Limit Theorems for the Non-Convex Multispecies Curie–Weiss Model

Abstract

We study the thermodynamic properties of the generalized non-convex multispecies Curie–Weiss model, where interactions among different types of particles (forming the species) are encoded in a generic matrix. For spins with a generic prior distribution, we compute the thermodynamic limit of the generating functional for the moments of the Boltzmann–Gibbs measure using simple interpolation techniques. For Ising spins, we further analyze the fluctuations of the magnetization in the thermodynamic limit under the Boltzmann–Gibbs measure. It is shown that a central limit theorem (CLT) holds for a rescaled and centered vector of species magnetizations, which converges to either a centered or non-centered multivariate normal distribution, depending on the rate of convergence of the relative sizes of the species.

1. Introduction

The Curie–Weiss model, also known as the mean-field Ising model, is one of the simplest models of magnetism that exhibits phase transitions [1]. In this model, the spins assume discrete binary values ($\pm 1$) and interact uniformly with one another. Due to its simplicity and analytical tractability, it has been applied in a variety of fields, including voting dynamics [2,3,4] and social collective behavior [4,5,6,7]. A multispecies extension of the Curie–Weiss model [8,9,10] has been proposed to capture the large-scale behavior of interacting systems involving multiple types of interacting particles, whose strength depends on which species each particle belongs to. These extensions were originally introduced in statistical physics as approximations of lattice models [11,12] and meta-magnets [13,14], which exhibit both ferromagnetic and antiferromagnetic interactions. From a mathematical physics perspective, the free energy of a two-species ferromagnet was rigorously derived in [15] and further investigated in [16,17,18] with the Hamilton–Jacobi formalism. Beyond the computation of the free energy, the fluctuations of the order parameter of the multispecies Curie–Weiss model were initially studied in [19], under a convexity assumption on the Hamiltonian, using the same approach as [20]. More recently, stronger results have been proved using different approaches. Notably, refs. [21,22] showed the validity of central limit theorems (CLTs) and provided their convergence rate via Stein’s method, while in [23,24,25] a moment generating functional approach was used. We stress that in all the aforementioned literature, the convexity assumption on the interaction matrix always plays a crucial role. A similar convexity condition appears in the context of disordered multispecies models, where a formula for the free energy is only known in the convex case [26,27], for spherical models [28,29], and on the Nishimori line [30,31]. The non-convex case remains an open problem (see [32,33] for partial results on the subject). Regarding models with random interactions, few fluctuation theorems for the order parameter are known: for the Sherrington–Kirkpatrick model, the validity of CLTs is limited to high-temperature regions [34,35,36], unless the model is on the Nishimori line [37], while the multispecies case was studied here [38].

In this paper, we introduce a multispecies Curie–Weiss model with non-convex energy arising from coupling constants, thereby allowing for arbitrary interaction matrices. This extension is motivated by the need to model systems in which interspecies interactions may be competitive—a scenario relevant not only in statistical physics but also in areas such as game theory [39], neural networks [40], and social science [41,42], where it may find applications. Furthermore, we allow for generic spin values within $[-1,1]$, extending the model beyond the classical Ising framework.

The aim of this work is to study the multispecies Curie–Weiss model in full generality, in particular, without any convexity assumption for the interaction matrix. The work is divided into two main parts. The first part examines the limiting free energy of the model for arbitrary spin distributions supported on $[-1,1]$. Using a combination of interpolation methods and decoupling techniques, we derive a variational formula for the free energy. The second part focuses on the asymptotic behavior of the vector of species magnetization in the case of Ising spins. The methods used in the latter are inspired by [43,44]. By generalizing these methods, we demonstrate that the rescaled vector of species magnetization follows a standard CLT in the region of the phase space, where the order parameter concentrates on a single value. On the other hand, when concentration occurs at multiple points, a conditional CLT still applies.

This paper is structured as follows: Section 2 gives a description for the generalized multispecies Curie–Weiss model with a generic coupling matrix. The main results are presented in Section 3, followed by detailed proofs in Section 4. Section 5 concludes the paper and discusses future directions. The Appendix A contains some technical results used in the proofs.

2. Model Description and Definitions

Let $\rho$ be a probability measure supported on $[-1,1]$, and K an integer representing the number of different species. Consider a Hamiltonian system consisting of N interacting spins, each labeled by an integer, ${\sigma }_i$, that lies in the set of indices $\Lambda =\{1,2,\cdots ,N\}$. To introduce a multispecies structure, we divide the set of indices $\Lambda$ into K disjoint subsets ${\Lambda }_p$ of cardinality $N_p\mathrm{for}p=1,\dots ,K$, namely

$${\Lambda }_p\cap {\Lambda }_l=\varnothing \forall p\ne l,\sum _{p=1}^K\left|{\Lambda }_p\right|=N_1+\cdots +N_K=N.$$

(1)

For future convenience, we also introduce the form factors, or relative sizes ratios ${\left({\alpha }_p\right)}_{p\le K}={(N_p/N)}_{p\le K}$, that shall be collected into a diagonal $K\times K$ matrix $\mathit{\alpha }=\mathrm{diag}{\left({\mathit{\alpha }}_p\right)}_{p\le K}$. Naturally, one has ${\sum }_{p\le K}{\alpha }_p=1$. In the following, we also allow these ratios to depend on N, ${\alpha }_{N,p}$, and we shall call their limit ${\alpha }_p$:

$$\underset{N\to \infty }{lim}{\alpha }_{N,p}=\underset{N\to \infty }{lim}{\displaystyle \frac{N_p}{N}}={\alpha }_p,p\le K.$$

(2)

However, for simplicity, we adopt the notation ${\alpha }_p$ instead of ${\alpha }_{N,p}$.

Let us turn back to our Hamiltonian system. The configuration space is ${[-1,1]}^N={\Omega }_N$. Now that the multispecies structure has been introduced, let the interaction be governed by an indefinite symmetric matrix $\mathbf{J}\in {\mathbb{R}}^{K\times K}$ and the species specific external field $\mathbf{h}\in {\mathbb{R}}^K$. A schematic representation of the interaction network is displayed in Figure 1.

The model under study is thus defined as follows: Let $\sigma ={\left({\sigma }_i\right)}_{i\in \Lambda }\in {\Omega }_N$, then for a given $(\mathbf{J},\mathbf{h})\in {\mathbb{R}}^{K\times K}\times {\mathbb{R}}^K$, the multispecies model is defined by the Hamiltonian:

$$\begin{array}{c}\hfill H_N\left(\sigma \right)=-{\displaystyle \frac{1}{2N}}\sum _{i,j=1}^N{J}_{ij}{\sigma }_i{\sigma }_j-\sum _{i=1}^N{h}_i{\sigma }_i=-{\displaystyle \frac{N}{2}}\sum _{p,l=1}^K{m}_p{\alpha }_p{J}_{pl}{\alpha }_l{m}_l-N\sum _{p=1}^K{\alpha }_p{h}_p{m}_p\end{array}$$

(3)

where for each species p the magnetization density is

$$\begin{array}{c}\hfill m_p\left(\sigma \right)={\displaystyle \frac{1}{N_p}}\sum _{i\in {\Lambda }_p}{\sigma }_i\end{array}$$

(4)

and we denote by ${\mathbf{m}}_N={\left(m_p\right)}_{p\le K}\in {[-1,1]}^K$ the magnetization vector. Here, $J_{pl}$ denotes the coupling strength between all species of types p and l, while $h_p$ is the external field acting on all species of type p. The Hamiltonian (3) can now be rewritten as

$$\begin{array}{c}\hfill H_N=-{\displaystyle \frac{N}{2}}({\mathbf{m}}_N,\Delta {\mathbf{m}}_N)-N(\tilde{\mathbf{h}},{\mathbf{m}}_N)\end{array}$$

(5)

where $(·,·)$ denotes the scalar product in ${\mathbb{R}}^K$ and

$$\begin{array}{c}\hfill \Delta =\mathit{\alpha }\mathbf{J}\mathit{\alpha }\mathrm{and}\tilde{\mathbf{h}}=\mathit{\alpha }\mathbf{h}.\end{array}$$

(6)

The joint distribution of $\sigma$ is governed by a Boltzmann–Gibbs measure

$${\mathcal{G}}_N\left(\sigma \right):={\displaystyle \frac{e^{-H_N\left(\sigma \right)}}{Z_N}}\prod _{i=1}^{N}d\rho \left({\sigma }_i\right),$$

(7)

where $Z_N={\int }_{{[-1,1]}^N}{e}^{-H_N\left(\sigma \right)}{\prod }_{i=1}^{N}d\rho \left({\sigma }_i\right)$ is the partition function. Note that, in Equation (7), the usual inverse temperature $\tilde{\beta }$ is absorbed into the model parameters $\mathbf{J}$ and $\mathbf{h}$. Averages with respect to ${\mathcal{G}}_N$ will be denoted by ${\omega }_N(·)$. The generating function for the moments of the measure in (7) is given by

$$\begin{array}{c}\hfill p_N={\displaystyle \frac{1}{N}}log{Z}_N.\end{array}$$

(8)

Note that $p_N$ coincides, up to a multiplicative factor, with the free energy of the model.

3. Main Results

In this section, we provide the variational formula for the large N limit of the generating functional (8) of the generalized multispecies Curie–Weiss model (5). Additionally, in the case of Ising spins, we present central limit theorems for the magnetization vector ${\mathbf{m}}_N$. We reiterate that the coupling matrix $\mathbf{J}$ used throughout this work is an arbitrary symmetric real matrix.

3.1. Thermodynamic Limit of the Generating Functional

Our first result expresses the large N limit of $p_N$ (8) as a variational problem in ${\mathbb{R}}^K$. Let us define the following variational functional:

$$p_{var}(\Delta ,\mathbf{h};\mathbf{x})\equiv p_{var}\left(\mathbf{x}\right)=-{\displaystyle \frac{1}{2}}(\mathbf{x},\Delta \mathbf{x})+\sum _{p=1}^K{\alpha }_{p}log{\int }_{\mathbb{R}}d\rho \left(\sigma \right)exp\left[\sigma \left(\sum _{l=1}^K{J}_{pl}{\alpha }_l{x}_l+h_p\right)\right].$$

(9)

Now, let O be the orthogonal matrix diagonalizing $\Delta$. Then, the following theorem holds:

Theorem 1. 

