# Multipopulation Spin Models: A View from Large Deviations Theoretic Window

Authors: Alex Akwasi Opoku; Godwin Osabutey
Journal: Journal of Mathematics
 (2018)

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## Abstract

This paper studies large deviations properties of vectors of empirical means and measures generated as follows. Consider a sequenceX1,X2,…,Xn of independent and identically distributed random variables partitioned into d-subgroups with sizes n1,…,nd. Further, consider a d-dimensional vector mn whose coordinates are made up of the empirical means of the subgroups. We prove the following. (1) The sequence of vector of empirical means mn satisfies large deviations principle with rate n and rate function I, when the sequence X1,X2,…,Xn is Rl valued, with l≥1. (2) Similar large deviations results hold for the corresponding sequence of vector of empirical measures Ln if Xi’s, i=1,2,…,n, take on finitely many values. (3) The rate functions for the above large deviations principles are convex combinations of the corresponding rate functions arising from the large deviations principles of the coordinates of mn and Ln. The probability distributions used in the convex combinations are given by α=(α1,…,αd)=limn→∞1/n(n1,…,nd). These results are consequently used to derive variational formula for the thermodynamic limit for the pressure of multipopulation Curie-Weiss (I. Gallo and P. Contucci (2008), and I. Gallo (2009)) and mean-field Pott’s models, via a version of Varadhan’s integral lemma for an equicontinuous family of functions. These multipopulation models serve as a paradigm for decision-making context where social interaction and other socioeconomic attributes of individuals play a crucial role.


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## Body

## 1. Introduction

The early 1970s saw the utilization of two-population mean-field models in the study of the phase transitions and critical behaviour of antiferromagnetic systems. Such mean-field models were used as mean-field approximations of bipartite lattice systems for studying metamagnets [1, 2]. The two-population mean-field ideology was used in [3] to investigate the gibbs-non-gibbs transitions in gibbs measures for Curie-Weiss model subjected to Glauber dynamics. Here the analysis was based on complete analysis of the phase diagram of the evolving spins constrained to having a given magnetization. Phase transitions in such a constrained system is an indication of loss of the gibbs property for the evolving system and it is preserved otherwise.Statistical mechanical models have seen applications in the socioeconomic literature. Here the focus is on how decisions of individuals are influenced by their socioeconomic environment. For instance, how one’s choice of employment, residence, eduction, etc. is influenced by the social and economic environments. Spin models have appeared as natural models for such discrete choice context where social interactions play a crucial role [4–7]. More recently, the authors of [8–10] have introduced two-population mean-field models for a binary choice context where the reference population is partitioned into subgroups of individuals sharing the same socioeconomic attributes. The key assumption here is that individuals with the same attributes tend to behave the same way. In these papers it is assumed that the fractions of the individuals in the subgroups are independent of the size of the reference population. The thermodynamic limit for the pressure of the models was proved for many-body interaction version of the interacting Curie-Weiss model. But variational expression for the pressure and almost sure factorization of correlation function were proved for the case of one and two-body interactions case [10].The aim of this paper is to set up large deviations machinery for assessing the variational formula for the pressure of the general model introduced in [10] and even extend the model there to the case where the fractions of the individuals in the subgroups are dependent on the size of the reference population. We also employ the tools developed here to derive variational formula for the pressure of a multibody multipopulation mean-field Potts’ model.We establish large deviation results for vectors of empirical means associated with a collection of random variables modelling the behaviour of interest of the subgroups that constitute the reference group. These empirical means are derived from uneven numbers of random variables. Thus the vector components are given by empirical mean of different numbers of independent and identically distributed random variables. Due to the variations in the sizes of the subgroups, the large populations asymptotics of the free energy results in proving a version of Varadhan’s integral lemma for a sequence of functionals of the vector of empirical measures instead of the usual case where this functional is fixed throughout the asymptotics. We provide a necessary condition for such a sequence to admit the desired asymptotic result.The rest of the paper is organized as follows: Section2 discusses the generalities on large deviations theory and main results of the paper. In Section 3 we introduce the multipopulation Curie-Weiss and mean-field Potts’ models to motivate the large deviations problem we address in this paper. The proofs of the results in Section 3 are given in Section 4.


## 2. Generalities on Large Deviations Theory and Main Results

Large deviation theory tells how, on an exponential scale, the probability for an atypical event decays to zero. More formerly, large deviations are defined as follows.Definition 1.
LetX be a complete separable metric space, B(X) the Borel σ-algebra of X, and {μn;n=1,2,…} a sequence of probability measures on B(X). (1)
{μn} is said to have a large deviation property if there exist a sequence of positive numbers {an;n=1,2,…} which tends to ∞ and a function I which maps X into [0,∞] such that the following hypothesis holds:
(a)
I is lower semicontinuous on X.(b)
I has compact level sets.(c)
limsupn→∞an-1logμnK≤-infx∈KI(x), for each closed set K in X.(d)
liminfn→∞an-1logμn(G)≥-infx∈GI(x), for each open set G in X.(2)
I is called the entropy/rate function of {μn}.In the above definition condition (b) implies that the rate functionI is good.


### 2.1. The Set Up

SupposeX1,X2,…,Xn,…, is a sequence of independent and identically distributed Rl-valued random variables, for a positive integer l≥1. Let In1,…,Ind be a partition of the set In={1,2,…,n}. The partition may be interpreted as the indexing set of the d≥1 subpopulations in a population of size n. Here we denote by ni the size of the ith subpopulation and we assume that(1)limn→∞αni=limn→∞nin=αi,for any i=1,2,…,d. For each subpopulation i we are interested in the empirical mean(2)Sni=1ni∑k∈IniXkand the vector of empirical means for the multipopulation is given by(3)Sn=Sn1,…,Snd.For the case l=1, it is clear that Sn∈Rd, but it is different from the empirical mean of a sequence of Rd-valued random variables. Note that in the latter each coordinate of the empirical mean is a sum of n random variables. In our case the coordinates of the empirical mean vector Sn are made up of sums of uneven numbers of random variables. In what follows we write μn for the distribution of Sn and the space X=(Rl)d.


### 2.2. TheR-Valued Case

Suppose the sequenceX1,X2,… of R-valued random variables is independent and identically distributed with common distribution μ. Suppose the logarithmic moment generating function associated with μ is given by(4)Λλ=logMλ=logEeλX1We assume Λ(λ)<∞, for all λ∈R. The Fenchel-Legendre transform of Λ(λ) is defined as(5)Λ∗x=supλ∈Rλx-Λλ.We now state our first large deviations result for the vector of empirical means Sn. Recall that μn is the law of Sn.Theorem 2.
The sequence of vector of empirical meansSn satisfies large deviations principle with rate n and rate function Λ¯∗(·), given by(6)Λ∗x=supλ∈Rd∑i=1dαiλixi-Λλi.


