastic analysis and large deviations are used in [34] to identify the limiting distribution of the partition function and also to obtain some information about the geometry of the Gibbs measure for small β . More recent treatments of the high temperature regime using the very different methods are due to Talagrand [27], see also [28,Subsection 2.13]. The importance of the SK model with multidimensional spins for understanding the ultrametricity of the original SK model [26] (which corresponds to d = 1 and µ being the Rademacher measure in the above notations) was emphasised in [33].
For the SK model, Guerra's scheme gave historically the first way to obtain the variational upper bound on the free energy in terms of the Parisi functional. The scheme is based on the comparison between two Gaussian processes: the first one being the original SK Hamiltonian (1.1) and the other one being a carefully chosen GREM inspired process indexed by σ ∈ Σ N . The second important ingredient is a recursively defined non-linear comparison functional acting on the Gaussian processes indexed by σ ∈ Σ N . The Aizenman-Sims-Starr (AS 2 ) scheme [1,2] gives an intrinsic way to obtain variational upper bounds on the free energy in the SK model. The scheme is also based on a comparison between two Gaussian processes. The first process is the sum of the original SK Hamiltonian X and a GREM-inspired process indexed by additional index space A ≡ AE n . The second one is another GREM-inspired process indexed by the extended configuration space Σ N × A . The scheme uses a comparison functional defined on Gaussian processes indexed by the extended configuration space equipped with the product measure between the original a priori measure and Ruelle's probability cascade (RPC) [24]. The role of the comparison functional in the AS 2 scheme is played by a free energy functional acting on the Gaussian processes indexed by the extended configuration space. In [22] Panchenko and Talagrand have reexpressed Guerra's scheme for the SK model using the RPC.
Talagrand [30] using Guerra's scheme and the wealth of other ingenious analytical insights showed that the variational upper bound is also the lower bound for the free energy in the SK model. This established, hence, the remaining half of the Parisi formula.
A particular case (d = 1, µ with bounded support) of the model we are considering here was treated by Panchenko in [21]. He used the techniques of [30] to prove that in the case d = 1 upper and lower bounds on the free energy coincide (cf. (1.14) and (1.22) in this chapter). However, the results of [21, Section 5 and the proofs of Theorems 2, 5 and 9] are based on relatively detailed differential properties of the optimal Lagrange multipliers in the saddle point optimisation problem of interest. These properties are harder to obtain in multidimensional situations such as that we are dealing with here. In fact, as we show in Theorems 1.1 and 1.2, one can obtain the same saddle point variational principles without invoking the detailed properties of the optimal Lagrange multipliers. This is achieved using a quenched large deviations principle (LDP) of the Gärtner-Ellis type.
The most advanced recent study of spin-glass models with multidimensional spins was attempted by Panchenko and Talagrand in [23], where the multidimensional spherical spin-glass model was considered. The authors combined the techniques of [30,21] to obtain partial results on the ultrametricity and also get some information on the local free energy for their model.
Main results. In this paper, we prove upper and lower bounds on the free energy in the SK model with multidimensional spins in terms of variational inequalities involving the corresponding multidimensional generalisation of the Parisi functional (Theorems 1.1, 1.2, 5. 1, 5.18). For this purpose, we generalise and unify the AS 2 and Guerra's schemes for the case of multidimensional spins, and employ a quenched LDP which may be of independent interest (Theorems 3.1 and 3.2). Both schemes are formulated in a unifying framework based on the same comparison functional. The functional acts on Gaussian processes indexed by an extended configuration space as in the original AS 2 scheme. As a by-product, we provide also a short derivation of the remainder term in multidimensional Guerra's scheme (Theorem 5.4) using wellknown properties of the RPC and the Bolthausen-Sznitman coalescent. This gives a clear meaning to the remainder in terms of averages with respect to a measure changed disorder. The change of measure is induced by a reweighting of the RPC using the exponentials of the GREM-inspired process 2 . See [22] for another approach in the case of the SK model (d = 1).
We study the properties of the multidimensional Parisi functional by establishing a link between the functional and a certain class of non-linear partial differential equations (PDEs), see Propositions 6.1, 6.2 and Theorem 6.2. We extend the Parisi functional to a continuous functional on a compact space (Theorems 6.1, 6.2). We show that the class of PDEs corresponds to the Hamilton-Jacobi-Bellman (HJB) equations induced by a linear problem of diffusion control (Proposition 6.4). Motivated by a problem posed by [31], we show the strict convexity of the local Parisi functional in some cases (Theorem 6.4).
We partially extend Talagrand's methodology of estimating the remainder term to the multidimensional setting (Theorem 5.4, Proposition 7.1, Theorem 7.1). In the case of multidimensional Gaussian a priori distribution of spins we prove the validity of the Parisi formula (Theorem 1.3).
We partially extend Talagrand's methodology of estimating the remainder term to the multidimensional setting (Theorem 5.4, Proposition 7.1, Theorem 7.1). Though the main technical problem of the methodology in the general multidimensional setting remains (Remark 7.5). In the case of the multidimensional Gaussian a priori distribution of spins we prove the validity of the Parisi formula (Theorem 1.3).
Below we introduce the notations, assumptions and formulate our main results. The other results (mentioned above) are formulated and proved in the subsequent sections. The examples listed below verify this assumption: 2 In d = 1 the latter fact was also known to the author of [3], private communication.
(3) Σ = σ ∈ Ê d : σ 2 ≤ 1 -the unit Euclidean ball. Remark 1.2. The boundedness assumption can be relaxed and replaced by concentration properties of the a priori measure. In Section 8 we will exemplify this in the case of a Gaussian a priori distribution. In general a subgaussian distribution will suffice.
We will call the set U the space of the admissible self-overlaps. In analogy to the usual overlap in the standard SK model, we define, for two configurations, σ (i) = (σ overlap matrix R N (σ (1) , σ (2) ) ∈ Ê d×d whose entries are given by R N (σ (1) , σ (2) Fix an overlap matrix U ∈ U . Given a subset V ⊂ U , define the set of the local configurations, Next, define the local free energy We also define where the existence of the limit follows from a result of Guerra and Toninelli [19,Theorem 1]. Consider a sequence of matrices where the ordering is understood in the sense of the corresponding quadratic forms. Consider in addition a partition of the unit interval where z (k) [·] denotes the expectation with respect to the σ -algebra generated by the random vector z (k) .  [14].
Define the local Parisi functional as   (1) and Q (2) of this paper (n = 1). (See also (1.25) below.) Furthermore, a straightforward application of the Cauchy-Schwarz inequality shows that the matrices q and ρ actually satisfy Assumption 1.2. We also note that in the simultaneous diagonalisation scenario in which the matrices in (1.7) are diagonalisable in the same orthogonal basis (see Sections 6.3 and 8.2) this assumption is also satisfied.
The first main result of the present paper uses the AS 2 scheme to establish the upper bound on the limiting free energy p(β ) in terms of the saddle point problem for the local Parisi functional (1.12). (1.14) where the infimum runs over all x satisfying (1.8), all Q satisfying both (1.7) and Assumption 1.2, and all Λ ∈ Sym(d).
We were not able to prove in general that the r.h.s. of (1.14) gives also the lower bound to the thermodynamic free energy. See, however, Theorem 1.3 for a positive example.
To formulate the lower bound on (1.3) we need some additional definitions.
Let the comparison index space be A ≡ AE n . Given α (1) , α (2) ∈ A , define where q L (α (1) , α (2) ) is the (normalised) lexicographic overlap defined as follows The matrix valued lexicographic overlap (1.15) can be used to construct the multidimensional (d ≥ 1) versions of the GREM (see, e.g., [8] and references therein for a review of the results on the one-dimensional case of the model). Here we shall need the following two GREM-inspired real-valued Gaussian processes: A ≡ {A(σ , α)} σ ∈Σ N ,α∈A and B ≡ {B(α)} α∈A with covariance structures A(σ (1) , α (1) )A(σ (2) , α (2) ) = 2 R(σ (1) , σ (2) ), Q(α (1) , α (2) ) , Note that the process A can be represented in the following form: (for different indices i) Gaussian Ê d -valued processes with the following covariance structure: for i ∈ [1; N] ∩ AE, for all α (1) , α (2) ∈ A and all u, v ∈ [1; d] ∩ AE assume that the following holds Given t ∈ [0; 1], we define the interpolating AS 2 Hamiltonian Next, we define the random probability measure [24]. We denote by {ξ (α)} α∈A the enumeration of the atom locations of the RPC and consider the enumeration as a random measure on A (independent of all other random variables around). Define the local AS 2 Gibbs measure G N (t, x, Q,U, V ) by where f : Σ N × A → Ê is an arbitrary measurable function for which the right-hand side of (1.19) is finite.
For V ⊂ U , define the AS 2 remainder term as We define also the limiting AS 2 remainder term where B(U, ε) is the ball with centre U and radius ε. (The existence of the limiting remainder term is proved in Theorem 1.2.) The second main result of this paper uses the AS 2 scheme to establish a lower bound on (1.3) in terms of the same saddle point Parisi-type functional as in the upper bound which includes, however, the nonpositive remainder term (1.21). In one-dimensional situations Talagrand [30] and Panchenko [21], respectively, have shown that the corresponding error term vanishes on the optimiser of the Parisi functional.  [17] (see also more recent accounts [32], [18] and [2]) is also applicable to our model and is covered by our quenched LDP approach, see Theorems 5.1 and 5.18 for the formal statements. Guerra's scheme seems to be more amenable (compared to the Aizenman-Sims-Starr one) for Talagrand's remainder estimates [30], see Section 7. The scheme is based on the following interpolation