For any $(\mathbf{J},\mathbf{h})\in {\mathbb{R}}^{K\times K}\times {\mathbb{R}}^K$, we have that

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim}{p}_N=\underset{z_1,\cdots ,z_a}{inf}\underset{z_{a+1},\cdots ,z_K}{sup}{p}_{var}\left(O\mathbf{z}\right).\end{array}$$

(10)

It is worth noting that the large N limit of $p_N$ can also be derived using large deviation techniques [45,46]. However, in this work, we adopt interpolation bounds. For further discussion, we refer interested readers to [47], where the phase diagram of the Curie–Weiss model with ferromagnetic interaction and generic compact spin distribution is analyzed.

3.2. Fluctuations of the Magnetization

Here, we state fluctuation results for the species magnetization in the binary spin case, i.e., $\sigma ={\left({\sigma }_i\right)}_{i\le N}\in {\{-1,1\}}^N$. Let us define the function

$$f\left(\mathbf{x}\right)={\displaystyle \frac{1}{2}}(\mathbf{x},\Delta \mathbf{x})+(\tilde{\mathbf{h}},\mathbf{x})-(\widehat{\mathit{\alpha }},I\left(\mathbf{x}\right)),\mathbf{x}\in {[-1,1]}^K$$

(11)

where $I\left(\mathbf{x}\right)={\left(I\left(x_p\right)\right)}_{p\le K}\in {\mathbb{R}}^K$ and

$$I\left(x\right)={\displaystyle \frac{1-x}{2}}log\left({\displaystyle \frac{1-x}{2}}\right)+{\displaystyle \frac{1+x}{2}}log\left({\displaystyle \frac{1+x}{2}}\right),x\in [-1,1]$$

(12)

and $\widehat{\mathit{\alpha }}$ is the vector associated with the diagonal matrix $\mathit{\alpha }$.

In what follows, we assume that the convergence in (2) is sufficiently fast. More precisely, setting ${\mathit{\alpha }}_N=\mathrm{diag}{\left({\mathit{\alpha }}_{N,p}\right)}_{p\le K}$, we assume that

$${\mathit{\alpha }}_N=\mathit{\alpha }+N^{-\mathit{\theta }}\mathrm{diag}\left(\mathit{\beta }\right)$$

(13)

where $\mathit{\beta }={\left({\beta }_p\right)}_{p\le K}$ with $0<{\beta }_p<\infty$ and $\theta \in \left[{\displaystyle \frac{1}{2}},\infty \right)$. The fluctuations of the magnetization vector ${\mathbf{m}}_N$ depend on the global maximum point(s) of the function f in (11). A detailed analysis of the critical points and the maximizers can be found in [15,21,48] for $K=2$. We also note that for general K and zero external fields, a high temperature condition [22,25] implies that the zero vector is the unique global maximum of f, meaning the system exhibits no spontaneous magnetization.

The following theorems present central limit theorems for the vector of global species magnetization ${\mathbf{m}}_N=(m_1,\dots ,m_K)$ under the measure ${\mathcal{G}}_N$ (7). We begin with the case of a unique, non-degenerate global maximum.

Theorem 2. 

Assume that $(\mathbf{J},\mathbf{h})\in {\mathbb{R}}^{K\times K}\times {\mathbb{R}}^K$ are such that f in (11) has a unique global maximizer $\mathit{\mu }={\left({\mu }_r\right)}_{r\le K}$ with Hessian ${\mathcal{H}}_f\left(\mathit{\mu }\right)\prec 0$. Then, under the measure ${\mathcal{G}}_N$, the following convergence in distribution holds:

$$\begin{array}{c}\hfill \left(\sqrt{N}\sqrt{{\mathit{\alpha }}_N}({\mathbf{m}}_N-\mathit{\mu })\right)\underset{N\to \infty }{\overset{\mathcal{D}}{\to }}\left\{\begin{array}{cc}\mathcal{N}(\mathit{\nu },\sqrt{\mathit{\alpha }}{\mathcal{H}}_f^{-1}\left(\mathit{\mu }\right)\sqrt{\mathit{\alpha }})\hfill & if\theta ={\displaystyle \frac{1}{2}}\hfill \\ \mathcal{N}(\mathbf{0},\sqrt{\mathit{\alpha }}{\mathcal{H}}_f^{-1}\left(\mathit{\mu }\right)\sqrt{\mathit{\alpha }})\hfill & if\theta >{\displaystyle \frac{1}{2}}\hfill \end{array}\right.\end{array}$$

(14)

where $\mathit{\nu }=-\sqrt{\mathit{\alpha }}{\mathcal{H}}_f^{-1}\left(\mathit{\mu }\right)\mathit{\alpha }\mathbf{J}\mathit{diag}\left(\mathit{\mu }\right)\mathit{\beta }$.

When multiple global maxima coexist, Gaussianity can be restored under suitable conditions:

Theorem 3. 

Assume that $(\mathbf{J},\mathbf{h})\in {\mathbb{R}}^{K\times K}\times {\mathbb{R}}^K$ are such that f in (11) has n global maximizers ${\mathit{\mu }}^1,\dots ,{\mathit{\mu }}^n$, each with Hessian ${\mathcal{H}}_f\left({\mathit{\mu }}^i\right)\prec 0$ for all $i=1,\dots ,n$. For any collection ${\left(A_i\right)}_{i\le n}$ of subsets of ${[-1,1]}^K$ such that ${\mathit{\mu }}^i\in \mathrm{int}\left(A_i\right)$ and $f\left({\mathit{\mu }}^i\right)>f\left(\mathbf{x}\right)$ for all $\mathbf{x}\in \mathrm{cl}\left(A_i\right)\setminus \left\{{\mathit{\mu }}^i\right\}$, then

$$\begin{array}{c}\hfill \left(\sqrt{N}\sqrt{{\mathit{\alpha }}_N}({\mathbf{m}}_N-{\mathit{\mu }}^i)\right)|\{{\mathbf{m}}_N\in A_i\}\underset{N\to \infty }{\overset{\mathcal{D}}{\to }}\left\{\begin{array}{cc}\mathcal{N}({\mathit{\nu }}^i,\sqrt{\mathit{\alpha }}{\mathcal{H}}_f^{-1}\left({\mathit{\mu }}^i\right)\sqrt{\mathit{\alpha }})\hfill & if\theta ={\displaystyle \frac{1}{2}}\hfill \\ \mathcal{N}(\mathbf{0},\sqrt{\mathit{\alpha }}{\mathcal{H}}_f^{-1}\left({\mathit{\mu }}^i\right)\sqrt{\mathit{\alpha }})\hfill & if\theta >{\displaystyle \frac{1}{2}}\hfill \end{array}\right.\end{array}$$

(15)

and ${\mathit{\nu }}^i=-\sqrt{\mathit{\alpha }}{\mathcal{H}}_f^{-1}\left({\mathit{\mu }}^i\right)\mathit{\alpha }\mathbf{J}\mathit{diag}\left({\mathit{\mu }}^i\right)\mathit{\beta }$ for all $i=1,\dots ,n$.

4. Proofs

4.1. Proof of Theorem 1

We begin by noting that any indefinite matrix can be decomposed as the sum of two semi-definite matrices:

$$\begin{array}{c}\hfill \Delta ={\Delta }_{+}+{\Delta }_{-}\mathrm{where}{\Delta }_{+}\ge 0,{\Delta }_{-}\le 0.\end{array}$$

(16)

Thus, the scalar products seen so far split into two terms with definite signs. For instance,

$$\begin{array}{c}\hfill ({\mathbf{m}}_N,\Delta {\mathbf{m}}_N)=({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{m}}_N)+({\mathbf{m}}_N,{\Delta }_{-}{\mathbf{m}}_N).\end{array}$$

(17)

Next, we order the species and eigenvectors so that the diagonal components satisfy

$$\begin{array}{cc}\hfill & {\Delta }^D=\mathrm{diag}({\lambda }_1,\cdots ,{\lambda }_a,{\lambda }_{a+1},\cdots ,{\lambda }_K),\mathrm{where}{\lambda }_1,\cdots ,{\lambda }_a\le 0,{\lambda }_{a+1},\cdots ,{\lambda }_K\ge 0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & {\Delta }^D={\Delta }_{+}^D+{\Delta }_{-}^D=\mathrm{diag}({\lambda }_1,\cdots ,{\lambda }_a,0,\cdots ,0)+\mathrm{diag}(0,\cdots ,0,{\lambda }_{a+1},\cdots ,{\lambda }_K).\hfill \end{array}$$

(19)

Here, the superscript D indicates a diagonal matrix. Additionally, note that the orthogonal matrix O simultaneously diagonalizes $\Delta ,{\Delta }_{+},\mathrm{and}{\Delta }_{-}$. We now establish the following key result:

Proposition 1 

(Hybrid Sum Rule). For every N and $\mathbf{x}\in {\mathbb{R}}^K$, the following holds:

$$\begin{array}{c}{p}_N=-{\displaystyle \frac{1}{2}}(\mathbf{x},{\Delta }_{-}\mathbf{x})+{\displaystyle \frac{1}{N}}log{\int }_{{\mathbb{R}}^K}d\rho \left(\sigma \right)e^{{\displaystyle \frac{\beta N}{2}}({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{m}}_N)+\beta N(\tilde{\mathbf{h}}+{\Delta }_{-}\mathbf{x},{\mathbf{m}}_N)}\hfill \\ \hfill +{\displaystyle \frac{1}{2}}{\int }_0^{1}dt{\omega }_{N,t}\left[({\mathbf{m}}_N-\mathbf{x},{\Delta }_{-}({\mathbf{m}}_N-\mathbf{x}))\right],\end{array}$$

(20)

where ${\omega }_{N,t}$ is the Gibbs state induced by a suitable interpolating Hamiltonian.

Proof. 