### 2.3. TheRl-Valued Case

In what follows we will consider the large deviations principle for the case of a sequence of independent and identically distributed (i.i.d.)Rl-valued random variables X1,X2,…. Here l≥2. As before, we denote by μ, a probability measure on Rl, the common law of the i.i.d. sequence X1,X2,…. Here again, we let(7)Sni=1ni∑j∈IniXj∈Rl,fori=1,2,…,d,and (8)Sn=Sn1,Sn2,…,Snd∈Rld.Further, we assume that(9)Λλ=logEeλ,X1<∞,foreveryλ∈Rl.In what follows we will write for every x∈(Rl)d, as(10)x=x1,x2,…,xd,wherexi=x1i,x2i,…,xli∈Rl,foreveryi=1,2,…,d.In view of the above, we will write the inner product for any pair x,y∈(Rl)d as follows:(11)x,y=∑i=1dxi,yi=∑i=1d∑j=1lxjiyji.We define the Rd-valued vectors n¯=(n1,n2,…,nd) and α=(α1,α2,…,αd). The large deviations principle for the Sn vector for l≥2 case is summarized in the results below.Theorem 3.
LetSn be defined as above. Then Sn satisfies a large deviations principle with rate n and rate function Λ¯∗ given by(12)Λ∗x=supλ∈Rld∑i=1dαiλi,xi-Λλi.


### 2.4. Large Deviations Principle for a Vector of Empirical Measures

In this section we consider the large deviations principle for a vector of empirical measures of the subgroups when the sequence of i.i.d. random variables takes finitely many values. Specifically, letΣ be a finite set and X1,X2,X3,… be an independent and identically distributed sequence of Σ-valued random variables with common law μ=(μ1,…,μ|Σ|)∈R|Σ|. Here μi=μ(ai), for any ai∈Σ. The empirical measure for the random variables indexed by Ini is given by(13)Lnix=LniXa1,…,LniXaΣ∈RΣ,whereLniXak=1ni∑j∈Ini1akXj.Here we have put 1ak(Xj)=1{a}(Xj), the indicator function for {ak}⊂Σ. Note that Lni[X](ak) is the relative frequency of ak among the random sample (Xj)j∈Ini.The vector of empirical measures is then given by(14)LnX=Ln1X,Ln2X,…,LndX∈RΣd.Note that Ln[X] is a vector whose coordinates are probability measures on Σ. To see the connection between our vector of empirical measures and the vector of empirical means we considered earlier, we introduce the following sequence of random variables: for positive integer k, define(15)Yk=1a1Xk,1a2Xk,…,1aΣXk.Then the sequence Y1,Y2,Y3,… is an i.i.d. R|Σ|-valued random variables are with the property that(16)Sni=1ni∑j∈IniXj=LniX,Sn=LnX.Thus the empirical mean of the Y-sequence is the same as the empirical measure of the X-sequence and the vector of empirical means associated with the Y-sequence coincides with the vector of empirical measures of the X-sequence.For everyλ∈R|Σ|, note that(17)Λλ=logEeλ,Y1=log∑j=1Σeλjμj.Further, for any probability measure ν on Σ, the relative entropy of ν relative to μ is given by(18)Hν∣μ=∑j=1Σνjlogνjμj.Suppose M1(Σ) is the set of all probability measures on Σ. Then M1(Σ)⊂R|Σ| and M1(Σ)d⊂(RΣ)d. For any ν∈M1(Σ), it can be written in the form ν=(ν1,ν2,…,νd), where νi∈(Σ), for any i=1,2,…,d. The following result is the large deviations principle for the vector of empirical measures Ln[X].Theorem 4.
The sequence of vectors of empirical measuresLn[X] satisfies a large deviations principle with rate n and rate function(19)Λ∗ν=supλ∈RΣd∑i=1dαiλi,νi-Λλi, for every ν∈M1(Σ)d. In particular,(20)Λ∗ν=∑i=1dαiHνi∣μ.


### 2.5. Varadhan’s Integral Lemma

In this section we consider Varadhan’s integral lemma for a sequence of equicontinuous functions. Here we putX=(Rl)d.Theorem 5.
Suppose{μn} satisfies a large deviation principle with rate n and a good rate function I:X→R. Further, let the sequence of functions ϕn:X→R be equicontinuous converging point-wise to a function ϕ:X→R. Assume either the tail condition(21)limM→∞limsupn→∞1nlogEenϕnSn1ϕnSn≥M=-∞or the following moment generating condition for some γ>1,(22)limsupn→∞1nlogEenγϕnSn<∞.Then(23)limn→∞1nlogEenϕnSn=supx∈Xϕx-Ix.


## 2.1. The Set Up

SupposeX1,X2,…,Xn,…, is a sequence of independent and identically distributed Rl-valued random variables, for a positive integer l≥1. Let In1,…,Ind be a partition of the set In={1,2,…,n}. The partition may be interpreted as the indexing set of the d≥1 subpopulations in a population of size n. Here we denote by ni the size of the ith subpopulation and we assume that(1)limn→∞αni=limn→∞nin=αi,for any i=1,2,…,d. For each subpopulation i we are interested in the empirical mean(2)Sni=1ni∑k∈IniXkand the vector of empirical means for the multipopulation is given by(3)Sn=Sn1,…,Snd.For the case l=1, it is clear that Sn∈Rd, but it is different from the empirical mean of a sequence of Rd-valued random variables. Note that in the latter each coordinate of the empirical mean is a sum of n random variables. In our case the coordinates of the empirical mean vector Sn are made up of sums of uneven numbers of random variables. In what follows we write μn for the distribution of Sn and the space X=(Rl)d.


## 2.2. TheR-Valued Case

Suppose the sequenceX1,X2,… of R-valued random variables is independent and identically distributed with common distribution μ. Suppose the logarithmic moment generating function associated with μ is given by(4)Λλ=logMλ=logEeλX1We assume Λ(λ)<∞, for all λ∈R. The Fenchel-Legendre transform of Λ(λ) is defined as(5)Λ∗x=supλ∈Rλx-Λλ.We now state our first large deviations result for the vector of empirical means Sn. Recall that μn is the law of Sn.Theorem 2.
The sequence of vector of empirical meansSn satisfies large deviations principle with rate n and rate function Λ¯∗(·), given by(6)Λ∗x=supλ∈Rd∑i=1dαiλixi-Λλi.