Theorem 1.2. For any open set
where the matrices Q * (2) = U * and Q * (1) solve the following system of equations:  Consider the function f : (0 : +∞) 2 → Ê given by (1. 27) The following result shows that, at least, in the highly symmetric situation (1.26) with h = 0 the multidimensional Parisi formula indeed holds true (see Lemma 8.7 for an explanation why the result is indeed a Parisi formula).

Remark 1.7.
Close results have previously been obtained in the case of the spherical model in [23], from where we borrow the general methodology of the proof of the Theorem 1.3. As noted in [23], another more straightforward way to obtain the Theorem 1.3 is to diagonalise the interaction matrix G and use the properties of the corresponding random matrix ensemble.
Organisation of the paper. The rest of the present paper is organised as follows. In Section 2 we record some basic properties of the covariance structure of the process X and establish the relevant concentration of measure results. The section contains also the tools allowing to compare and interpolate between the free energy-like functionals of different Gaussian processes. In Section 3 we derive a quenched LDP of the Gärtner-Ellis type under measure concentration assumptions. Section 4 contains the derivation (based on the AS 2 scheme) of the upper and lower bounds on the free energy of the SK model with multidimensional spins in terms of the saddle point of the Parisi-like functional. In Section 5 we employ the ideas of Guerra's comparison scheme in order to obtain the upper and lower bounds on the free energy and we also get a useful analytic representation of the remainder term. In Section 6 we study the properties of the multidimensional Parisi functional. Section 7 contains the partial extension of Talagrand's remainder term estimates to the case of multidimensional spins. In Section 8 a case of Gaussian a priori distribution of spins is considered and the corresponding local Parisi formula is proved. In the appendix we prove the almost super-additivity of the local free energy, as an application of the Gaussian comparison results of Subsection 2.3.
2. SOME PRELIMINARY RESULTS 2.1. Covariance structure. Our definition of the overlap matrix in (1.4) is motivated by the fact that, as can be seen from a straightforward computation that is, the the covariance structure of the process X N (σ ) is given by the square of the Frobenius (Hilbert-Schmidt) norm of the matrix R N (σ (1) , σ (2) ). The basic properties of the overlap matrix are summarised in the following proposition. 0.