Consider the following interpolating Hamiltonian:

$$\begin{array}{c}\hfill H_N\left(t\right)=-{\displaystyle \frac{N}{2}}({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{m}}_N)-{\displaystyle \frac{Nt}{2}}({\mathbf{m}}_N,{\Delta }_{-}{\mathbf{m}}_N)-(1-t)({\Delta }_{-}\mathbf{x},{\mathbf{m}}_N)-N(\tilde{\mathbf{h}},{\mathbf{m}}_N).\end{array}$$

(21)

The associated Boltzmann–Gibbs average is ${\omega }_{N,t}$, and the interpolating generating functional $p_N\left(t\right)$ satisfies

$$\begin{array}{cc}\hfill & p_N\left(1\right)=p_N\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & p_N\left(0\right)={\displaystyle \frac{1}{N}}log{\int }_{{\mathbb{R}}^K}d\rho \left(\sigma \right)e^{{\displaystyle \frac{\beta N}{2}}({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{m}}_N)+\beta N(\tilde{\mathbf{h}}+{\Delta }_{-}\mathbf{x},{\mathbf{m}}_N)}.\hfill \end{array}$$

(23)

Differentiating and completing the square yields

$$\begin{array}{c}\hfill p_N^{\prime }\left(t\right)=-{\displaystyle \frac{1}{N}}{\omega }_{N,t}\left(H_N\left(t\right)\right)=-{\displaystyle \frac{1}{2}}(\mathbf{x},{\Delta }_{-}\mathbf{x})+{\displaystyle \frac{1}{2}}{\omega }_{N,t}\left[({\mathbf{m}}_N-\mathbf{x},{\Delta }_{-}({\mathbf{m}}_N-\mathbf{x}))\right].\end{array}$$

The result follows by an application of the fundamental theorem of calculus.  □

Lemma 1. 

Assume the eigenvalue ordering in (19). Then, for any $(\mathbf{J},h)\in {\mathbb{R}}^{K\times K}\times {\mathbb{R}}^K$, one has

$$\begin{array}{c}\hfill p_N(\Delta ,\mathbf{h})\le \mathcal{O}\left({\displaystyle \frac{logN}{N}}\right)+\underset{z_1\cdots z_a}{inf}\underset{z_{a+1}\cdots z_K}{sup}{p}_{var}(\Delta ,\mathbf{h};O\mathbf{z}).\end{array}$$

(24)

Proof. 

We define the grid of hypercubes as $A_k^p=\left[y_k^p,y_k^p+{\displaystyle \frac{2}{N_p}}\right]$, where $y_k^p=-1+2(k-1)/N$ for $k\in \Lambda$, and p is the species label. The vertices of this grid are identified by a multi-index $\gamma =({\gamma }_1,\cdots ,{\gamma }_K)\in {\Lambda }^K$, and and we denote them as ${\mathbf{y}}_{\gamma }$ in the following.

Next, observe that

$$\begin{array}{l}{\displaystyle \frac{1}{N}}log{\int }_{{\mathbb{R}}^N}d\rho \left(\sigma \right)e^{{\displaystyle \frac{N}{2}}({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{m}}_N)+N(\tilde{\mathbf{h}}+{\Delta }_{-}\mathbf{x},{\mathbf{m}}_N)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{1}{N}}log{\int }_{{\mathbb{R}}^N}d\rho \left(\sigma \right)\prod _{p=1}^K\sum _{k=1}^{N_p}\mathbb{1}(m_p\in A_k^p)e^{{\displaystyle \frac{N}{2}}({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{m}}_N)+N(\tilde{\mathbf{h}}+{\Delta }_{-}\mathbf{x},{\mathbf{m}}_N)}\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{1}{N}}log\sum _{{\gamma }_1,\cdots ,{\gamma }_K=1}^{N_1,\cdots ,N_K}{\int }_{{\mathbb{R}}^N}d\rho \left(\sigma \right)\prod _{p=1}^K\mathbb{1}(m_p\in A_{{\gamma }_p}^p)e^{{\displaystyle \frac{N}{2}}({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{m}}_N)+N(\tilde{\mathbf{h}}+{\Delta }_{-}\mathbf{x},{\mathbf{m}}_N)}.\end{array}$$

(25)

Enforcing the constraint ${\prod }_{p=1}^K\mathbb{1}(m_p\in A_{{\gamma }_p}^p)$, we obtain

$$\begin{array}{c}\hfill 0\le \left({\mathbf{m}}_N-{\mathbf{y}}_{\gamma },{\Delta }_{+}({\mathbf{m}}_N-{\mathbf{y}}_{\gamma })\right)\le {\displaystyle \frac{C}{N^2}}\end{array}$$

(26)

where $C>0$, and the multi-index $\gamma =({\gamma }_1,\cdots ,{\gamma }_K)$. Using this bound, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{N}}log& {\int }_{{\mathbb{R}}^N}d\rho \left(\sigma \right)e^{{\displaystyle \frac{N}{2}}({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{m}}_N)+N(\tilde{\mathbf{h}}+{\Delta }_{-}\mathbf{x},{\mathbf{m}}_N)}\le {\displaystyle \frac{C}{N^2}}\hfill \\ \hfill & +{\displaystyle \frac{1}{N}}log\sum _{{\gamma }_1,\cdots ,{\gamma }_K=1}^{N_1,\cdots ,N_K}{\int }_{{\mathbb{R}}^N}d\rho \left(\sigma \right)\prod _{p=1}^K\mathbb{1}(m_p\in A_{{\gamma }_p}^p)e^{-{\displaystyle \frac{N}{2}}({\mathbf{y}}_{\gamma },{\Delta }_{+}{\mathbf{y}}_{\gamma })+N({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{y}}_{\gamma })+N(\tilde{\mathbf{h}}+{\Delta }_{-}\mathbf{x},{\mathbf{m}}_N)}\hfill \\ \hfill & \le {\displaystyle \frac{C}{N^2}}+{\displaystyle \frac{1}{N}}log\sum _{{\gamma }_1,\cdots ,{\gamma }_K=1}^{N_1,\cdots ,N_K}{\int }_{{\mathbb{R}}^N}d\rho \left(\sigma \right)e^{-{\displaystyle \frac{N}{2}}({\mathbf{y}}_{\gamma },{\Delta }_{+}{\mathbf{y}}_{\gamma })+N({\mathbf{m}}_N,{\Delta }_{+}{\mathbf{y}}_{\gamma })+N(\tilde{\mathbf{h}}+{\Delta }_{-}\mathbf{x},{\mathbf{m}}_N)}\hfill \\ \hfill & =\mathcal{O}\left({\displaystyle \frac{logN}{N}}\right)+{\displaystyle \frac{1}{N}}log\underset{\mathbf{y}}{sup}{\int }_{{\mathbb{R}}^N}d\rho \left(\sigma \right)e^{-{\displaystyle \frac{N}{2}}(\mathbf{y},{\Delta }_{+}\mathbf{y})+N(\tilde{\mathbf{h}}+{\Delta }_{-}\mathbf{x}+{\Delta }_{+}\mathbf{y},{\mathbf{m}}_N)}\hfill \\ \hfill & =\mathcal{O}\left({\displaystyle \frac{logN}{N}}\right)+{\displaystyle \frac{1}{N}}log\underset{\mathbf{y}}{sup}{e}^{-{\displaystyle \frac{N}{2}}(\mathbf{y},{\Delta }_{+}\mathbf{y})}{\int }_{{\mathbb{R}}^N}d\rho \left(\sigma \right)e^{N{\sum }_{p=1}^K{\alpha }_p{\displaystyle \frac{1}{N_p}}{\sum }_{i\in {\Lambda }_p}{\sigma }_i(h_p+{\widehat{\mathit{\alpha }}}^{-1}{({\Delta }_{-}\mathbf{x}+{\Delta }_{+}\mathbf{y})}_p)}\hfill \\ \hfill & =\underset{\mathbf{y}}{sup}\left\{-{\displaystyle \frac{(\mathbf{y},{\Delta }_{+}\mathbf{y})}{2}}+\sum _{p=1}^K{\alpha }_{p}log{\int }_{\mathbb{R}}d\rho \left(\sigma \right)exp\left\{\sigma {[\mathbf{h}+{\widehat{\mathit{\alpha }}}^{-1}({\Delta }_{-}\mathbf{x}+{\Delta }_{+}\mathbf{y})]}_p\right\}\right\}+\mathcal{O}\left({\displaystyle \frac{logN}{N}}\right).\hfill \end{array}$$

(27)

Due to the decomposition of $\Delta$, certain combinations of the components of $\mathbf{x}$ and $\mathbf{y}$ do not appear, i.e., those corresponding to the zero eigenvalues of ${\Delta }_{+}\mathrm{and}{\Delta }_{-}$. Therefore, we have

$$\begin{array}{cc}\hfill {\Delta }_{+}\mathbf{y}+{\Delta }_{-}\mathbf{x}=& O{\Delta }_{+}^D{\mathbf{y}}^D+O{\Delta }_{-}^D{\mathbf{x}}^D=O{\Delta }^D{\mathbf{z}}^D=\Delta \mathbf{z}\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \mathrm{where}& {\mathbf{z}}^D=(x_1^D,\cdots ,x_a^D,y_{a+1}^D,\cdots ,y_K^D)\hfill \end{array}$$

(29)

and D, when accompanying vectors, indicates vectors read in the basis of the eigenvectors of $\Delta$ (i.e., they denote vectors expressed in the eigenbasis of $\Delta$). Hence, only K degrees of freedom remain. Now, by Equation (23), we have

$$\begin{array}{l}{p}_N\left(0\right)\le \mathcal{O}\left({\displaystyle \frac{logN}{N}}\right)+\underset{y_{a+1}^D,\cdots ,y_K^D}{sup}{p}_{var}\left({\Delta }_{+},{\widehat{\mathit{\alpha }}}^{-1}{\Delta }_{-}\mathbf{x}+\mathbf{h};O{\mathbf{y}}^D\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\mathcal{O}\left({\displaystyle \frac{logN}{N}}\right)+\underset{y_{a+1}^D,\cdots ,y_K^D}{sup}\left\{-{\displaystyle \frac{1}{2}}\left(O{\mathbf{y}}^D,{\Delta }_{+}O{\mathbf{y}}^D\right)+\sum _{p=1}^K{\alpha }_{p}log{\int }_{\mathbb{R}}d\rho \left(\sigma \right)e^{\sigma {({\widehat{\mathit{\alpha }}}^{-1}\Delta O{\mathbf{z}}^D+\mathbf{h})}_p}\right\}.\end{array}$$

(30)

Note that this holds for any $x_1^D,\cdots ,x_a^D$ corresponding to the negative definite matrix. Adding the missing quadratic term (${\Delta }_{-}$) from the derivative and optimize over the remaining degrees of freedom, we obtain

$$\begin{array}{c}\hfill p_N\le \mathcal{O}\left({\displaystyle \frac{logN}{N}}\right)+\underset{z_1,\cdots ,z_a}{inf}\underset{z_{a+1},\cdots ,z_K}{sup}{p}_{var}\left(O\mathbf{z}\right).\end{array}$$

We omitted D since $\mathbf{z}$ is now a dummy variable.  □

Now we need to find the other bound.