## 2.3. TheRl-Valued Case

In what follows we will consider the large deviations principle for the case of a sequence of independent and identically distributed (i.i.d.)Rl-valued random variables X1,X2,…. Here l≥2. As before, we denote by μ, a probability measure on Rl, the common law of the i.i.d. sequence X1,X2,…. Here again, we let(7)Sni=1ni∑j∈IniXj∈Rl,fori=1,2,…,d,and (8)Sn=Sn1,Sn2,…,Snd∈Rld.Further, we assume that(9)Λλ=logEeλ,X1<∞,foreveryλ∈Rl.In what follows we will write for every x∈(Rl)d, as(10)x=x1,x2,…,xd,wherexi=x1i,x2i,…,xli∈Rl,foreveryi=1,2,…,d.In view of the above, we will write the inner product for any pair x,y∈(Rl)d as follows:(11)x,y=∑i=1dxi,yi=∑i=1d∑j=1lxjiyji.We define the Rd-valued vectors n¯=(n1,n2,…,nd) and α=(α1,α2,…,αd). The large deviations principle for the Sn vector for l≥2 case is summarized in the results below.Theorem 3.
LetSn be defined as above. Then Sn satisfies a large deviations principle with rate n and rate function Λ¯∗ given by(12)Λ∗x=supλ∈Rld∑i=1dαiλi,xi-Λλi.


## 2.4. Large Deviations Principle for a Vector of Empirical Measures

In this section we consider the large deviations principle for a vector of empirical measures of the subgroups when the sequence of i.i.d. random variables takes finitely many values. Specifically, letΣ be a finite set and X1,X2,X3,… be an independent and identically distributed sequence of Σ-valued random variables with common law μ=(μ1,…,μ|Σ|)∈R|Σ|. Here μi=μ(ai), for any ai∈Σ. The empirical measure for the random variables indexed by Ini is given by(13)Lnix=LniXa1,…,LniXaΣ∈RΣ,whereLniXak=1ni∑j∈Ini1akXj.Here we have put 1ak(Xj)=1{a}(Xj), the indicator function for {ak}⊂Σ. Note that Lni[X](ak) is the relative frequency of ak among the random sample (Xj)j∈Ini.The vector of empirical measures is then given by(14)LnX=Ln1X,Ln2X,…,LndX∈RΣd.Note that Ln[X] is a vector whose coordinates are probability measures on Σ. To see the connection between our vector of empirical measures and the vector of empirical means we considered earlier, we introduce the following sequence of random variables: for positive integer k, define(15)Yk=1a1Xk,1a2Xk,…,1aΣXk.Then the sequence Y1,Y2,Y3,… is an i.i.d. R|Σ|-valued random variables are with the property that(16)Sni=1ni∑j∈IniXj=LniX,Sn=LnX.Thus the empirical mean of the Y-sequence is the same as the empirical measure of the X-sequence and the vector of empirical means associated with the Y-sequence coincides with the vector of empirical measures of the X-sequence.For everyλ∈R|Σ|, note that(17)Λλ=logEeλ,Y1=log∑j=1Σeλjμj.Further, for any probability measure ν on Σ, the relative entropy of ν relative to μ is given by(18)Hν∣μ=∑j=1Σνjlogνjμj.Suppose M1(Σ) is the set of all probability measures on Σ. Then M1(Σ)⊂R|Σ| and M1(Σ)d⊂(RΣ)d. For any ν∈M1(Σ), it can be written in the form ν=(ν1,ν2,…,νd), where νi∈(Σ), for any i=1,2,…,d. The following result is the large deviations principle for the vector of empirical measures Ln[X].Theorem 4.
The sequence of vectors of empirical measuresLn[X] satisfies a large deviations principle with rate n and rate function(19)Λ∗ν=supλ∈RΣd∑i=1dαiλi,νi-Λλi, for every ν∈M1(Σ)d. In particular,(20)Λ∗ν=∑i=1dαiHνi∣μ.


## 2.5. Varadhan’s Integral Lemma

In this section we consider Varadhan’s integral lemma for a sequence of equicontinuous functions. Here we putX=(Rl)d.Theorem 5.
Suppose{μn} satisfies a large deviation principle with rate n and a good rate function I:X→R. Further, let the sequence of functions ϕn:X→R be equicontinuous converging point-wise to a function ϕ:X→R. Assume either the tail condition(21)limM→∞limsupn→∞1nlogEenϕnSn1ϕnSn≥M=-∞or the following moment generating condition for some γ>1,(22)limsupn→∞1nlogEenγϕnSn<∞.Then(23)limn→∞1nlogEenϕnSn=supx∈Xϕx-Ix.


## 3. Applications

Let us now introduce the models that motivated the large deviations questions addressed in this paper. We are interested in a model that will capture how individual decisions or choices are influenced by the choices of the rest of the people in their reference group. Additionally, individuals do have attributes, such as gender, place of residence, level of education, and ethnicity, that also influence their decisions. The Curie-Weiss case introduced in [10] is discussed first and the corresponding mean-field Potts’ case will be discussed after that. The method discussed here could also apply to continuous spin models with compact support. In particular, it will apply to the mean-field versions of the following models: the O(n) model [11], the spherical model [12, 13], the liquid crystal model [11, 14, 15], and the Kuramoto model [16]. The detail analysis for continuous spin models and their phase diagram will be carried out in a future paper. We present the results for the Curie-Weiss and Potts’s models here because we have already studied the thermodynamic limits of these models in [17, 18].