2.2.
Concentration of measure. The following concentration of measure result for the free energy is standard. Proposition 2.2. Let (Σ, S) be a Polish space. Suppose µ is a random finite measure on Σ. Suppose, moreover, that X(σ ), σ ∈ Σ is the family of Gaussian random variables independent of µ which possesses a bounded covariance, i.e., there exists K > 0 such that sup Remark 2.1. An analogous result was given in a somewhat more specialised case in [21].
Thus, thanks to (2.2), we obtainφ We now apply this general result to the our model and also to the free energy-like functional of the GREM-inspired process A.

Gaussian comparison inequalities for free energy-like functionals.
We begin by recalling wellknown integration by parts formula which is the source of many comparison results for functionals of Gaussian processes.
Let F : X → Ê be a functional on a linear space X. Given x ∈ X and e ∈ X, a directional (Gâteaux) . (2.8) With this notation the following lemma holds.
is either locally absolute continuous or everywhere differentiable on Ê. Moreover, assume that the random variables hF(g) and ∂ g e F(g) are in L 1 . Then (2.10) The previous proposition coincides with [21,Lemma 4] (modulo the differentiability condition on (2.9) and the integrability assumptions which are needed, e.g., for [5, Theorem 5.1.2]).
The following proposition connects the computation of the derivative of the free energy with respect to the parameter that linearly occurs in the Hamiltonian with a certain Gibbs average for a replicated system.
where G (u) is the random element of M 1 (Σ) which, for any measurable f : Σ → Ê , satisfies Proof. We write The main ingredient of the proof is the Gaussian integration by parts formula. Denote, for τ ∈ Σ, e(τ) ≡ [X(σ )H u (τ)]. By (2.10), we have Due to the independence, we have Henceforth, the computation of the directional derivative in (2.12) amounts to ∂ ∂t Substituting the r.h.s. of (2.13) into (2.11), we obtain the assertion of the proposition.

QUENCHED GÄRTNER-ELLIS TYPE LDP
In this section, we derive a quenched LDP under measure concentration assumptions. Theorems 3.1 and 3.2 give the corresponding LDP upper and lower bounds, respectively. The proofs of the LDP bounds will be adapted to get the proofs of the upper and lower bounds on the free energy of the SK model with multidimensional spins. However, they may be of independent interest. Note that the existing "level-2" quenched large deviation results of Comets [10] are applicable only to a certain class of mean-field random Hamiltonians which are required to be "macroscopic" functionals of the joint empirical distribution of the random variables representing the disorder and the independent spin variables. The SK Hamiltonian can not be represented in such form, since the interaction matrix consists of i.i.d. random variables. Moreover, it is assumed in [10] that the Hamiltonian has the form H N (σ ) = NV (σ ), where {V (σ )} σ ∈Σ N is a random process taking values in some fixed bounded subset of Ê. Since the Hamiltonian of our model is a Gaussian process, this assumption is also not satisfied, due to the unboundedness of the Gaussian distribution.

Quenched LDP upper bound.
The following assumption will be satisfied for the applications we have in mind. As is clear from what follows, much weaker concentration functions are also allowed.
N=1 is a sequence of random measures on a Polish space (X , X). Assume that there exists some L > 0 such that for any Q N -measurable set A ⊂ X we have Note that Assumption 3.1 will hold in the cases we are interested in due to Proposition 2.2.
is a sequence of random measures on a Polish space (X , X) and for {A r ⊂ X : r ∈ {1, . . . , p}} is a sequence of Q N -measurable sets such that, for some absolute constant L > 0 and some concentration function η N (t) : Ê + → Ê + with the property we have However, our subsequent results hold for substantially worse concentration functions satisfying (3.2).
Since, for a, b ∈ Ê p , the following elementary inequality holds max r a r − max The last equation in turn implies that and the r.h.s. of the previous formula vanishes as N ↑ +∞ due to (3.2).
Let Q N ∈ M (X ), N ∈ AE be a family of random measures on (X , X). Define the Laplace transform Suppose that, for all Λ ∈ Ê d , we have Define the Legendre transform Define, for δ > 0,
Then, for any closed set V ⊂ Ê d , we have Proof.
(1) Suppose at first that V is a compact. Thanks to (3.7), for any x ∈ X , there exists Λ(x) ∈ X such that By compactness, the covering x∈Y A(x) ⊃ V has the finite subcovering, say p r=1 A(x r ) ⊃ V . Hence, Applying condition (3.4), we get By the Chebyshev inequality, Hence, (3.14) together with (3.11) yields Combining (3.12), (3.13), (3.15), we obtain Taking δ ↓ +0 limit, we get the assertion of the theorem. (2) Let us allow now the set V to be unbounded. We first prove that the family Q N is quenched exponentially tight. For that purpose, let and denote We want to prove that Fix some u ∈ {1, . . ., d}. Suppose δ u,p ∈ {0, 1} is the standard Kronecker symbol. Let e u ∈ Ê d be an element of the standard basis of Ê d , i.e., for all p ∈ {1, . . . , d}, we have Thanks to the Chebyshev inequality, we have , a.s. (3.18) Hence, combining (3.17) and (3.18), we obtain Using the same argument, we also get 1 (3.20) We obviously have the assertion of the theorem follows from (3.16) by taking the lim M↑+∞ in the bound (3.24).