Lemma 2. 

For any $(\mathbf{J},h)\in {\mathbb{R}}^{K\times K}\times {\mathbb{R}}^K$, we have

$$\begin{array}{c}\hfill p_N\ge \underset{z_1,\cdots ,z_a}{inf}\underset{z_{a+1},\cdots ,z_K}{sup}{p}_{var}\left(O\mathbf{z}\right)+\mathcal{O}\left({\displaystyle \frac{1}{N}}\right).\end{array}$$

(31)

Proof. 

For any $\mathbf{z}\in {\mathbb{R}}^K$, consider the interpolating Hamiltonian:

$$\begin{array}{c}\hfill H_N\left(t\right)=-{\displaystyle \frac{N}{2}}({\mathbf{m}}_N,\Delta {\mathbf{m}}_N)+{\displaystyle \frac{Nt}{2}}({\mathbf{m}}_N-\mathbf{z},\Delta ({\mathbf{m}}_N-\mathbf{z}))-N(\tilde{\mathbf{h}},{\mathbf{m}}_N).\end{array}$$

(32)

The corresponding interpolating generating functional $p_N\left(t\right)$ satisfies

$$\begin{array}{cc}\hfill & p_N\left(0\right)=p_N\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & p_N\left(1\right)=p_{var}(\Delta ,\mathbf{h};\mathbf{z})\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & p_N^{\prime }\left(t\right)=-{\displaystyle \frac{1}{2}}{\omega }_{N,t}\left[({\mathbf{m}}_N-\mathbf{z},\Delta ({\mathbf{m}}_N-\mathbf{z}))\right].\hfill \end{array}$$

(35)

By the convexity of the interpolating generating functional $p_N\left(t\right)$, $p_N\left(1\right)\le p_N\left(0\right)+p_N^{\prime }\left(0\right)$. Thus, for every $\mathbf{z}\in {\mathbb{R}}^K$, we obtain

$$\begin{array}{c}\hfill p_N\ge {\displaystyle \frac{1}{2}}{\omega }_{N,0}\left[({\mathbf{m}}_N-\mathbf{z},\Delta ({\mathbf{m}}_N-\mathbf{z}))\right]+p_{var}(\Delta ,\mathbf{h};\mathbf{z}).\end{array}$$

(36)

Now suppose that $\mathbf{z}$ equals a critical point ${\overline{\mathbf{z}}}^D=O^{-1}\overline{\mathbf{z}}$ of $p_{var}$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial {\mathbf{z}}^D}}{p}_{var}\left(O{\overline{\mathbf{z}}}^D\right)=O{\displaystyle \frac{\partial }{\partial \mathbf{z}}}{p}_{var}\left(\overline{\mathbf{z}}\right)=O\left[-\Delta \overline{\mathbf{z}}+{\displaystyle \frac{\Delta {\int }_{\mathbb{R}}\sigma exp\sigma ({\mathit{\alpha }}^{-1}\Delta \overline{\mathbf{z}}+\mathbf{h})d\mathit{\rho }\left(\mathit{\sigma }\right)}{{\int }_{\mathbb{R}}exp\tau ({\mathit{\alpha }}^{-1}\Delta \overline{\mathbf{z}}+\mathbf{h})d\mathit{\rho }\left(\mathit{\tau }\right)}}\right]=0\end{array}$$

(37)

where the above exponentials and division are applied component-wise. Criticality implies from (37) that

$$\begin{array}{c}\hfill \overline{\mathbf{z}}-{\displaystyle \frac{{\int }_{\mathbb{R}}\sigma exp\sigma ({\mathit{\alpha }}^{-1}\Delta \overline{\mathbf{z}}+\mathbf{h})d\mathit{\rho }\left(\mathit{\sigma }\right)}{{\int }_{\mathbb{R}}exp\tau ({\mathit{\alpha }}^{-1}\Delta \overline{\mathbf{z}}+\mathbf{h})d\mathit{\rho }\left(\mathit{\tau }\right)}}\in \mathrm{Ker}(\Delta ).\end{array}$$

(38)

Observe that the above can also be recast as

$$\begin{array}{cc}\hfill & \overline{\mathbf{z}}-{\omega }_{N,0}\left({\mathbf{m}}_N\right)\in \mathrm{Ker}(\Delta ).\hfill \end{array}$$

Expanding the quadratic form and separating diagonal and off-diagonal terms, we obtain

$$\begin{array}{c}{\omega }_{N,0}\left[({\mathbf{m}}_N-\overline{\mathbf{z}},\Delta ({\mathbf{m}}_N-\overline{\mathbf{z}}))\right]=\sum _{p\ne l,1}^K{\Delta }_{pl}{\omega }_{N,0}\left[(m_p-{\overline{z}}_p)\right]{\omega }_{N,0}\left[(m_{N_l}-{\overline{z}}_l)\right]\hfill \\ \hfill +\sum _{p=1}^K{\Delta }_{pp}{\omega }_{N,0}\left[{(m_p-{\overline{z}}_p)}^2\right].\end{array}$$

(39)

Now, observe that

$$\begin{array}{ll}{\omega }_{N,0}\left[{(m_p-{\overline{z}}_p)}^2\right]=& {\omega }_{N,0}\left[{\left(m_p\right)}^2\right]-2{\omega }_{N,0}\left[m_p\right]{\overline{z}}_p+{\left({\overline{z}}_p\right)}^2\\ & ={\displaystyle \frac{1}{N_p^2}}\left(\sum _{i\in {\Lambda }_p}{\omega }_{N,0}\left[{\sigma }_i^2\right]+\sum _{i\ne j}{\omega }_{N,0}\left[{\sigma }_i{\sigma }_j\right]\right)-2{\omega }_{N,0}\left[m_p\right]{\overline{z}}_p+{\left({\overline{z}}_p\right)}^2.\end{array}$$

(40)

For $i\ne j$, the measure ${\omega }_{N,0}$ factorizes, and naturally ${\sum }_{i\in {\Lambda }_p}{\sigma }_i^2\le N_p$. Therefore,

$${\omega }_{N,0}\left[{(m_p-{\overline{z}}_p)}^2\right]={\omega }_{N,0}^2\left[(m_p-{\overline{z}}_p)\right]+\mathcal{O}\left(N^{-1}\right).$$

(41)

This finally implies

$${\omega }_{N,0}\left[({\mathbf{m}}_N-\overline{\mathbf{z}},\Delta ({\mathbf{m}}_N-\overline{\mathbf{z}}))\right]=\mathcal{O}\left({\displaystyle \frac{1}{N}}\right)+({\omega }_{N,0}({\mathbf{m}}_N-\overline{\mathbf{z}}),\Delta {\omega }_{N,0}({\mathbf{m}}_N-\overline{\mathbf{z}})).$$

(42)

The last term vanishes because ${\omega }_{N,0}({\mathbf{m}}_N-\overline{\mathbf{z}})\in \mathrm{Ker}(\Delta )$.  □

The proof of Theorem 1 thus follows from Lemma 1 and Lemma 2.

4.2. Proof of Theorem 2

The proof of the CLT in the case of Ising spins, where $\rho ={\displaystyle \frac{1}{2}}({\delta }_{-1}+{\delta }_{+1})$, is based on careful control of the asymptotic expansion of the partition function $Z_N$, following the methods of [43,44]. For any integer N and $x\in [-1,1]$, we define the quantity

$$A_N\left(x\right)=\mathrm{card}\left\{\sigma \in {\{-1,1\}}^N:{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\sigma }_i=x\right\}=\left({\displaystyle \genfrac{}{}{0pt}{}{N}{\frac{N(1+x)}{2}}}\right).$$

(43)

We state the following useful standard bounds on $A_N$, whose proof can found in [36].

Lemma 3. 

For any $x\in [-1,1]$, the following inequality holds:

$${\displaystyle \frac{1}{C\sqrt{N}}}{e}^{-NI\left(x\right)}\le A_N\left(x\right)\le e^{-NI\left(x\right)}$$

(44)

where C is a universal constant and,

$$I\left(x\right)={\displaystyle \frac{1-x}{2}}log\left({\displaystyle \frac{1-x}{2}}\right)+{\displaystyle \frac{1+x}{2}}log\left({\displaystyle \frac{1+x}{2}}\right).$$

(45)

Moreover, for any $x\in (-1,1)$, one has

$$A_N\left(x\right)=\sqrt{{\displaystyle \frac{2}{\pi N(1-x^2)}}}exp\left(-NI\left(x\right)\right)·\left(1+\mathcal{O}\left(N^{-1}\right)\right).$$

(46)

We begin by dividing the configuration space ${\{-1,1\}}^N$ into microstates of equal local magnetization. For a given species $l\le K$ with spins configuration ${\sigma }^{\left(l\right)}$, the local magnetization $m_l$ takes values in $S_l=\{-1+{\displaystyle \frac{2n}{N_l}},n=0,\dots ,N_l\}$, with $|S_l|=(N_l+1)$. The magnetization vector ${\mathbf{m}}_N={\left(m_l\right)}_{l\le K}$ thus takes values in $S_N={{\displaystyle \times }}_{l=1}^K{S}_l$. Thus, the partition function can be expressed as

$$Z_N=\sum _{\mathbf{x}\in S_N}\prod _{l=1}^K{A}_{N_l}\left(x_l\right)exp\left(-H_N\left(\mathbf{x}\right)\right).$$

(47)

Here, $A_{N_l}\left(x_l\right)$ counts the number of configurations of ${\sigma }^{\left(l\right)}\in {\{-1,1\}}^{N_l}$ that share the same magnetization $x_l$. From Lemma 3, we can obtain the following bound for the generating function (8), by substituting (44) into (47):

$$-{\displaystyle \frac{1}{N}}\left(logC+{\displaystyle \frac{1}{2}}\sum _{l=1}^{K}log{N}_l\right)+\underset{\mathbf{x}}{max}{f}_N\left(\mathbf{x}\right)\le p_N\le {\displaystyle \frac{1}{N}}\sum _{l=1}^{K}log(N_l+1)+\underset{\mathbf{x}}{max}{f}_N\left(\mathbf{x}\right)$$

(48)

where

$$f_N\left(\mathbf{x}\right)={\displaystyle \frac{1}{2}}\sum _{p,l=1}^K{\alpha }_{N,l}{J}_{pl}{\alpha }_{N,p}{x}_p{x}_l+\sum _{p=1}^K{\alpha }_{N,p}{h}_p{x}_p-\sum _{p=1}^K{\alpha }_{N,p}I\left(x_p\right).$$