### 3.1. Multipopulation Curie-Weiss Model

Suppose each individual in a population ofN agents chooses a binary action, such as voting YES or NO on some issue at some common time. This binary action is coded by σi∈{1,-1} with(24)σi=+1,ifindividualivotesYES-1,ifindividualivotesNO,fori∈1,…,N.The choices made by all the N individuals are also coded by σ∈ΩN={-1,+1}N. The level of satisfaction of the population for deciding on σ∈ΩN is given by the Curie-Weiss Hamiltonian(25)HNσ=12N∑i,l=1NJilσiσl+∑i=1Nhiσi.The functionHN on the configurations σ represents the utility of individuals as a result of their choices and the influences on them while making those choices [19]. HN(σ) measures the level of satisfaction of the entire population for making the choice σ. The higher is the HN(σ), the higher is the level of satisfaction of the population. It has two parts; the first part models the social incentive of individuals in the population and the second part models the private incentive of individuals. Here Jil measures the influence of individual i on individual l. When Jil is positive means conformity or imitation is rewarded and conformity is not rewarded when Jil is negative. hi controls the part of the utility that is specific to individual i.Next we reparametrize the parametersJil and hi as follows: suppose that each individual i in the reference population has k attributes ai=(ai(1),…,ai(k))∈{0,1}k. For instance, suppose that attributes 1 and 2 are, respectively, employment ai(1) and marital status ai(2), then(26)ai1=1,ifindividualiisemployed0,ifindividualiisunemployed,and(27)ai2=1,ifanindividualimarried0,ifanindividualiisunmarried.Therefore, with respect to the attributes, the reference group can be partitioned into2k nonoverlapping subgroups. Members in a given subgroup share the same attributes and it is therefore reasonable to assume that they also behave the same way. In view of this, we shall assume in what follows that Jil=Jgg′, for all choices of i coming from subgroup g and all l taken from subgroup g′. Further, we assume that hi=hg for all individuals in group g. In the sequel we will let INg be the set of individuals in subgroup g, for g=1,…,2k, with |INg| =Ng. Therefore, N=N1+⋯+N2k. Note that IN={1,2,…,N}=⋃g=12kINg with INg∩INg′=∅, for g≠g′. Define for every N∈N(28)γNg=NgN,mNg=1Ng∑i∈INgσi.Note that γNg is the proportion of the individuals found in subgroup g and mNg is the magnetization of the spins in subgroup g.Therefore, it follows from (25) and the above parametrization of Jil and hi that(29)HNσ=12N∑g=12k∑g′=12k∑i∈INg∑l∈INg′Jilσiσl+∑g=12k∑i∈INghiσi=N∑g=12k∑g′=12kJgg′2γNgγNg′mNgmNg′+∑g=12khgγNgmNg.Note that if k=0, we get the original Curie-Weiss Hamiltonian. For the case k≥1, we end up with 2k Curie-Weiss models on the subgroups IN1,…,IN2k, that are interacting with one another. Here we have 2k subgroups because we have attributes that are binary. We could allow the attributes to have any finite number of alternatives and the alternatives for the attributes need not to come from the same set. Therefore, in what follows we will assume there are d≥1 subgroups and that the Hamiltonian takes the form(30)HNσ=N∑g=1d∑g′=1dJgg′2γNgγNg′mNgmNg′+∑g=1dhgγNgmNg.The Hamiltonian in (30) consists of one-body and two-body interactions. In what follows we will extend the number of bodies in the interaction to range from 1 to d, where d is the number of subgroups [10]. We consider Hamiltonian of the form(31)HNσ=N∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgkmNgk.=∑r=1d1Nr-1∑g1,…,gr=1dJg1…grr∑i1∈INg1…∑ir∈INgrσi1…σir.Note from (30) and (31) that for r=1, Jg1=hg and for r=2, Jg1g2=Jgg′. The Jg1,…,gr are interaction coefficients associated with the r-body interaction among individuals coming from the subgroups g1,…,gr, respectively. Thus the interaction is defined with the help of a tensor Jg1,…,gr of rank r for each of the r-body interactions [10].Further, we assume that there is a probability measureγ on the set {1,2,…,d}, such that γ=(γ1,…,γd) and(32)limN→∞γNg=γg,foranyg=1,2,…,d.Note that the model we consider here is more general than the cases in [8, 10], in that for any finite N the fractions of the subgroups γN1,…,γNd are dependent on N. In [8, 10] these fractions are chosen to be independent of N. This simplifies the proofs, especially the existence of the thermodynamic limit.In the sequel we will use the following notation: letΔ=[-1,1]⊂R. Note that mN=mNg1,…,mNgd∈Δd. Further, for every positive integer N, define a map FN:Δd→R as(33)FNm=∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgkmgk.Note that the FN’s are uniformly bounded by ∑r=1d‍∑g1,…,gr=1d‍|Jg1…gr/r|. It is also clear that(34)Fm=∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγgkmgk=limN→∞∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgkmgk.Since the maps m↦FN(m) are continuous for every N, and Δd is a compact subset of Rd, it follows from Theorems 7.13 and 7.24 of [20] that the sequence {FN} is equicontinuous. Further, the Hamiltonian HN in (31) become(35)HNσ=NFNmN.Suppose the spinsσ1,σ2,… are independent and identically distributed sequence of random variables with(36)Pσ1=1=Pσ1=-1=12.We denote by PN the corresponding product measure on ΩN={-1,1}N. The equilibrium state ωN associated with the Hamiltonian HN in (31) is given by(37)ωNσ=eHNσPNσZN,forσ∈ΩN,where(38)ZN=∑σ~∈ΩNeHNσ~PNσ~=∫ΔdeNFNmNQNmis the partition function of the model and QN is the law of vector of empirical means mN=(mN1,…,mNd) under PN. In (38) we have used (35). In what follows μ, h, and J shall be as follows:(39)γ=γ1,…,γd,h=h1,…,hd∈Rd,J=Jg1,…,gr1≤g1,…,gr,r≤d.The pressure function of the model is then given by(40)pN=1NlogZN.The large N behaviour of the model is governed by the pressure function. It is known from [18] that the thermodynamic limit(41)pγ,h,J=limN→∞1NlogZNexists. The proof of the case γNg=γg was earlier given in [8, 10].Theorem 6.
For choice of the parametersγ, h, and J, the limiting pressure admits the following variational representation:(42)pγ,h,J=supμ∈-1,1dFμ-∑g=1dγgIμg,where(43)Iμg=1-μg2log1-μg+1+μg2log1+μgand F is given in (34).Proof.
The proof of this theorem follows from (38), (40), and (41) and Theorems 2 and 5 upon setting σi=Xi, Ng=ng, N=n, γg=αg, γ=α, and FN=ϕn, and noting that the FN’s form an equicontinuous family and they are uniformly bounded.