Quenched LDP lower bound.
Suppose that, for some Λ ∈ Ê d and all N ∈ AE, we have Let Q N,Λ ∈ M (X ) be the random measure defined by (1) Measure concentration. For all N ∈ AE, there exists some L > 0 and η N : Ê + → Ê + such that, for Assume, in addition, that, for some p > 0, the concentration function satisfies (2) Tails decay condition. Let There exists p ∈ AE such that (3) Non-degeneracy. The family of the sets B j ⊂ X : j ∈ {1, . . . , q} satisfies the following condition there exists some j 0 ∈ {1, . . . , q} such that lim Then, for any Λ ∈ Ê d , we have Proof of Lemma 3.3. We fix some j ∈ {1, . . . , q}. Take an arbitrary ε > 0, Consider, for i ∈ J M,ε , the following closed sets We get We have Due to condition (1), we have (3.33) We put M ≡ M N ≡ N p , and we get Due to property (3.28), there exists K > 0 such that we have Thanks to (3.32), we have For t > ε, we apply (3.33) and (3.34) to (3.37) to obtain (3.38) Combining (3.30) and (3.31), we get LetQ N,Λ be the (random) probability measure defined bŷ .

Lemma 3.4.
Suppose that the measure Q N satisfies the assumptions of the previous lemma. Then (3.29) is valid also forQ N,Λ .
Proof. Similar to the one of the previous lemma.

Remark 3.3.
Recall that a point x ∈ X is called an exposed point of the concave mapping I * if there exists is the set of the exposed points of the mapping I * .
Proof. Let B(x, ε) be a ball of radius ε > 0 around some arbitrary x ∈ X . It suffices to prove that Indeed, since we have (3.45) applying 1 N log(·), taking the expectation, taking lim N↑+∞ , ε ↓ +0 and taking the supremum over x ∈ G in (3.45), we get (3.43).
Take any x ∈ G ∩ E . Then we can find the corresponding vector Λ e = Λ e (x) ∈ Ê d orthogonal to the exposing hyperplane at the point x, as in (3.42). Define the new ("tilted") random probability measureQ N on Ê d by demanding that Hence, Since we have in order to show (3.44) it remains to prove that Hence, we arrive atÎ Moreover, we haveÎ By the assumptions of the theorem, the family Q N satisfies the assumptions of Lemma 3.3. Hence, due to Lemma 3.4, the familyQ N satisfies (3.4). Thus we can apply Theorem 3.1 to obtain Since Λ e is an exposing hyperplane, using (3.48), we get and hence, combining (3.49) and (3.50), we get Therefore, due to the concentration of measure, we have almost surely which implies that, for all ε > 0, we have almost surely and (3.47) follows by yet another application of the concentration of measure.

Corollary 3.1. Suppose that in addition to the assumptions of previous Theorem 3.2 we have
Then E (I * ) = Ê d , consequently Proof. The proof is the same as in the classical Gärtner-Ellis theorem (see, e.g., [13]).

THE AIZENMAN-SIMS-STARR COMPARISON SCHEME
In this section, we shall extend the AS 2 scheme to the case of the SK model with multidimensional spins and prove Theorems 1.1 and 1.2, as stated in the introduction. We use the Gaussian comparison results of Section 2.3 in the spirit of AS 2 scheme in order to relate the free energy of the SK model with multidimensional spins with the free energy of a certain GREM-inspired model. Comparing to [1], due to more intricate nature of spin configuration space, some new effects occur. In particular, the remainder term of the Gaussian comparison non-trivially depends on the variances and covariances of the Hamiltonians under comparison. To deal with this obstacle, we use the idea of localisation to the configurations having a given overlap (cf. (1.5)). This idea is formalised by adapting the proofs of the quenched Gärtner-Ellis type LDP obtained in Section 3.
4.1. Naive comparison scheme. We start by recalling the basic principles of the AS 2 comparison scheme (see, e.g., [7,Chapter 11]). It is a simple idea to get the comparison inequalities by adding some additional structure into the model. However, the way the additional structure is attached to the model might be suggested by the model itself. Later on we shall encounter a real-world use of this trick. Let (Σ, S) and (A , A) be Polish spaces equipped with measures µ and ξ , respectively. Furthermore, let is a suitable real-valued Gaussian process.
which we can obtain, e.g., from Proposition 2.5. Combining (4.2) and (4.3), we get the bound  Let is a suitable Gaussian process. Let us consider the following family (N ∈ AE) of random measures on the Borell subsets of Sym(d) generated by the SK Hamiltonian, and consider also the following family of the random measures generated by the Hamiltonian A(σ , α) where the parameters Q and U are taken from the definition of the process A(α) (cf. (1.7)). The vector x defines the random measure ξ ∈ M (A ) (cf. (1.8)), and, hence, also the measure π N ∈ M (Σ × A ).

Remark 4.2.
To lighten the notation, most of the time we shall not indicate explicitly the dependence of the following quantities on the parameters x, Q, U.
Consider (if it exists) the Laplace transform of the measure (4.7) Define the following Legendre transform Denote, for δ > 0, Note that the result of [19] assures the existence of the limit in the previous formula.
Moreover, for all N ∈ AE, we have that is I N (·) in fact does not depend on N. Proof.
(1) We prove (4.13), the proof of (4.12) is similar. Since U is a compact, it follows that, for arbitrary ε > 0, there exists the following ε-partition of U For small enough ε, we have Therefore, if we denote Hence, Let ε ↓ +0 in (4.15) and we arrive at That is, the existence of L N (Λ) and the representation (4.13) are proved.
(2) For all N ∈ AE, we have, by the RPC averaging property (see, e.g., [2,Theorem 5.4] or Theorem 5.3, Proof of Theorem 1.1. In essence, the proof follows almost literally the proof of Theorem 3.1. The notable difference is that we apply the Gaussian comparison inequality (Proposition 2.5) in order to "compute" the rate function in a somewhat more explicit way. Due to (4.5), we can without loss of generality suppose that V is compact. For any δ > 0 and U ∈ V , by (4.10), there exists Fix some ε > 0. Without loss of generality, we can suppose that all the neighbourhoods satisfy additionally the condition diam V (U) ≤ ε. By compactness, the covering U∈V V (U) ⊃ V has a finite subcovering, say p r=1 V (U (r) ) ⊃ V . We denote the corresponding to this covering approximants in (4.16) by Due to the concentration of measure Proposition 2.3, we can apply Lemma 3.1 and get In fact, since we know that (4.11) exists, (4.18) implies that where K > 0 is an absolute constant.