(49)

Therefore,

$$\underset{N\to \infty }{lim}{p}_N=\underset{\mathbf{x}\in {[-1,1]}^K}{max}f\left(\mathbf{x}\right)$$

(50)

where

$$f\left(\mathbf{x}\right)=\underset{N\to \infty }{lim}{f}_N\left(\mathbf{x}\right)={\displaystyle \frac{1}{2}}(\mathbf{x},\Delta \mathbf{x})+(\tilde{\mathbf{h}},\mathbf{x})-(\widehat{\mathit{\alpha }},I\left(\mathbf{x}\right))$$

(51)

and $I\left(\mathbf{x}\right)={\left(I\left(x_l\right)\right)}_{l\le K}$ is defined in (45). The stationarity conditions are

$$x_l=tanh\left(h_l+\sum _{p=1}^K{\alpha }_p{J}_{lp}{x}_p\right)\mathrm{for}l=1,\dots ,K.$$

(52)

The solutions of the fixed point Equation (52) correspond to the stationary points of f, among which we are interested in those that achieve the supremum. There may be more than one global maximizer, depending on the parameters $(\mathbf{J},\mathbf{h})$ of the model. In general, even for $K=2$, the landscape of critical points of f, depending on the values $(\mathbf{J},\mathbf{h})$, can be rather complicated; see [48,49] for a detailed study of this case. The Hessian of f can be expressed as

$$\begin{array}{c}\hfill {\mathcal{H}}_f\left(\mathbf{x}\right)=\mathit{\alpha }\left(\mathbf{J}-{\mathit{\alpha }}^{-1}\mathrm{diag}{(1-x_p^2)}_{p\le K}^{-1}\right)\mathit{\alpha }.\end{array}$$

Thus, the Hessian is negative definite whenever $\mathbf{J}-{\mathit{\alpha }}^{-1}diag{(1-x_p^2)}_{p\le K}^{-1}$ is. Using a test vector, it is not difficult to see that if $\mathbf{J}-{\mathit{\alpha }}^{-1}$ is negative definite, then ${\mathcal{H}}_f\left(\mathbf{x}\right)$ remains negative definite (and hence f concave) over the entire optimization domain. This regime, in which Equation (52) has only one solution, can be identified with the case where interactions are not strong enough to induce a phase transition. On the contrary, if $\mathbf{J}-{\mathit{\alpha }}^{-1}$ has even a single positive eigenvalue, the variational potential develops an unstable saddle around the origin, and there exists some $\mathbf{x}\in {[-1,1]}^K$ where convexity and concavity are exchanged again. What we can say for sure, however, is that for $\mathbf{h}=0$, the point $\mathbf{x}=0$ is always critical, and is the solution to (52) when $\mathbf{J}-{\mathit{\alpha }}^{-1}\prec 0$. On the other hand, if $\mathbf{J}-{\mathit{\alpha }}^{-1}$ has a positive eigenvalue, $\mathbf{x}=0$ becomes a saddle, and the potential necessarily develops other maxima. Therefore, for $\mathbf{h}=0$ for instance, a phase transition must necessarily occur when the maximum eigenvalue (with sign, not spectral radius) of $\mathbf{J}-{\mathit{\alpha }}^{-1}$ vanishes.

Here, we assume that $(\mathbf{J},\mathbf{h})$ are such that f admits a unique maximizer $\mathit{\mu }={\left({\mathit{\mu }}^{\left(l\right)}\right)}_{l\le K}$ with negative definite Hessian, i.e., ${\mathcal{H}}_f\left(\mathit{\mu }\right)\prec 0$. Moreover, f is smooth around $\mathit{\mu }$ since the latter lies in the interior of ${[-1,1]}^K$ for any choice of $(\mathbf{J},\mathbf{h})$ (see Lemma (A1)).

The main idea is to derive the asymptotic expansion of the partition function for a perturbed system, where a small external field is added. For any $\mathbf{t}\in {\mathbb{R}}^K$, we define

$$H_{N,\mathbf{t}}\left({\mathbf{m}}_N\right)=H_N\left({\mathbf{m}}_N\right)-\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathbf{m}}_N\right)$$

(53)

where $\sqrt{{\mathit{\alpha }}_N}=\mathrm{diag}{\left(\sqrt{N_p/N}\right)}_{p\le K}$, and $H_N\left({\mathbf{m}}_N\right)$ is the unperturbed Hamiltonian defined in (5). The associated partition function is denoted by $Z_{N,\mathbf{t}}$. We also define the function

$$f_{N,\mathbf{t}}\left(\mathbf{x}\right)=f_N\left(\mathbf{x}\right)+{\displaystyle \frac{1}{\sqrt{N}}}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}\mathbf{x}\right).$$

(54)

Observe that from (54), the function $f_{N,\mathbf{t}}$, and all its partial derivatives, with respect to $\mathbf{x}$, of any order converge uniformly to those of f on the interior of ${[-1,1]}^K$. Hence, by Lemma A2, for large enough N, the function $f_{N,\mathbf{t}}$ admits a unique maximizer ${\mathit{\mu }}_{N,\mathbf{t}}={\left({\mathit{\mu }}_{N,\mathbf{t}}^{\left(l\right)}\right)}_{l\le K}\in {\mathbb{R}}^K$ that converges to $\mathit{\mu }$, and satisfies ${\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)\prec 0$. Next, we demonstrate that the magnetization vector concentrates around ${\mathit{\mu }}_{N,\mathbf{t}}$ with overwhelming probability under the Boltzmann–Gibbs measure ${\mathcal{G}}_{N,\mathbf{t}}$ induced by $H_{N,\mathbf{t}}$. For $\delta >0$, we denote by $B_{N,\delta }$ a poly-interval centered at ${\mathit{\mu }}_{N,\mathbf{t}}$, with each coordinate ${\mu }_{N,\mathbf{t}}^{\left(l\right)}$ as the center and $N_l^{-{\displaystyle \frac{1}{2}}+\delta }$ as the length of the interval in the $l^{th}$ direction:

$$B_{N,\delta }=\left\{\mathbf{x}\in {\mathbb{R}}^K:\left|x_l-{\mu }_{N,\mathbf{t}}^{\left(l\right)}\right|<N_l^{-{\displaystyle \frac{1}{2}}+\delta },\forall l\in \{1,\dots ,K\}\right\}.$$

(55)

Lemma 4. 

Let $\mathbf{t}\in {\mathbb{R}}^K$ and assume that $f\left(\mathbf{x}\right)$ admits a unique global maximizer $\mathit{\mu }$ satisfying ${\mathcal{H}}_f\left(\mathit{\mu }\right)\prec 0$. Then, for a sufficiently large N, $f_{N,\mathbf{t}}$ has a unique maximizer ${\mathit{\mu }}_{N,\mathbf{t}}\stackrel{N\to \infty }{\to }\mathit{\mu }$ and ${\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)\prec 0$. Moreover, for $\delta \in (0,{\displaystyle \frac{1}{2K+4}})$, we have that

$${\mathcal{G}}_{N,\mathbf{t}}({\mathbf{m}}_N\in B_{N,\delta }^c\left({\mathit{\mu }}_{N,\mathbf{t}}\right))\le exp\left\{{\displaystyle \frac{1}{2}}{N}^{2\delta }{\lambda }_{N,\mathbf{t}}\right\}\mathcal{O}\left(N^{{\displaystyle \frac{3K}{2}}}\right).$$

(56)

where ${\lambda }_{N,\mathbf{t}}<0$ and the partition function (47) can be expanded as

$$Z_{N,\mathbf{t}}={\displaystyle \frac{e^{N{f}_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)}}{\sqrt{det\left(-{\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)\right){\prod }_{l=1}^K(1-{\left({\mu }_{N,\mathbf{t}}^{\left(l\right)}\right)}^2)}}}·\left(1+O\left(N^{-{\displaystyle \frac{1}{2}}+(K+2)\delta }\right)\right).$$

(57)

Proof. 

By Lemma A2, for a sufficiently large N, $f_{N,\mathbf{t}}$ has a unique non-degenerate global maximizer. For the concentration bound, we observe

$$\begin{array}{cc}\hfill {\mathcal{G}}_{N,\mathbf{t}}({\mathbf{m}}_N\in B_{N,\delta }^c)& ={\displaystyle \frac{{\sum }_{\mathbf{x}\in S_N\cap B_{N,\delta }^c}{\prod }_{l=1}^K{A}_{N_l}\left(x_l\right)exp\left(-H_{N,\mathbf{t}}\left(\mathbf{x}\right)\right)}{{\sum }_{\mathbf{x}\in S_N}{\prod }_{l=1}^K{A}_{N_l}\left(x_l\right)exp\left(-H_{N,\mathbf{t}}\left(\mathbf{x}\right)\right)}}\hfill \\ \hfill & \le {\displaystyle \frac{C{\prod }_{l=1}^K\sqrt{N_l}(N_l+1){sup}_{\mathbf{x}\in B_{N,\delta }^c}{e}^{N{f}_{N,\mathbf{t}}\left(\mathbf{x}\right)}}{{sup}_{\mathbf{x}\in {[-1,1]}^K}{e}^{N{f}_{N,\mathbf{t}}\left(\mathbf{x}\right)}}}\hfill \\ \hfill & =exp\left\{N\left(\underset{\mathbf{x}\in B_{N,\delta }^c}{sup}{f}_{N,\mathbf{t}}\left(\mathbf{x}\right)-f_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)\right)\right\}\mathcal{O}\left(N^{{\displaystyle \frac{3K}{2}}}\right).\hfill \end{array}$$

(58)

The sup in the denominator of the inequality appearing in the second line has been extended over ${[-1,1]}^K$ instead of $S_N$ due to the fact that $f_{N,\mathbf{t}}$ is Lipschitz continuous away from the boundaries of ${[-1,1]}^K$. The difference between ${sup}_{\mathbf{x}\in S_N}{f}_{N,\mathbf{t}}\left(\mathbf{x}\right)$ and ${sup}_{\mathbf{x}\in {[-1,1]}^K}{f}_{N,\mathbf{t}}\left(\mathbf{x}\right)$ is small due to the Lipschitz property of $f_{N,\mathbf{t}}$ and bounded by a constant due to the mesh size of $S_N$. Let us consider the simplest case, where $K=1$. Then, the set $S_N=S_1$ and $S_1=\left\{-1+{\displaystyle \frac{2n}{N_1}},n=0,\dots ,N_1\right\}$ with mesh size ${\displaystyle \frac{2}{N_1}}$ and $\mathbf{t}\in {\mathbb{R}}^K=t$. Now, the Lipschitz continuity of $f_{N,t}$ means there exists a constant L such that $\left|f_{N,t}\left(x\right)-f_{N,t}\left(y\right)\right|\le L|x-y|$ for all $x,y$ away from $\pm 1$. Given that the maximum distance between any point $x\in S_N$ and the nearest point in $(-1,1)$ is at most ${\displaystyle \frac{2}{N_1}}$, the difference between the supremum of $f_{N,t}$ over $S_N$ and over $(-1,1)$ can be bounded by $\left|{sup}_{x\in S_N}{f}_{N,t}\left(x\right)-{sup}_{x\in (-1,1)}{f}_{N,t}\left(x\right)\right|\le {\displaystyle \frac{2L}{N_1}}$. Thus, the difference is small and goes to zero as $N_1\to \infty$. This justifies substituting the sum over the discrete set $S_N$ with the supremum over $[-1,1]$, with the error being controlled by the mesh size and the Lipschitz constant L.