### 3.2. Multipopulation Mean-Field Potts’ Model

Suppose this time round that the individuals in the population ofN agents choose from finite number of alternatives, say various alternatives of employment, at some common time. This discrete choice action of individual i is coded by σi∈E={1,…,q}, where q≥2, with(44)σi=1,ifindividualichoosesemploymentalternative1⋮q,ifindividualichoosesemploymentalternativeqfori∈1,…,N.The choices made by all the N individuals are also coded by σ∈ΩN={1,…,q}N. The level of satisfaction of the population for deciding on σ∈ΩN is given by the mean-field Potts’ Hamiltonian(45)HNσ=12N∑i,l=1NJilδσi,σl+∑i=1Nhiδσi,1.Here δσi,σj is the Dirac-delta measure. The function HN on the configurations σ represents the utility of individuals as a result of their choices and the influences on them while making a decision [19]. HN and its parameters have the usual interpretation given for the Curie-Weiss model.Using the partitionIN1,…,INd of IN, we define for every N∈N(46)LNgσs=1Ng∑i∈INgδσi,s,foranys=1,2,…,q.Therefore(47)LNgσ=LNgσ1,…,LNgσqis the empirical measure for the choices of the individuals in subgroup g. It is an element in P(E), the set of probability measures on E={1,…,q}. Further,(48)LNσ=LN1σ,…,LN2kσis the vector of empirical measures. Note that LN[σ]∈P(E)2k and γNg is the proportion of the individuals found in subgroup g.Therefore, it follows from (45) and the above parametrization of Jil and hi that(49)HNσ=12N∑g=12k∑g′=12k∑i∈INg∑l∈INg′Jilδσi,σl+∑g=12k∑i∈INghiδσi,1=N∑g=12k∑g′=12kJgg′2γNgγNg′LNgσ,LNg′σ+∑g=12khgγNgLNgσ,e1.In the second equation above we have used that(50)∑s=1q∑i∈INgδσi,s∑l∈INg′δσl,s=∑i∈INg∑l∈INg′δσi,σl,x,y=∑i=12kxiyi,foranyx,y∈R2k,e1=1,0,0,…,0∈R2k.Note that if k=0, we get the original mean-field Potts’ Hamiltonian. For the case k≥1, we end up with 2k mean-field Potts’ models on the subgroups IN1,…,IN2k, that are interacting with one another. Here we have 2k subgroups because we have attributes that are binary. We could allow the attributes to have any finite number of alternatives and the alternatives for the attributes need not come from the same set. Therefore, in what follows we will assume there are d≥1 subgroups and this gives rise to the Hamiltonian(51)HNσ=N∑g=1d∑g′=1dJgg′2γNgγNg′LNgσ,LNg′σ+∑g=1dhgγNgLNgσ,e1.The Hamiltonian in (51) consists of one-body and two-body interactions. In what follows we will extend the number of bodies in the interaction to range from 1 to d, where d is the number of subgroups. We consider Hamiltonian of the form(52)HNσ=∑r=1d1Nr-1∑g1,…,gr=1dJg1…grr∑i1∈INg1…∑ir∈INgrδσi1,…,σir=N∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgkLNg1σ,…,LNgrσ..In the above we have used that(53)∑s=1q∑i1∈INg1δσi1,s…∑ir∈INgrδσir,s=∑i1∈INg1…∑ir∈INgrδσi1,…,σir,x1,…,xr=∑i=1d∏k=1rxik,foranyx1,…,xr∈Rd.Note from (51) and (52) that for r=1, Jg1=hg and 〈x1〉=〈x1,e1〉. The Jg1,…,gr are interaction coefficients associated with the r-body interaction among individuals coming from subgroups g1,…,gr, respectively. Thus the interaction is defined with the help of a tensor Jg1,…,gr of rank r for each of the r-body interactions. The model considered here is mean-field Pott’s version of the Curie-Weiss model considered in [8, 10, 17]. Further, for any finite N the fractions of the subgroups γN1,…,γNd are dependent on N. In [8, 10] these fractions were chosen to be independent of N, which simplified the proofs, especially the proof of the existence of the thermodynamic limit.Note thatLN[σ]=LN1[σ],…,LNd[σ]∈P(E)d. Further, for every positive integer N, define a map FN:P(E)d→R as(54)FNν=∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgk∑i=1d∏k=1rνigk.Note that the FN’s are uniformly bounded by ∑r=1d‍∑g1,…,gr=1d‍|Jg1…gr/r|. It is also clear that(55)Fν=∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγgk∑i=1d∏k=1rνigk=limN→∞∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgk∑i=1d∏k=1rνigk.Since the maps ν↦FN(ν) are continuous for every N, and Δd is a compact subset of (Rq)d, it follows from Theorems 7.13 and 7.24 of [20] that the sequence {FN} is equicontinuous. Further, the Hamiltonian HN in (52) become(56)HNσ=NFNLNσ.Suppose the spinsσ1,σ2,… are independent and identically distributed sequence of E-valued random variables with(57)Pσ1=s=ρs=1q,foreverys∈E.We denote by PN the corresponding product measure ρN on ΩN=EN. The equilibrium state ωN associated with the Hamiltonian HN in (52) is given by(58)ωNσ=eHNσPNσZN,forσ∈ΩN,where(59)ZN=∑σ~∈ΩNeHNσ~PNσ~=∫PEdeNFNνQNdνis the partition function of the model. Here QN is the law of the vector of empirical measures LN(σ). The pressure function of the model is then given by(60)pN=1NlogZN.It follows from [17] that the thermodynamic limit(61)pγ,h,J=limN→∞1NlogZNexists.The limiting pressure admits the following variational formula representation.Theorem 7.
For any choice of the parametersγ, h, and J, the limiting pressure admits the following variational representation:(62)pγ,h,J=supν∈ΔdFν-∑g=1dγgHνg∣ρ,where(63)Hνg∣ρ=∑s=1qνsglogνsgρs is the relative entropy of νg with respect to ρ and F is given in (55).Proof.
The proof of this theorem follows from (59) to (61) and Theorems 4 and 5 upon setting σi=Xi, Ng=ng, N=n, γg=αg, γ=α, and FN=ϕn, and noting that the FN’s form an equicontinuous family and they are uniformly bounded.