Free energy lower bound.
In this subsection, we return to the notations of Section 4.2.

Lemma 4.2. For any
Proof. In view of (1.10), iterative application of the Jensen inequality with respect to z (k) leads to the following Performing the Gaussian integration, we get Define the following Legendre transform Proof of Theorem 1.2. As it is the case with the proof of Theorem 1.1, this proof also follows in essence almost literally the proof of Theorem 3.2. The notable difference is that we apply the Gaussian comparison in order to "compute" the rate function in a somewhat more explicit way. In notations of Theorem 3.2 we are in the following situation: X ≡ Sym(d) and X is the topology induced by any norm on Sym(d).
Let B(U, ε) be the ball (in the Hilbert-Schmidt norm) of radius ε > 0 around some arbitrary U ∈ V . Let us prove at first that Similarly to (4.20), for any (x, Q), we have The random measure P N satisfies the assumptions of Corollary 3.1. Indeed: (1) Due to representation (4.14), mapping I(·) is differentiable with respect to Λ. Henceforth assumption (1) of the corollary is also fulfilled.
(2) Let us note at first that, thanks to Proposition 2.3, we have D(I) = Ê d . Thus, the assumption (2) of Corollary 3.1 is satisfied, as is condition (3.9).
Moreover, the assumptions of Lemma 3.3 are satisfied: (1) The concentration of measure condition is satisfied due to Proposition 2.3.
(2) The tail decay is obvious since the family { P N : N ∈ AE} has compact support. Namely, for all N ∈ AE, we have supp P N = U . Thus the measure Q N,Λ (cf. Hence, due to (4.27), arguing in the same way as in Theorem 3.2, we arrive at (4.26). Note that the N ↑ +∞ limit of R N (x, Q,U, B(U, ε)) exists, since in (4.27) the limits of the other two N-dependent quantities exist due to [19]. The subsequent ε ↓ +0 limit of the remainder term exists due to the monotonicity.

GUERRA'S COMPARISON SCHEME
In this section, we shall apply Guerra's comparison scheme (see the recent accounts by [18,32,2]) to the SK model with multidimensional spins. However, we shall use also the ideas (and the language) of [1]. In particular, we shall use the same local comparison functional (4.6) as in the AS 2 scheme, see (5.4). The section contains the proofs of the upper (5.16) and lower (5.18) bounds on the free energy without Assumption 1.2. The proofs use the GREM-like Gaussian processes, RPCs as in the AS 2 scheme. We also obtain an analytic representation of the remainder term (which is an artifact of this scheme) using the properties of the Bolthausen-Sznitman coalescent. 5.1. Multidimensional Guerra's scheme. Let ξ = ξ (x 1 , . . . , x n ) be an RPC process. Theorem 5.3 of [2] guarantees that there exists a rearrangement ξ = {ξ (i)} i∈AE of the ξ 's atoms in a decreasing order. Recall (1.16) and define a (random) limiting ultrametric overlap q L : AE 2 → [0; n] ∩ as follows where we use the convention that max / 0 = 0. This overlap valuation induces a sequence of random partitions of AE into equivalence classes. Namely, given a k ∈ AE ∩ [0; n], we define, for any i, j ∈ AE, the Bolthausen-Sznitman equivalence relation as follows Given n ∈ AE, assume that x and Q satisfy (1.8) and (1.7), respectively. Recall the definitions of the Gaussian processes X and A which satisfy (1.1) and (1.17), respectively. We consider, for t ∈ [0; 1], the following interpolating Hamiltonian on the configuration space Given U ⊂ Sym + (d), the Hamiltonian (5.3) in the usual way induces the following local free energy where we use the same local comparison functional (4.6) as in the AS 2 scheme. Using (1.5), we obtain then Now, we are going to disintegrate the Gibbs measure defined on U × A into two Gibbs measures acting on U and A separately. For this purpose we define the correspondent (random) local free energy on U as follows For α ∈ A , we can define the (random) local Gibbs measure G (t, Q, α, U ) ∈ M 1 (Σ N ) by demanding that the following holds Let us define a certain reweighting of the RPC ξ with the help of (5.5). We define the random point process {ξ } α∈A in the following wayξ We also define the normalisation operation N : .
We introduce the local Gibbs measure G (t, x, Q, U ) ∈ M 1 (U × A ), for any V ⊂ U × A , as follows Finally, we introduce, what shall call Guerra's remainder term: Note that (5.7) coincides with (1.20) after substituting (1.18) with (1.23).

Local comparison.
We recall for completeness the following.
Proof. The proof is identical to the one of Theorem 1.1.
Define the local limiting Guerra remainder term R(x, Q,U) as follows The existence of the limits in (5.17) is proved similar to the case of the AS 2 scheme, see the proof of Theorem 1.2. Define the natural inverse ρ −1 :

Theorem 5.2. For any open set
be the space of all piece-wise constant paths in Q(U, d) with finite (but arbitrary) number of jumps with an additional requirement that they have a jump at x = 1. Given some ρ ∈ Q ′ (U, d), we enumerate its jumps and define the finite collection of matrices where ρ(x k ) = Q (k) . Let us associate to ρ ∈ Q ′ (U, d) a new pathρ ∈ Q(U, d) which is obtained by the linear interpolation of the path ρ. Namely, let Proof. The proof is straightforward. where ∇ 2 g(y) denotes the matrix of second derivatives of the function g at y ∈ Ê d .