The supremum of $f_{N,\mathbf{t}}$ restricted to $B_{N,\delta }^c$, in the last equality of (58), is attained at some ${\mathbf{y}}_{N,\mathbf{t},\delta }$ on the boundary of $B_{N,\delta }$. A formal proof can be obtained through a straightforward generalization of Lemma B.11 in [43]. In fact, if $\mathbf{x}\in B_{N,\delta }^c$, then at least one of the coordinates $x_l$ is outside $\left({\mu }_{N,\mathbf{t}}^{\left(l\right)}-N_l^{-{\displaystyle \frac{1}{2}}+\delta },{\mu }_{N,\mathbf{t}}^{\left(l\right)}+N_l^{-{\displaystyle \frac{1}{2}}+\delta }\right)$. The function $f_{N,\mathbf{t}}$ decreases outside $B_{N,\delta }$, and hence the supremum must be on the boundary. Bearing this in mind, we have

$$\begin{array}{c}|{\mathbf{y}}_{N,\mathbf{t},\delta }-{\mathit{\mu }}_{N,\mathbf{t}}{|}^2\le \sum _{l\le K}{N}_l^{-1+2\mathit{\delta }}=\sum _{l\le K}{\left(N{\mathit{\alpha }}_{N,l}\right)}^{-1+2\mathit{\delta }}=N^{-1+2\mathit{\delta }}\underset{=c_{{\mathit{\alpha }}_N,\mathit{\delta }}}{\underbrace{\sum _{l\le K}{\left({\mathit{\alpha }}_{N,l}\right)}^{-1+2\mathit{\delta }}}}\hfill \\ \hfill =N^{-1+2\delta }{c}_{{\alpha }_N,\delta }\end{array}$$

(59)

and, hence, using a Taylor expansion up to third order with Lagrange type remainder,

$$\begin{array}{c}{f}_{N,\mathbf{t}}\left({\mathbf{y}}_{N,\mathbf{t},\delta }\right)=f_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)+{\displaystyle \frac{1}{2}}\left(({\mathbf{y}}_{N,\mathbf{t},\mathit{\delta }}-{\mathit{\mu }}_{N,\mathbf{t}}){\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right),({\mathbf{y}}_{N,\mathbf{t},\mathit{\delta }}-{\mathit{\mu }}_{N,\mathbf{t}})\right)\hfill \\ +{\displaystyle \frac{1}{6}}\sum _{l,p,s=1}^K{\displaystyle \frac{{\partial }^3{f}_{N,\mathbf{t}}}{\partial x_l\partial x_p\partial x_s}}\left({\vartheta }_{N,\mathbf{t}}\right){({\mathbf{y}}_{N,\mathbf{t},\delta }-{\mathit{\mu }}_{N,\mathbf{t}})}_l{({\mathbf{y}}_{N,\mathbf{t},\mathit{\delta }}-{\mathit{\mu }}_{N,\mathbf{t}})}_p{({\mathbf{y}}_{N,\mathbf{t},\mathit{\delta }}-{\mathit{\mu }}_{N,\mathbf{t}})}_s\\ \le f_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)+{\displaystyle \frac{1}{2}}{N}^{-1+2\mathit{\delta }}\underset{>1}{\underbrace{c_{{\mathit{\alpha }}_N,\mathit{\delta }}}}{\mathit{\lambda }}_{N,\mathbf{t}}+{\displaystyle \frac{1}{6}}\sum _{l,p,s=1}^K\left|{\displaystyle \frac{{\partial }^3{f}_{N,\mathbf{t}}}{\partial x_l\partial x_p\partial x_s}}\left({\mathit{\vartheta }}_{N,\mathbf{t}}\right)\right|·N^{-3/2+3\mathit{\delta }}{\left(c_{{\mathit{\alpha }}_N,\mathit{\delta }}\right)}^{3/2}\\ \hfill \le f_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)+{\displaystyle \frac{1}{2}}{N}^{-1+2\mathit{\delta }}{\mathit{\lambda }}_{N,\mathbf{t}}+\mathcal{O}\left(N^{-3/2+3\mathit{\delta }}\right)\end{array}$$

(60)

where ${\vartheta }_{N,\mathbf{t}}$ is an intermediate point of the segment $[{\mathit{\mu }}_{N,\mathbf{t}},{\mathbf{y}}_{N,\mathbf{t},\mathit{\delta }}]$, ${\lambda }_{N,\mathbf{t}}<0$ is the largest eigenvalue of ${\mathcal{H}}_{f_{N,\mathbf{t}}}$, and in the third term of the last inequality of (60) we have that ${\alpha }_{N,l}\to {\alpha }_l$ as $N\to \infty$. Hence, (56) follows from (58) and (60).

We now prove the asymptotic expansion of the partition function $Z_{N,\mathbf{t}}$ (57). Observe that the concentration results, Equation (56), imply that almost all the contribution to $Z_{N,\mathbf{t}}$ comes from spin configurations with magnetization ${\mathbf{m}}_N$ in a vanishing neighborhood of ${\mathit{\mu }}_{N,\mathbf{t}}$, i.e., ${\mathcal{G}}_{N,\mathbf{t}}({\mathbf{m}}_N\in B_{N,\delta })=1-\mathcal{O}\left(e^{-c{N}^{2\delta }}\right)$ for some $c>0$. Hence,

$$Z_{N,\mathbf{t}}=\left(1+\mathcal{O}\left(e^{-c{N}^{2\delta }}\right)\right)\sum _{\mathbf{x}\in S_N\cap B_{N,\delta }}{\zeta }_{N,\mathbf{t}}\left(\mathbf{x}\right),{\zeta }_{N,\mathbf{t}}\left(\mathbf{x}\right):=\prod _{l=1}^K\left({\displaystyle \genfrac{}{}{0pt}{}{N_l}{\frac{N_l(1+x_l)}{2}}}\right)exp\left(-H_{N,\mathbf{t}}\left(\mathbf{x}\right)\right).$$

(61)

Following the same argument as [43,44], we approximate the sum in (61) by an integral using Lemma A5 over the set $B_{N,\delta }$ with shrinking interval containing the unique vector of the global maximizer of $f_{N,\mathbf{t}}$, which are elements of $S_N$:

$$\begin{array}{c}|{\int }_{B_{N,\delta }}{\zeta }_{N,\mathbf{t}}\left(\mathbf{x}\right)d\mathbf{x}-{\displaystyle \frac{2^K}{{\prod }_{l=1}^K{N}_l}}\sum _{\mathbf{x}\in S_N\cap B_{N,\delta }}{\zeta }_{N,\mathbf{t}}\left(\mathbf{x}\right)|\le N^{\left({\displaystyle \frac{1}{2}}+\delta \right)\left(K-1\right)}·N^{-{\displaystyle \frac{1}{2}}+\delta }·N^{-K}\underset{\mathbf{x}\in B_{N,\delta }}{sup}|\nabla {\zeta }_{N,t}\left(\mathbf{x}\right)|\hfill \\ \hfill =\mathcal{O}\left(N^{\left(-{\displaystyle \frac{1}{2}}+\delta \right)\left(K+1\right)}\right){\zeta }_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)\end{array}$$

(62)

where ${sup}_{\mathbf{x}\in B_{N,\delta }}|\nabla {\zeta }_{N,\mathbf{t}}\left(\mathbf{x}\right)|$ is bounded in Lemma A3. Now, following from (62) and applying the results of Lemma A6 to approximate the integral, we have that

$$\begin{array}{c}\sum _{\mathbf{x}\in S_N\cap B_{N,\delta }}{\zeta }_{N,\mathbf{t}}\left(\mathbf{x}\right)={\displaystyle \frac{{\prod }_{l=1}^K{N}_l}{2^K}}{\int }_{B_{N,\delta }}{\zeta }_{N,\mathbf{t}}\left(\mathbf{x}\right)d\mathbf{x}+\mathcal{O}\left(N^{\left(-{\displaystyle \frac{1}{2}}+\delta \right)\left(K+1\right)+K}\right){\zeta }_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)\hfill \\ ={\displaystyle \frac{{\prod }_{l=1}^K{N}_l}{2^K}}\sqrt{{\displaystyle \frac{2^K}{{\pi }^K{\prod }_{l=1}^K{N}_l(1-{\left({\mu }_{N,\mathbf{t}}^{\left(l\right)}\right)}^2)}}}\sqrt{{\displaystyle \frac{{\left(2\pi \right)}^K}{{\prod }_{l=1}^K{N}_{l}det\left(-{\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)\right)}}}{e}^{N{f}_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)}·\\ ·\left(1+\mathcal{O}\left(N^{-1/2+(K+2)\delta }\right)\right)\\ \hfill ={\displaystyle \frac{e^{N{f}_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)}}{\sqrt{det\left(-{\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)\right){\prod }_{l=1}^K(1-{\left({\mu }_{N,\mathbf{t}}^{\left(l\right)}\right)}^2)}}}·\left(1+\mathcal{O}\left(N^{-{\displaystyle \frac{1}{2}}+(K+2)\delta }\right)\right)\end{array}$$