## 3.1. Multipopulation Curie-Weiss Model

Suppose each individual in a population ofN agents chooses a binary action, such as voting YES or NO on some issue at some common time. This binary action is coded by σi∈{1,-1} with(24)σi=+1,ifindividualivotesYES-1,ifindividualivotesNO,fori∈1,…,N.The choices made by all the N individuals are also coded by σ∈ΩN={-1,+1}N. The level of satisfaction of the population for deciding on σ∈ΩN is given by the Curie-Weiss Hamiltonian(25)HNσ=12N∑i,l=1NJilσiσl+∑i=1Nhiσi.The functionHN on the configurations σ represents the utility of individuals as a result of their choices and the influences on them while making those choices [19]. HN(σ) measures the level of satisfaction of the entire population for making the choice σ. The higher is the HN(σ), the higher is the level of satisfaction of the population. It has two parts; the first part models the social incentive of individuals in the population and the second part models the private incentive of individuals. Here Jil measures the influence of individual i on individual l. When Jil is positive means conformity or imitation is rewarded and conformity is not rewarded when Jil is negative. hi controls the part of the utility that is specific to individual i.Next we reparametrize the parametersJil and hi as follows: suppose that each individual i in the reference population has k attributes ai=(ai(1),…,ai(k))∈{0,1}k. For instance, suppose that attributes 1 and 2 are, respectively, employment ai(1) and marital status ai(2), then(26)ai1=1,ifindividualiisemployed0,ifindividualiisunemployed,and(27)ai2=1,ifanindividualimarried0,ifanindividualiisunmarried.Therefore, with respect to the attributes, the reference group can be partitioned into2k nonoverlapping subgroups. Members in a given subgroup share the same attributes and it is therefore reasonable to assume that they also behave the same way. In view of this, we shall assume in what follows that Jil=Jgg′, for all choices of i coming from subgroup g and all l taken from subgroup g′. Further, we assume that hi=hg for all individuals in group g. In the sequel we will let INg be the set of individuals in subgroup g, for g=1,…,2k, with |INg| =Ng. Therefore, N=N1+⋯+N2k. Note that IN={1,2,…,N}=⋃g=12kINg with INg∩INg′=∅, for g≠g′. Define for every N∈N(28)γNg=NgN,mNg=1Ng∑i∈INgσi.Note that γNg is the proportion of the individuals found in subgroup g and mNg is the magnetization of the spins in subgroup g.Therefore, it follows from (25) and the above parametrization of Jil and hi that(29)HNσ=12N∑g=12k∑g′=12k∑i∈INg∑l∈INg′Jilσiσl+∑g=12k∑i∈INghiσi=N∑g=12k∑g′=12kJgg′2γNgγNg′mNgmNg′+∑g=12khgγNgmNg.Note that if k=0, we get the original Curie-Weiss Hamiltonian. For the case k≥1, we end up with 2k Curie-Weiss models on the subgroups IN1,…,IN2k, that are interacting with one another. Here we have 2k subgroups because we have attributes that are binary. We could allow the attributes to have any finite number of alternatives and the alternatives for the attributes need not to come from the same set. Therefore, in what follows we will assume there are d≥1 subgroups and that the Hamiltonian takes the form(30)HNσ=N∑g=1d∑g′=1dJgg′2γNgγNg′mNgmNg′+∑g=1dhgγNgmNg.The Hamiltonian in (30) consists of one-body and two-body interactions. In what follows we will extend the number of bodies in the interaction to range from 1 to d, where d is the number of subgroups [10]. We consider Hamiltonian of the form(31)HNσ=N∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgkmNgk.=∑r=1d1Nr-1∑g1,…,gr=1dJg1…grr∑i1∈INg1…∑ir∈INgrσi1…σir.Note from (30) and (31) that for r=1, Jg1=hg and for r=2, Jg1g2=Jgg′. The Jg1,…,gr are interaction coefficients associated with the r-body interaction among individuals coming from the subgroups g1,…,gr, respectively. Thus the interaction is defined with the help of a tensor Jg1,…,gr of rank r for each of the r-body interactions [10].Further, we assume that there is a probability measureγ on the set {1,2,…,d}, such that γ=(γ1,…,γd) and(32)limN→∞γNg=γg,foranyg=1,2,…,d.Note that the model we consider here is more general than the cases in [8, 10], in that for any finite N the fractions of the subgroups γN1,…,γNd are dependent on N. In [8, 10] these fractions are chosen to be independent of N. This simplifies the proofs, especially the existence of the thermodynamic limit.In the sequel we will use the following notation: letΔ=[-1,1]⊂R. Note that mN=mNg1,…,mNgd∈Δd. Further, for every positive integer N, define a map FN:Δd→R as(33)FNm=∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgkmgk.Note that the FN’s are uniformly bounded by ∑r=1d‍∑g1,…,gr=1d‍|Jg1…gr/r|. It is also clear that(34)Fm=∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγgkmgk=limN→∞∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgkmgk.Since the maps m↦FN(m) are continuous for every N, and Δd is a compact subset of Rd, it follows from Theorems 7.13 and 7.24 of [20] that the sequence {FN} is equicontinuous. Further, the Hamiltonian HN in (31) become(35)HNσ=NFNmN.Suppose the spinsσ1,σ2,… are independent and identically distributed sequence of random variables with(36)Pσ1=1=Pσ1=-1=12.We denote by PN the corresponding product measure on ΩN={-1,1}N. The equilibrium state ωN associated with the Hamiltonian HN in (31) is given by(37)ωNσ=eHNσPNσZN,forσ∈ΩN,where(38)ZN=∑σ~∈ΩNeHNσ~PNσ~=∫ΔdeNFNmNQNmis the partition function of the model and QN is the law of vector of empirical means mN=(mN1,…,mNd) under PN. In (38) we have used (35). In what follows μ, h, and J shall be as follows:(39)γ=γ1,…,γd,h=h1,…,hd∈Rd,J=Jg1,…,gr1≤g1,…,gr,r≤d.The pressure function of the model is then given by(40)pN=1NlogZN.The large N behaviour of the model is governed by the pressure function. It is known from [18] that the thermodynamic limit(41)pγ,h,J=limN→∞1NlogZNexists. The proof of the case γNg=γg was earlier given in [8, 10].Theorem 6.
For choice of the parametersγ, h, and J, the limiting pressure admits the following variational representation:(42)pγ,h,J=supμ∈-1,1dFμ-∑g=1dγgIμg,where(43)Iμg=1-μg2log1-μg+1+μg2log1+μgand F is given in (34).Proof.
The proof of this theorem follows from (38), (40), and (41) and Theorems 2 and 5 upon setting σi=Xi, Ng=ng, N=n, γg=αg, γ=α, and FN=ϕn, and noting that the FN’s form an equicontinuous family and they are uniformly bounded.