Remark 5.3. It is easy to recognise that the definition of f is a continuous "algorithmisation" of (1.11).
Namely, Markovian properties, see [6].
and also Proof.
(1) To prove (5.23) we notice that  where the last equality is due to Proposition 5.1.
(2) Similarly, (5.24) follows from the following observation where the last equality is due to Proposition 5.1.
Proof. This is a direct consequence of (5.26) and the fact that under the assumptions of the theorem Q(i, j) = Q (k) .

Remark 5.4.
It is obvious from the previous theorem that µ k is a probability measure.
The main result of this subsection is an "analytic projection" of the probabilistic RPC representation which integrates out the dependence on the RPC. Comparing to (1.20), it has a more analytic flavor which will be exploited in the remainder estimates (Section 7). This is also a drawback in some sense, since the initial beauty of the RPCs is lost.

THE PARISI FUNCTIONAL IN TERMS OF DIFFERENTIAL EQUATIONS
In this section, we study the properties of the multidimensional Parisi functional. We derive the multidimensional version of the Parisi PDE. This allows to represent the Parisi functional as a solution of a PDE evaluated at the origin. We also obtain a variational representation of the Parisi functional in terms of a HJB equation for a linear problem of diffusion control. As a by-product, we arrive at the strict convexity of the Parisi functional in 1-D which settles a problem of uniqueness of the optimal Parisi order parameter posed by [31,20].
In particular, the function B is differentiable with respect to the t-variable on (0; 1) and C 2 (Ê d ) with respect to the y-variable.

21) satisfies the final-value problem for the controlled semi-linear parabolic Parisi-type PDE
Proof. A successive application of Lemma 6.1 to (5.21) on the intervals D starting from (x n ; 1) gives (6.5).
Proof. This is an immediate consequence of the RPC averaging property (5.27).
The proof is concluded similarly to the proof of Proof. This proof is the same as in [30].
We now generalise the PDE (6.5). Given a piece-wise continuous x ∈ Q(1, 1) and Q ∈ Q(U, d), consider the following terminal value problem We say that f ∈ C([0; 1] × Ê d → Ê) is a piece-wise viscosity solution of (6.9), if there exists the partition of the unit segment 0 =: is a viscosity solution (see, e.g., [9] . Proof. This is an adaptation of the proof of [31, Theorem 3.1] to the multidimensional case. Assume without loss of generality that the paths ρ (1) and ρ (2) have same jump times {x k } n+1 k=0 . Denote the corresponding overlap matrices as {Q (1,k) } n+1 k=0 and {Q (2,k) } n+1 k=0 . Given s ∈ [0; 1], define the new path ρ(s) ∈ Q ′ (U, d) by assuming that it has the same jump times {x k } n+1 k=0 as the paths ρ (1) , ρ (2) and defining its overlap matrices as Q (k) (s) ≡ sQ (1,k) + (1 − s)Q (2,k) . On the one hand, we readily have On the other hand, using Lemma 6.3, we have Finally, we have Combining the last three formulae, we get the theorem.

Remark 6.2.
Note that using the same argument and notations as in the previous theorem we get that, for  0) is Lipschitzian and can be uniquely extended by continuity to the whole Q(U, d). Proof.
(1) The topology induced by the norm (6.10) coincides with the topology of weak convergence of the above-defined vector measures. Since Q(U, d) is a bounded set, it is compact in the weak topology.
(2) This is an immediate consequence of Proposition 6.2.
In the next result, we summarise some results on the PDE (6.9) for the non-discrete parameters, cf. Proposition 6.1.
(2) By the approximation argument (cf. Theorem 6.1), it is enough to assume that x (1) and Q ∈ Q ′ (U, d). Then Proposition 6.1 gives the existence of the corresponding piece-wise classical solutions of (6.9): f Q,x (1) , f Q,x (2) . These solutions are obviously also the (unique) piece-wise viscosity solutions of (6.9). The comparison result [9, Theorem 5] and the non-linear Feynman-Kac formula [9,Proposition 8 ] give then the claim. (3) This can be seen either from the representation (6.6) and an approximation argument, or exactly as in (2) by invoking the results of [9].
6.1. The Parisi functional. We consider now a specific terminal condition in the system (6.5) given in (5.22).
the solution of (6.5) with the specific terminal condition given by (5.22). Following the ideas in the physical literature, we now define the Parisi functional P(β , ρ, Λ) : The integral in (6.11) is understood in the usual Lebesgue-Stiltjes sense.

Remark 6.4.
Note that the path integral term in (6.11) equals f (0, 0), where f (t, y) is the solution of (6.9) with the following boundary condition Theorem 6.3. We have Proof. The bound (6.12) is a straightforward consequence of Theorem 5.1.

On strict convexity of the Parisi functional and its variational representation.
In this subsection, we derive a variational representation for Parisi's functional. As a consequence, for d = 1, we prove that the functional is strictly convex with respect to the x ∈ Q(1, 1), if the terminal condition g (cf. (6.9)) is strictly convex and increasing. This result is related to the problem of strict convexity of the Parisi functional in the case of the SK model.
Let W ≡ {W (s)} s∈Ê + be the standard Ê d -valued Brownian motion and let {F t } t∈Ê + be the correspondent filtration. Define Proof. We have By an approximation argument, it is enough to prove the strict convexity for the continuous x 1 , x 2 ∈ Q(1, 1) (x 1 = x 2 ). For any γ ∈ (0; 1), we have = γY (Q,x 1 ,u,t,y) (1) + (1 − γ)Y (Q,x 2 ,u,t,y) (1), (6.14) where the strict inequality above is due to the strict concavity of the square root function. The strict convexity and monotonicity of g combined with the representation (6.14) implies that (6.13) is strictly convex as a function of x, since a supremum of a family of convex functions is convex.
Proof. In a way similar to the proof of Theorem 6.2, we successively use [

Simultaneous diagonalisation scenario.
In the setups with highly symmetric state spaces Σ N (such as the spherical spin models of [23] or the Gaussian spin models, see Section 8 below), less complex order parameter spaces as Q(U, d) suffice.