(63)

for $\delta \in \left(0,{\displaystyle \frac{1}{2K+4}}\right)$. Therefore, we obtain the asymptotic form of the partition function:

$$Z_{N,\mathbf{t}}={\displaystyle \frac{e^{N{f}_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)}}{\sqrt{det\left(-{\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}\right)\right){\prod }_{l=1}^K(1-{\left({\mu }_{N,\mathbf{t}}^{\left(l\right)}\right)}^2)}}}·\left(1+\mathcal{O}\left(N^{-{\displaystyle \frac{1}{2}}+(K+2)\delta }\right)\right).$$

(64)

This completes the proof.  □

Now we are ready to proof Theorem 2. In order to approximate the distribution of the scaled difference between the vector of global species magnetization and the limiting global maximizers, we will compute the limiting moment generating function of ${\mathbf{m}}_N={\left(m_l\right)}_{l\le K}$ for some $\mathbf{t}\in {\mathbb{R}}^K$ using the expanded form of the partition function in (64):

$$\begin{array}{c}\mathbb{E}\left[e^{\sqrt{N}(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}({\mathbf{m}}_N-\mathit{\mu }))}\right]=e^{-\sqrt{N}(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}\mathit{\mu })}{\int }_{{\mathbb{R}}^K}{e}^{\sqrt{N}(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathbf{m}}_N)}{\mathcal{G}}_N\left({\mathbf{m}}_N\right)d{\mathbf{m}}_N\hfill \\ \hfill =e^{-\sqrt{N}(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}\mathit{\mu })}{\displaystyle \frac{Z_{N,\mathbf{t}}}{Z_N}}.\end{array}$$

(65)

From the last equality on the right of (65) and using (64),

$$\begin{array}{cc}\hfill {\displaystyle \frac{Z_{N,\mathbf{t}}}{Z_N}}\sim & {\displaystyle \frac{exp\left(N{max}_{\mathbf{x}}{f}_{N,\mathbf{t}}\left(\mathbf{x}\right)\right)}{exp\left(N{max}_{\mathbf{x}}{f}_N\left(\mathbf{x}\right)\right)}}\hfill \\ \hfill =& exp\left(N[f_N\left({\mathit{\mu }}_{N,\mathbf{t}}\right)-f_N\left({\mathit{\mu }}_N\right)]+\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathit{\mu }}_{N,\mathbf{t}}\right)\right)\hfill \end{array}$$

(66)

where ∼ denotes an equality up to $(1+o(1\left)\right)$. In the last equality above, we had that ${\mathit{\mu }}_{N,\mathbf{t}}$ and ${\mathit{\mu }}_N$ are unique global maximizers of $f_{N,\mathbf{t}}\left(\mathbf{x}\right)$ and $f_N\left(\mathbf{x}\right)$, respectively. Now, following from Lemma A4, Equation (66) becomes

$${\displaystyle \frac{Z_{N,\mathbf{t}}}{Z_N}}\sim exp\left(-{\displaystyle \frac{1}{2}}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathcal{H}}_{f_N}^{-1}\left({\mathit{\mu }}_N\right)\sqrt{{\mathit{\alpha }}_N}\mathbf{t}\right)+\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathit{\mu }}_N\right)\right).$$

(67)

Therefore,

$$\begin{array}{cc}\hfill \mathbb{E}\left[e^{\sqrt{N}(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}({\mathbf{m}}_N-\mathit{\mu }))}\right]\sim & e^{-\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}\mathit{\mu }\right)}·e^{\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathit{\mu }}_N\right)}·e^{-{\displaystyle \frac{1}{2}}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathcal{H}}_{f_N}^{-1}\left({\mathit{\mu }}_N\right)\sqrt{{\mathit{\alpha }}_N}\mathbf{t}\right)}\hfill \\ \hfill =& e^{\sqrt{N}\left[\mathbf{t}\sqrt{{\mathit{\alpha }}_N}\left({\mathit{\mu }}_N-\mathit{\mu }\right)\right]}·e^{-{\displaystyle \frac{1}{2}}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathcal{H}}_{f_N}^{-1}\left({\mathit{\mu }}_N\right)\sqrt{{\mathit{\alpha }}_N}\mathbf{t}\right)}.\hfill \end{array}$$

(68)

Now, from (49) we have that

$${\mathit{\mu }}_N=tanh\left(\mathbf{J}{\mathit{\alpha }}_N{\mathit{\mu }}_N+\mathbf{h}\right).$$

(69)

Let $\mathit{\beta }={\left({\mathit{\beta }}_p\right)}_{p\le K}$ with $|{\beta }_p|<\infty$, and $\theta \in \left[{\displaystyle \frac{1}{2}},\infty \right)$. Assume that ${\mathit{\alpha }}_N\equiv \mathit{\alpha }\left(\mathit{\beta }\right)=\mathit{\alpha }+N^{-\mathit{\theta }}\mathrm{diag}\left(\mathit{\beta }\right)$. If we set ${\mathbf{b}}_N=\mathbf{J}{\mathit{\alpha }}_N{\mathit{\mu }}_N+\mathbf{h}$, then

$${\displaystyle \frac{\partial {\mathit{\mu }}_N}{\partial \mathit{\beta }}}={\displaystyle \frac{\partial {\mathit{\mu }}_N}{\partial {\mathbf{b}}_N}}·{\displaystyle \frac{\partial {\mathbf{b}}_N}{\partial \mathit{\beta }}}$$

(70)

where

$${\displaystyle \frac{\partial {\mathit{\mu }}_N}{\partial {\mathbf{b}}_N}}=\mathrm{diag}{(1-{\mu }_{N,l}^2)}_{l\le K}\mathrm{and}{\displaystyle \frac{\partial {\mathbf{b}}_N}{\partial \mathit{\beta }}}=\mathbf{J}{\displaystyle \frac{\partial {\mathit{\alpha }}_N}{\partial \mathit{\beta }}}{\mathit{\mu }}_N+\mathbf{J}{\mathit{\alpha }}_N{\displaystyle \frac{\partial {\mathit{\mu }}_N}{\partial \mathit{\beta }}}.$$

(71)

Let $\mathbf{M}=\mathrm{diag}{(1-{\mu }_{N,l}^2)}_{l\le K}$ then

$${\displaystyle \frac{\partial {\mathit{\mu }}_N}{\partial \mathit{\beta }}}=\mathbf{M}\left[\mathbf{J}{\displaystyle \frac{\partial {\mathit{\alpha }}_N}{\partial \mathit{\beta }}}{\mathit{\mu }}_N+\mathbf{J}{\mathit{\alpha }}_N{\displaystyle \frac{\partial {\mathit{\mu }}_N}{\partial \mathit{\beta }}}\right]$$

(72)

which entails

$${\displaystyle \frac{\partial {\mathit{\mu }}_N}{\partial \mathit{\beta }}}\left[{\mathbf{M}}^{-1}-\mathbf{J}{\mathit{\alpha }}_N\right]=N^{-\mathit{\theta }}\mathbf{J}\mathrm{diag}\left({\mathit{\mu }}_N\right).$$

(73)

Since ${\mathcal{H}}_{f_N}\left({\mathit{\mu }}_N\right)=-\left[{\mathbf{M}}^{-1}-\mathbf{J}{\mathit{\alpha }}_N\right]·{\mathit{\alpha }}_N$, hence, (72) yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\mathit{\mu }}_N}{\partial \mathit{\beta }}}=& {\left[{\mathbf{M}}^{-1}-\mathbf{J}{\mathit{\alpha }}_N\right]}^{-1}{N}^{-\mathit{\theta }}\mathbf{J}\mathrm{diag}\left({\mathit{\mu }}_N\right)=-N^{-\mathit{\theta }}{\mathcal{H}}_{f_N}^{-1}\left({\mathit{\mu }}_N\right){\mathit{\alpha }}_N\mathbf{J}\mathrm{diag}\left({\mathit{\mu }}_N\right).\hfill \end{array}$$

(74)

Finally, a Taylor expansion of ${\mathit{\mu }}_N$ around $\mathit{\beta }=\mathbf{0}$ gives

$${\mathit{\mu }}_N-\mathit{\mu }=-N^{-\mathit{\theta }}{\mathcal{H}}_f^{-1}\left(\mathit{\mu }\right)\mathit{\alpha }\mathbf{J}\mathrm{diag}\left(\mathit{\mu }\right)\mathit{\beta }+\mathcal{O}\left(N^{-2\mathit{\theta }}\right).$$

(75)

Therefore, Theorem 2 follows from (75) and taking the limit as $N\to \infty$ of (68).

4.3. Proof of Theorem 3

We will now address the case where f reaches its maximum in more than one point.

Lemma 5. 

Suppose $f\left(\mathbf{x}\right)$ has n global maximizers ${\mathit{\mu }}^i$ for $i\le n$ satisfying ${\mathcal{H}}_f\left({\mathit{\mu }}^i\right)\prec 0$. For each $i\le n$, let $A_i\subset {[-1,1]}^K$ be a poly-interval such that ${\mathit{\mu }}^i\in \mathrm{int}\left(A_i\right)$ is the unique maximizer of f on $\mathrm{cl}\left(A_i\right)$. Then, for each $i\le n$ and for a sufficiently large N, $f_{N,\mathbf{t}}$ has a unique global maximizer ${\mathit{\mu }}_{N,\mathbf{t}}^i\to \mathit{\mu }$ on $A_i$ with ${\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)<0$. Moreover, for $\delta \in \left(0,{\displaystyle \frac{1}{2K+4}}\right)$, one has

$${\mathcal{G}}_{N,\mathbf{t}}({\mathbf{m}}_N\in B_{N,\delta }^c\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)|{\mathbf{m}}_N\in A_i)=exp\left\{{\displaystyle \frac{1}{2}}{N}^{2\delta }{\lambda }_{N,\mathbf{t}}^i\right\}\mathcal{O}\left(N^{{\displaystyle \frac{3K}{2}}}\right)$$

(76)

where ${\lambda }_{N,\mathbf{t}}^i<0$ is the largest eigenvalue of ${\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)$ for ${\mathit{\mu }}_{N,\mathbf{t}}^i\in int\left(A_i\right)$ and

$${\mathcal{G}}_{N,\mathbf{t}}\left({\mathbf{m}}_N\in B_{N,\delta ,n}^c\right)=exp\left\{{\displaystyle \frac{1}{2}}{N}^{2\delta }\underset{1\le i\le n}{max}{\lambda }_{N,\mathbf{t}}^i\right\}\mathcal{O}\left(N^{{\displaystyle \frac{3K}{2}}}\right)$$

(77)

where $B_{N,\delta ,n}^c={\bigcup }_{i\le n}{B}_{N,\delta }^c\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right).$ The partition function restricted to the interval $A_i$ can be expanded as

$$\begin{array}{cc}\hfill Z_{N,\mathbf{t}}{|}_{A_i}& =\sum _{\mathbf{x}\in S_N\cap A_i}\prod _{l=1}^K{A}_{N_l}\left(x_l\right)expN\left(f_{N,\mathbf{t}}\left(\mathbf{x}\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{e^{N{f}_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)}}{\sqrt{det\left(-{\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)\right){\prod }_{l=1}^K\left(1-{\left({\mu }_{N,\mathbf{t}}^{i,\left(l\right)}\right)}^2\right)}}}·\left(1+O\left(N^{-1/2+(K+2)\delta }\right)\right).\hfill \end{array}$$

(78)

Proof. 