## 3.2. Multipopulation Mean-Field Potts’ Model

Suppose this time round that the individuals in the population ofN agents choose from finite number of alternatives, say various alternatives of employment, at some common time. This discrete choice action of individual i is coded by σi∈E={1,…,q}, where q≥2, with(44)σi=1,ifindividualichoosesemploymentalternative1⋮q,ifindividualichoosesemploymentalternativeqfori∈1,…,N.The choices made by all the N individuals are also coded by σ∈ΩN={1,…,q}N. The level of satisfaction of the population for deciding on σ∈ΩN is given by the mean-field Potts’ Hamiltonian(45)HNσ=12N∑i,l=1NJilδσi,σl+∑i=1Nhiδσi,1.Here δσi,σj is the Dirac-delta measure. The function HN on the configurations σ represents the utility of individuals as a result of their choices and the influences on them while making a decision [19]. HN and its parameters have the usual interpretation given for the Curie-Weiss model.Using the partitionIN1,…,INd of IN, we define for every N∈N(46)LNgσs=1Ng∑i∈INgδσi,s,foranys=1,2,…,q.Therefore(47)LNgσ=LNgσ1,…,LNgσqis the empirical measure for the choices of the individuals in subgroup g. It is an element in P(E), the set of probability measures on E={1,…,q}. Further,(48)LNσ=LN1σ,…,LN2kσis the vector of empirical measures. Note that LN[σ]∈P(E)2k and γNg is the proportion of the individuals found in subgroup g.Therefore, it follows from (45) and the above parametrization of Jil and hi that(49)HNσ=12N∑g=12k∑g′=12k∑i∈INg∑l∈INg′Jilδσi,σl+∑g=12k∑i∈INghiδσi,1=N∑g=12k∑g′=12kJgg′2γNgγNg′LNgσ,LNg′σ+∑g=12khgγNgLNgσ,e1.In the second equation above we have used that(50)∑s=1q∑i∈INgδσi,s∑l∈INg′δσl,s=∑i∈INg∑l∈INg′δσi,σl,x,y=∑i=12kxiyi,foranyx,y∈R2k,e1=1,0,0,…,0∈R2k.Note that if k=0, we get the original mean-field Potts’ Hamiltonian. For the case k≥1, we end up with 2k mean-field Potts’ models on the subgroups IN1,…,IN2k, that are interacting with one another. Here we have 2k subgroups because we have attributes that are binary. We could allow the attributes to have any finite number of alternatives and the alternatives for the attributes need not come from the same set. Therefore, in what follows we will assume there are d≥1 subgroups and this gives rise to the Hamiltonian(51)HNσ=N∑g=1d∑g′=1dJgg′2γNgγNg′LNgσ,LNg′σ+∑g=1dhgγNgLNgσ,e1.The Hamiltonian in (51) consists of one-body and two-body interactions. In what follows we will extend the number of bodies in the interaction to range from 1 to d, where d is the number of subgroups. We consider Hamiltonian of the form(52)HNσ=∑r=1d1Nr-1∑g1,…,gr=1dJg1…grr∑i1∈INg1…∑ir∈INgrδσi1,…,σir=N∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgkLNg1σ,…,LNgrσ..In the above we have used that(53)∑s=1q∑i1∈INg1δσi1,s…∑ir∈INgrδσir,s=∑i1∈INg1…∑ir∈INgrδσi1,…,σir,x1,…,xr=∑i=1d∏k=1rxik,foranyx1,…,xr∈Rd.Note from (51) and (52) that for r=1, Jg1=hg and 〈x1〉=〈x1,e1〉. The Jg1,…,gr are interaction coefficients associated with the r-body interaction among individuals coming from subgroups g1,…,gr, respectively. Thus the interaction is defined with the help of a tensor Jg1,…,gr of rank r for each of the r-body interactions. The model considered here is mean-field Pott’s version of the Curie-Weiss model considered in [8, 10, 17]. Further, for any finite N the fractions of the subgroups γN1,…,γNd are dependent on N. In [8, 10] these fractions were chosen to be independent of N, which simplified the proofs, especially the proof of the existence of the thermodynamic limit.Note thatLN[σ]=LN1[σ],…,LNd[σ]∈P(E)d. Further, for every positive integer N, define a map FN:P(E)d→R as(54)FNν=∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgk∑i=1d∏k=1rνigk.Note that the FN’s are uniformly bounded by ∑r=1d‍∑g1,…,gr=1d‍|Jg1…gr/r|. It is also clear that(55)Fν=∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγgk∑i=1d∏k=1rνigk=limN→∞∑r=1d∑g1,…,gr=1dJg1…grr∏k=1rγNgk∑i=1d∏k=1rνigk.Since the maps ν↦FN(ν) are continuous for every N, and Δd is a compact subset of (Rq)d, it follows from Theorems 7.13 and 7.24 of [20] that the sequence {FN} is equicontinuous. Further, the Hamiltonian HN in (52) become(56)HNσ=NFNLNσ.Suppose the spinsσ1,σ2,… are independent and identically distributed sequence of E-valued random variables with(57)Pσ1=s=ρs=1q,foreverys∈E.We denote by PN the corresponding product measure ρN on ΩN=EN. The equilibrium state ωN associated with the Hamiltonian HN in (52) is given by(58)ωNσ=eHNσPNσZN,forσ∈ΩN,where(59)ZN=∑σ~∈ΩNeHNσ~PNσ~=∫PEdeNFNνQNdνis the partition function of the model. Here QN is the law of the vector of empirical measures LN(σ). The pressure function of the model is then given by(60)pN=1NlogZN.It follows from [17] that the thermodynamic limit(61)pγ,h,J=limN→∞1NlogZNexists.The limiting pressure admits the following variational formula representation.Theorem 7.
For any choice of the parametersγ, h, and J, the limiting pressure admits the following variational representation:(62)pγ,h,J=supν∈ΔdFν-∑g=1dγgHνg∣ρ,where(63)Hνg∣ρ=∑s=1qνsglogνsgρs is the relative entropy of νg with respect to ρ and F is given in (55).Proof.
The proof of this theorem follows from (59) to (61) and Theorems 4 and 5 upon setting σi=Xi, Ng=ng, N=n, γg=αg, γ=α, and FN=ϕn, and noting that the FN’s form an equicontinuous family and they are uniformly bounded.


## 4. Proofs

The proofs of the results of this paper are given in this section. In the proof below we will use the following properties of the functionsΛ and Λ∗. For proof of these properties we refer the reader to the proof of Lemma 2.2.5 of [21].Lemma 8.
(1)
Λ is a convex function and Λ∗ is a convex rate function. (2)
IfΛ is only finite at λ=0, then Λ∗ is identically zero. If Λ(λ)<∞ for some λ>0, then x¯=E(X1)<∞ (possibly x¯=-∞), and for all x≥x¯,(64)Λ∗x=supλ≥0λx-Λλ is forx>x¯, a nondecreasing function. Similarly, if Λ(λ)<∞ for some λ<0, then x¯=E(X1)>-∞ (possibly x¯=∞), and for all x≤x¯,(65)Λ∗x=supλ≤0λx-Λλ is forx<x¯, a nonincreasing function. Whenx¯ is finite, Λ∗(x¯)=0, and always(66)infx∈RΛ∗x=0.