REMAINDER ESTIMATES
In this section, we partially extend Talagrand's remainder estimates to the multidimensional setting. Due to Proposition 5.2, to prove the validity of Parisi's formula it is enough to show that all the µ k terms in (5.30) almost vanish for the almost optimal parameters of the optimisation problem in (5.16). This can be done if the free energy of two coupled replicas of the system (7.3) is strictly smaller than twice the free energy of the uncoupled single system (5.4), see inequality (7.2). However, the systems involved in (7.2) are effectively at least as complex as the SK model itself. In Section 7.2, we again apply Guerra's scheme to obtain the upper bounds on (7.3) in terms of the free energy of the corresponding comparison GREM-inspired model. One might then hope that by a careful choice of the comparison model one can prove inequality (7.2). In Sections 7.3 and 7.4, we formulate some conditions on the comparison system which would suffice to get inequality (7.2), giving, hence, the conditional proof of the Parisi formula, see Theorem 7.1. 7.1. A sufficient condition for µ k -terms to vanish. In this subsection, we are going to establish a sufficient condition for the measures µ k to vanish. This condition states roughly the following. Whenever the free energy of a certain replicated system uniformly in N strictly less then twice the free energy of the single system, the measure µ k vanishes in N → +∞ limit (see Lemma 7.2).
x, Q, Σ(B(U, ε))). (7.4) Proof. The first inequality in (7.4) is obvious, since the expression under the integral in (7.2) is positive. The equality in (7.4) is an immediate consequence of the RPC averaging property (5.27).
In what follows, we shall be looking for the sharper (in particular, strict) versions of the inequality (7.4) because of the following observation due to Talagrand [30].

7.2.
Upper bounds on ϕ (2) : Guerra's scheme revisited. In this subsection, we shall develop a mechanism to obtain upper bounds on ϕ (2) defined in (7.3). This will be achieved in the full analogy to Guerra's scheme by using a suitable Gaussian comparison system.
Hereinafter without further notice we assume that U ∈ Sym + (2d) has the form (7.6), where V is some admissible mutual overlap matrix for U.
(7.8) Remark 7.3. For σ (1) σ (2) ∈ Σ N , we shall use the notation σ (1) σ (2) ∈ Ê 2d N to denote the vector obtained by the following concatenation of the vectors σ (1) and σ (2) Let us observe that the process is actually an instance of the 2d-dimensional Gaussian process defined in (1.1). Hence, it has the following correlation structure, for τ 1 , τ 2 ∈ Σ The path ρ induces also the following two new (independent of everything before) comparison process
Define the corresponding local free energy-like quantity as (cf. (5.4)) t,s . (7.12) To lighten the notation, we indicate hereinafter only the dependence of χ on s. Denote

Lemma 7.3.
There exists C = C(Σ) > 0 such that, for any U as above, we have ∂ ∂ s χ(s,t, k, x, Q, Q, Σ N (L, U, ε, δ )) ≤ −B x,Q + Cε, (7.13) Consequently, N (L, U, ε, δ )) ≤Φ Proof. The idea is the same as in the proof of Theorem 5.1 and is based on Proposition 2.5. Since we are considering the localised free energy-like quantities (7.12), the variance terms induced by the interpolation (7.10) in (2.14) cancel out (up to the correction O(ε)) and we are left with the non-positive contribution of the covariance terms.

(7.19)
Define recursively, for l ∈ [n; 0] ∩ AE, the following quantities Lemma 7.5. We have Proof. This is an immediate consequence of the RPC averaging property (5.27).

Adjustment of the upper bounds on ϕ
Assume r = k. (Other cases are similar or easier as shown for 1-D in [30].) We make the following tuning of the upper bounds of the previous subsection. Set n ≡ n + 1. Let w ∈ [x r−1 /2; x r ]. Define x l , l ∈ [k + 1; n + 1] ∩ AE.
Such Q exists due to (7.22). Moreover, if d ≥ 2, then it is obviously non-unique.
Lemma 7.6. In the above setup, we have Proof. The claim is a straightforward consequence of (7.23) and (7.24).
Finally, for l ∈ [0; k − 2], we recursively obtain X (2) l (y (1) Remark 7.5. Motivated by Lemmata 7.2 and 7.8 (see also Section 7.4), we pose the following problem. Is it true that, as in 1-D (see [30,21]), there exists Q ∈ Q ′ (U, 2d) satisfying the assumption (7.24) such that the following inequality holds Similar problems have at first been posed in [33]. The resolution of the above problem seems to require more detailed information on the behaviour of the Parisi functional (6.11) or, equivalently, of the solution of (6.9) as a function of Q ∈ Q(U, d) .

Talagrand's a priori estimates.
We start from defining a class of the almost optimal paths for the optimisation problem in (6.12). Recall the following convenient definition from [21].
Definition 7.2. Given U ∈ Sym + (d), we shall call the triple (n, ρ * , Λ * ) ∈ AE×Q ′ n (U, d)× Ê d a θ -optimiser of the Parisi functional (6.11), if it satisfies the following two conditions (7.27) Remark 7.6. It is obvious that for any θ > 0 such a θ -optimiser exists. The main convenient feature of this definition (as pointed out in [30]) is that n (the number of jumps of ρ * ) is finite and fixed.
Recalling (5.13), we set Under the following assumption (at first proposed in 1-D in [30]), we shall effectively prove that remainder term almost vanishes on the θ minimisers of (6.11), see Theorem 7.1.