For a sufficiently large N, $B_{N,\delta }\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)\subset A_i$ for all $i\le n$ and equation (76) is obtained following the step-by-step argument used to prove Equation (56). Hence, it follows that

$${\mathcal{G}}_{N,\mathbf{t}}({\mathbf{m}}_N\in B_{N,\delta }^c\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)|{\mathbf{m}}_N\in A_i)=exp\left\{{\displaystyle \frac{1}{2}}{N}^{2\delta }{\lambda }_{N,\mathbf{t}}^i\right\}\mathcal{O}\left(N^{{\displaystyle \frac{3K}{2}}}\right).$$

(79)

Now, let us observe that, for a sufficiently large N, $A_i\setminus B_{N,\delta }\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)=A_i\setminus B_{N,\delta ,n}$ for $i\le n$. Hence, for all $1\le i\le n$ and N large, it follows that

$${\mathcal{G}}_{N,\mathbf{t}}({\mathbf{m}}_N\in B_{N,\delta }^c\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)|{\mathbf{m}}_N\in A_i)={\mathcal{G}}_{N,\mathbf{t}}\left({\mathbf{m}}_N\in B_{N,\delta ,n}^c|{\mathbf{m}}_N\in A_i\right).$$

(80)

Therefore, we have that

$$\begin{array}{cc}\hfill {\mathcal{G}}_{N,\mathbf{t}}\left({\mathbf{m}}_N\in B_{N,\delta ,n}^c\right)=& \sum _{1\le i\le n}{\mathcal{G}}_{N,\mathbf{t}}\left({\mathbf{m}}_N\in B_{N,\delta ,n}^c|{\mathbf{m}}_N\in A_i\right){\mathcal{G}}_{N,\mathbf{t}}\left({\mathbf{m}}_N\in A_i\right)\hfill \\ \hfill \le & exp\left\{{\displaystyle \frac{1}{2}}{N}^{2\delta }\underset{1\le i\le n}{max}{\lambda }_{N,\mathbf{t}}^i\right\}\mathcal{O}\left(N^{{\displaystyle \frac{3K}{2}}}\right)\sum _{1\le i\le n}{\mathcal{G}}_{N,\mathbf{t}}\left({\mathbf{m}}_N\in A_i\right)\hfill \\ \hfill =& exp\left\{{\displaystyle \frac{1}{2}}{N}^{2\delta }\underset{1\le i\le n}{max}{\lambda }_{N,\mathbf{t}}^i\right\}\mathcal{O}\left(N^{{\displaystyle \frac{3K}{2}}}\right).\hfill \end{array}$$

(81)

This complete the result in (77) following from (81).

The proof for the asymptotic expansion of the partition function when there are multiple vectors of global maximizers of $f_{N,\mathbf{t}}$ follows exactly the same argument for the case with a unique vector of global maximizer when conditioned on an interval containing only one of the global maximizers. Notice that for fixed $i\le n$ and N large, ${\mathbf{m}}_N$ concentrates around ${\mathit{\mu }}_{N,\mathbf{t}}^i\in A_i$ as stated in Equation (76). Hence,

$${\mathcal{G}}_{N,\mathbf{t}}({\mathbf{m}}_N\in B_{N,\delta }\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)|{\mathbf{m}}_N\in A_i)={\displaystyle \frac{1}{Z_{N,\mathbf{t}}{|}_{A_i}}}\sum _{\mathbf{x}\in S_N\cap B_{N,\delta }}\prod _{l=1}^K\left({\displaystyle \genfrac{}{}{0pt}{}{N_l}{\frac{N_l(1+x_l)}{2}}}\right)exp\left\{-H_{N,\mathbf{t}}\left(\mathbf{x}\right)\right\}.$$

(82)

Now, following the exact computation and argument in the uniqueness case, the restricted partition function for each of the global maximizers ${\mathit{\mu }}_{N,\mathbf{t}}^i$ for $i\le n$ can be expanded as

$$Z_{N,\mathbf{t}}{|}_{A_i}={\displaystyle \frac{e^{N{f}_{N,\mathbf{t}}\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)}}{\sqrt{det\left(-{\mathcal{H}}_{f_{N,\mathbf{t}}}\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)\right){\prod }_{l=1}^K\left(1-{\left({\mu }_{N,\mathbf{t}}^{i,\left(l\right)}\right)}^2\right)}}}·\left(1+\mathcal{O}\left(N^{-1/2+(K+2)\delta }\right)\right).$$

(83)

This completes the proof of Lemma 5.  □

The conditional moment generating function for a certain parameter $\mathbf{t}\in {\mathbb{R}}^K$ can be computed as

$$\begin{array}{c}\mathbb{E}\left[e^{\sqrt{N}(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}({\mathbf{m}}_N-{\mathit{\mu }}^i))}|\left\{{\mathbf{m}}_N\in A_i\right\}\right]\hfill \\ =e^{-\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathit{\mu }}^i\right)}{\int }_{{\mathbb{R}}^K}{e}^{-\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathbf{m}}_N\right)}{\mathcal{G}}_N\left({\mathbf{m}}_N\right){|}_{\left\{{\mathbf{m}}_N\in A_i\right\}}d{\mathbf{m}}_N\\ \hfill =e^{-\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathit{\mu }}^i\right)}{\displaystyle \frac{Z_{N,\mathbf{t}|A_i}}{Z_{N|A_i}}}.\end{array}$$

(84)

Using the asymptotic expansion of the perturbed partition function in (83), the proof follows the same arguments as in Theorem 2. Hence,

$${\displaystyle \frac{Z_{N,\mathbf{t}|A_i}}{Z_{N|A_1}}}\sim exp\left(N[f_N\left({\mathit{\mu }}_{N,\mathbf{t}}^i\right)-f_N\left({\mathit{\mu }}_N^i\right)]+\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathit{\mu }}_{N,\mathbf{t}}^i\right)\right)$$

(85)

where ${\mathit{\mu }}_{N,\mathbf{t}}^i\in int\left(A_i\right)$ is the unique maximizer of $f_{N,\mathbf{t}}$ and ${\mathit{\mu }}_N^i\in int\left(A_i\right)$ is the unique maximizer of $f_N$. Now, following the argument of Lemma A4 and the proof of Theorem 2, we have that

$$\begin{array}{c}\mathbb{E}\left[e^{\sqrt{N}(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}({\mathbf{m}}_N-{\mathit{\mu }}^i))}|\left\{{\mathbf{m}}_N\in A_i\right\}\right]\sim e^{-\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathit{\mu }}^i\right)}·e^{\sqrt{N}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathit{\mu }}_N^i\right)}·e^{-{\displaystyle \frac{1}{2}}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathcal{H}}_{f_N}^{-1}\left({\mathit{\mu }}_N^i\right)\sqrt{{\mathit{\alpha }}_N}\mathbf{t}\right)}\hfill \\ \hfill =e^{\sqrt{N}\left[\mathbf{t}\sqrt{{\mathit{\alpha }}_N}\left({\mathit{\mu }}_N^i-{\mathit{\mu }}^i\right).\right]}·e^{-{\displaystyle \frac{1}{2}}\left(\mathbf{t},\sqrt{{\mathit{\alpha }}_N}{\mathcal{H}}_{f_N}^{-1}\left({\mathit{\mu }}_N^i\right)\sqrt{{\mathit{\alpha }}_N}\mathbf{t}\right)}.\end{array}$$

(86)

Similarly, if we set ${\mathit{\alpha }}_N\equiv \mathit{\alpha }\left(\mathit{\beta }\right)=\mathit{\alpha }+N^{-\mathit{\theta }}\mathrm{diag}\left(\mathit{\beta }\right)$, it follows from (75) that

$${\mathit{\mu }}_N^i-{\mathit{\mu }}^i=-N^{-\mathit{\theta }}{\mathcal{H}}_f^{-1}\left({\mathit{\mu }}^i\right)\mathit{\alpha }\mathbf{J}\mathrm{diag}\left({\mathit{\mu }}^i\right)\mathit{\beta }+\mathcal{O}\left(N^{-2\mathit{\theta }}\right)$$

(87)

for $\mathit{\beta }={\left({\mathit{\beta }}_p\right)}_{p\le K}$ where $0<{\beta }_p<\infty$ and $\theta \in \left[{\displaystyle \frac{1}{2}},\infty \right)$. Hence, Theorem 3 follows from (87) by taking the limit as $N\to \infty$ of (86). This completes the proof of Theorem 3.

5. Conclusions

In this work, we introduced a generalized version of the multispecies Curie–Weiss model featuring arbitrary spins and a non-definite interaction matrix. Specifically, we computed the large-number limit of the generating functional associated with the Hamiltonian (3) and established the validity of the Central Limit Theorem (CLT) for a suitably rescaled vector of global magnetization in the Ising spin case. Our results demonstrate that the rescaled global magnetization vector follows either a centered or non-centered multivariate normal distribution, depending on the rate at which the relative particle densities converge to their limiting values as the system size tends to infinity.

This generalized framework holds significant promise for applications in systems where spin particles or units can exhibit arbitrary states, moving beyond traditional binary or discrete-valued models. Such flexibility is particularly relevant in fields like statistical physics, network theory, and social dynamics, where elements often have a continuum of possible states or operate under complex interactions. Furthermore, the inclusion of an indefinite interaction matrix enables the modeling of systems with both cooperative and antagonistic interactions [50], broadening the scope of potential applications to include meta-magnets, financial systems, and multi-agent interactions.

Future work could explore extending these results to models with more complex topologies or additional constraints, such as time-varying interaction matrices or non-equilibrium dynamics [51,52,53]. Additionally, investigating the robustness of the CLT under perturbations in the interaction matrix or in the distribution of spin states could yield further insights into the stability and universality of the model’s behavior and broaden its applicability [54].