Gronwall's inequality and the Parisi formula.
Proof. The proof follows the argument of [30] (see also [21]) with the adaptations to the case of multidimensional spins. The main ingredients are the Gronwall inequality and Lemma 7.2. Theorem 5.1 implies that We now turn to the proof of the matching lower bound. As in the proof of Theorem 1.2, it is enough to show that (1) We fix an arbitrary U ∈ Sym + (d). Fix also some t 0 ∈ [0; 1). By Assumption 7.1, we can find the corresponding θ (t 0 ,V,U) > 0 with the properties listed in the assumption. We pick any θ ∈ (0; θ (t 0 ,V,U)] and let (n, ρ * , Λ * ) be a correspondent θ -optimiser. Note that, by definition (7.28), we have φ (x * ,Q * ,Λ * ) (1) = P(β , ρ * ,U, Λ * ) and, by Definition 7.2, ε)).
Note that, due to (5.12), we obviously have The definition (7.28) and Theorem 5.4 yield (3) Let us set D ≡ sup σ ∈Σ σ 2 . We note that, for any σ (1) , σ (2) ∈ Σ N , we have Given the constant K from (7.29), for any c > 0, we define the set It is easy to see that by compactness we can find a finite covering of Σ (2),k N (U, ε) by the neighbourhoods (7.11) with centres, e.g., in the corresponding set of admissible overlap matrices That is, there exists M = M(ε, δ ) ∈ AE and the finite collections of matrices where and L * (i) is the corresponding δ -minimal Lagrange multiplier.
(7) In particular, the previous formula yields Note that ϕ N (1, x, Q, B(U, ε)) does not depend on the choice of x and Q. Hence, by (7.32), we obtain The proof of (7.31) is finished by noticing that the θ can be made arbitrary small.

PROOF OF THE LOCAL PARISI FORMULA FOR THE SK MODEL WITH MULTIDIMENSIONAL GAUSSIAN SPINS
In this section, we prove Theorem 1.3. The rich symmetries of the Gaussian a priori distribution allow rather explicit computations of the X 0 terms (see (1.11)). This allows us to prove that the analogon of Assumption 7.1 is satisfied, implying the Parisi formula for the local free energy (Theorem 1.3).
Remark 8.1. The case of Gaussian spins is very tractable due to the (unusually) good symmetry (i.e., the rotational invariance) of the Gaussian measure. Therefore, it is not surprising that in this case the calculus resembles the one for the spherical SK model, cf. [23,29].
We start from the estimates under a generic (i.e., no simultaneous diagonalisation, cf. Section 6.3) scenario.
8.1. The case of positive increments. Let, for k ∈ [0; n] ∩ AE, We define, for Λ ∈ Sym(d), a family of matrices D (l) ∈ Ê d×d n+1 l=0 as follows and, further, for k ∈ [0; n] ∩ AE, We assume that the matrices Λ and C are such that, for all l ∈ [1; n + 1] ∩ AE, we have We need the following two small (and surely known) technical Lemmata which exploit the symmetries of our Gaussian setting. We include their statements for reader's convenience.
Recall that we have and note that That is, the vectors d (l) are (component-wise) increasingly ordered and non-negative.

Lemma 8.4.
We have Proof. This is a standard argument which relies on the standard invariance properties of the determinant and the matrix trace.
Define the 1-D Parisi functional for the case (1.26) as  In this subsection, we adapt the proof of [29] to obtain the equivalence between the (very tractable) Crisanti-Sommers functional [11] and the Parisi one (8.8) in the case of the Gaussian a priori measure (1.26). Similar ideas based on the symmetry of the a priori measure were exploited in the case of the spherical models by [4,23]. We restrict the consideration to 1-D situation for a moment. Given u ≥ 0, consider ρ ∈ Q ′ n (u, 1), λ ∈ Ê, h ∈ Ê and let {d (l) ∈ Ê} n+1 l=1 be the scalars playing the role of matrices D (l) (cf. (8.1)). That is, We define, for k ∈ [1; n] ∩ AE, the family of vectors {s (k) ∈ Ê d } n k=0 by We also define the Crisanti-Sommers functional as follows (8.12) Lemma 8.5. If (ρ, λ ) is an optimiser for (8.8), that is, then, for all k ∈ [1; n] ∩ AE, the pair (ρ, λ ) satisfies (8.14) 15) and, for all k ∈ [1; n] ∩ AE, we have and also (8.17)
Proof. The strategy is the same as in the previous lemma. We rearrange the summands in (8.12) to get We have, for k, l ∈ [1; n] ∩ AE, Hence, (8.28) (2) To handle the case k = n, we note that and, hence, the argument in the previous item shows that (8.28) is also valid for k = n.
N (L, U, ε, δ )) ≤ In this case, as a straightforward calculation shows, the expression in the curly brackets in (8.45) equals By the definitions and the general properties of matrix trace, we have where in the last equality we used (8.47).

The local low temperature Parisi formula.
Proof of Theorem 1.3. The result follows from Theorem 8.1 and Theorem 7.1. Note that the proof of Theorem 7.1 requires a minor modification to cope with the fact that the a priori distribution (1.26) is unbounded. This minor problem can be fixed by considering the pruned Gaussian distribution and using the elementary estimates to bound the tiny Gaussian tails.
APPENDIX A.
The general result of Guerra and Toninelli [19] implies that the thermodynamic limit of the local free energy (1.6) exists almost surely and in L 1 . The following existence of the limiting average overlap is an immediate consequence of this. The following super-additivity result is an application of the Gaussian comparison inequalities obtained in Subection 2.3. Note that the result does not provide enough information for the cavity-like argument of [1]. where X (1) and X (2) are two independent copies of the process X. Given some Gaussian process {C(σ )} σ ∈Σ N , let us introduce the functional Φ N (β )[C] as follows  where the last inequality is due to the fact that, for all α ∈ Σ N (V ) and all τ ∈ Σ N (V ), we have α τ ∈ Σ N+M (V ).
Moreover, for σ = α τ with α ∈ Σ N (V ) and σ ∈ Σ M (V ) we have Applying 1 0 dt to (A.1) and using the previous two formulae, we get the claim.