### 4.1. Proof of Theorem2

Proof.
The proof comes in two steps. In step one we will establish a large deviations upper bound and the corresponding lower bound is proved in step two. The proof is an adaptation of arguments used to prove the large deviations principle for the empirical mean of i.i.d sequence ofRd-valued random vectors.
Step 1. Let I(x)=Λ∗(x) be the intended rate function of the problem and define the δ-rate function as follows:(67)Iδx=minIx-δ,1δ. The proof of the upper bound will follow if we can show for every δ>0 and every closed subset F of Rd that(68)limn→∞sup1nlogμnF≤δ-infx∈FIδx. We will first of all prove this inequality for compact subsets and extend it to closed subsets with the help of exponential tightness argument. Suppose K is a compact subset of Rd. Then for every δ>0 and q∈K, we can find a λq∈Rd such that(69)∑i=1qαiqiλqi-Λλqi≥Iδq. It follows from the definition of Λ∗ that such a λq exists. For any λq,u∈Rd, define(70)λq,u=u1λq1,…,udλqd. Therefore if n¯=(n1,…,nd), then 1/nλq,n¯=(αn1λq1,…,αndλqd). For each q∈Rd, choose ρq>0 such that ρq|λq|≤δ and define(71)Bq,ρq=x∈Rd:x-q<ρq, where(72)x=x,x,λ,q=∑i=1dλiqi. Note that for any q∈K(73)-infx∈Bq,ρqλq,α,x≤ρqλq,α-λq,α,q≤δ-λq,α,qsince 0≤αi≤1 for each i=1,2,…,d. Therefore it follows from the exponential Chebycheff inequality that (74)1nlogμnBq,ρq≤1nlogEexpλq,n¯,Sn-infx∈Bq,ρqλq,n¯,x=-infx∈Bq,ρq1nλq,n¯,x+1nlogEexp∑i=1dλqi∑j∈IniXj=-infx∈Bq,ρq1nλq,n¯,x+1n∑i=1dniΛλqi≤δ-∑i=1dαniλqiqi-∑i=1dαniΛλqi.The second equality uses that the sequence X1,X2,… is i.i.d. and the second inequality uses that ni/n=αni. Since K is compact, it has a finite covering consisting of N=N(K,δ) open balls Bql,ρql centred at q1,…qN. It follows from the subadditivity property of probability measures and the choice of λql’s that(75)logμnK≤1nlogN+δ-minl=1,…,N∑i=1dαniλqliqli-∑i=1dαniΛλqli. Therefore(76)limsupn→∞logμnK≤δ-limsupn→∞minl=1,…,N∑i=1dαniλqliqli-Λλqli=δ-minl=1,…,N∑i=1dαiλqliqli-Λλqli≤δ-minl=1,…,NIδql≤δ-infq∈KIδqThe equality above uses that the quantity minl=1,…,N∑i=1d‍αniλqliqli-Λ(λqli) is finite, by the finiteness assumption on Λ, which is a concave function of αn and consequently it is continuous of αn.
To extend the proof of the above to all closed subsets ofRd, we need to establish exponential tightness of the measures μn. Let ρ>0 and define Hρ=[-ρ,ρ]d. Then Hρc⊂∪i=1d{x∈Rd:|xi|>ρ} and(77)μnHρc≤∑i=1dμniρ,∞+∑i=1dμni-∞,-ρ,where μni is the law of the ith coordinate of Sn, i.e., the law of Sni=1/ni∑k∈Ini‍Xk. Thus μni=μni. Therefore for any λi≥0(78)μniρ,∞≤E1Sni-ρ≥0≤EeniλiSni-ρ=e-niλiρ∏k∈IniEeλiXk=e-niλiρ-Λλi.Therefore, for every ρ≥x¯, it follows from (64) that(79)μniρ,∞≤e-niΛ∗ρ.Similarly, using (65) we get that(80)μni-∞,-ρ≤e-niΛ∗ρ.It follows from the fact that Λ(λ)<∞, for every λ∈R, and Lemma 2.2.20 of [21] that Λ∗(x)→∞ and |x|→∞. Therefore it follows from (78), (79), and (80) that(81)limρ→∞limsupn→∞1nlogμnHρc=-∞.This then implies that the sequence of measures μn is exponentially tight. Hence the upper bound (76) holds for every closed set F of Rd.
Step 2. Suppose y∈DΛ¯∗={x∈Rd:Λ∗(x)<∞} and for some η∈Rd, yi=Λ′(ηi)=dΛ/dλ(ηi), for any i=1,2,…,d. Define, for any i=1,…,d, a probability measure μ~i on R, as(82)dμ~idμzi=eηizi-Λ′ηi and suppose X1,X2,…,Xn be an independent sequence of random variables with the law of Xj being μ~i if j∈Ini. Let μ~n be the corresponding law for Sn. Then (83)dμ~ndz=∏i=1deniηizi-Ληidμnidzi. This then implies that (84)dμnz=∏i=1deniΛηi-ηizidμ~nz=∏i=1deniΛηi-yiηi+yiηi-ηizidμ~nz.
Therefore(85)1nlogμBy,δ=1nlog∫By,δe∑i=1dniΛηi-yiηi+yiηi-ηizidμ~nz=1nloge∑i=1dniΛηi-yiηi+1nlog∫z∈By,δe∑i=1dniηiyi-zidμn~z=∑i=1dαniΛηi-yiηi+1nlog∫z∈By,δeηn¯,y-zdμ~nz≥∑i=1dαniΛηi-yiηi-1nηn¯δ+1nlogμ~nBy,δ=∑i=1dαniΛηi-yiηi-∑i=1dαni2ηi2δ+1nlogμ~nBy,δ≥∑i=1dαniΛηi-yiηi-ηδ+1nlogμ~nBy,δ,
whereηn¯=(n1η1,…,ndηd). For any i=1,…,d, let Xj be such that j∈Ini. Then(86)Eμ~iXj=1Mηi∫xeηixdμx and by the dominated convergence theorem(87)Eμ¯iXj=Λ′ηi=yi. Therefore by the weak law of large numbers, limn→∞μ~n(By,δ)=1 for all δ>0, since the ith component of S^n converges in probability to yi. Since(88)liminfn→∞∑αniΛηi-yiηi=∑αiΛηi-yiηi we have that(89)liminfn→∞1nlogμnBy,δ≥∑i=1dαiΛηi-yiηi-ηδ, and by the definition of Λ∗(90)Λ∗y≥∑i=1dαi-Ληi+yiηi. Hence(91)liminfn→∞1nlogμnBy,δ≥Λ¯∗y-ηδ. Therefore(92)liminfn→∞1nlogμnBy,δ≥limδ→0liminfn→∞1nlogμnBy,δ≥-Λ¯∗y. Next, suppose that y∈DΛ¯∗, but there is no η∈Rd such that yi=Λ′(ηi) for all i=1,…,d. Here we change the common law μ of the X1,X2,… sequence by adding a small normal random variable so that the perturbed μ will admit logarithmic moment generating function ΛM, for which we can find an η∈Rd that satisfies ΛM′(ηi)=yi, for every i=1,…,d. With this we get a lower bound for the large deviations principle which is later used to deduce the lower for the unperturbed case. Formerly, let Y1,Y2,… be an i.i.d sequence given by(93)Yi=Xi+ViM, where Vi is a sequence of independent and identically distributed standard normal random variables that are independent of the X1,X2,… sequence. The quantity M<∞. Let ν be the common law of the Yi’s and Sn(M)=(Sn,Mi,…,Sn,Md), where(94)Sn,Mi=1ni∑j∈IniYj,fori=1,2,…,d. Then(95)ΛMλ1=logEeλ1Y1=Λλ1+12Mλ12≥Λλ1. Therefore (96)Λ¯∗y=supλ∈Rd∑αiyiλi-Λλi≥supλ∈Rd∑αiyiλi-ΛMλi. Recall that x¯=E(X1)<∞. Hence by the Jensen’s inequality we have that Λ(λ)≥λx¯, for all λ∈R. Therefore the function(97)gλ=∑αiλiyi-ΛMλi is finite, differentiable and it satisfies(98)limρ→∞supλ>ρgλ=-∞. Therefore g(·) attains its supremum over Rd at some η∈Rd, for which(99)0=∂g∂λiηi=αiyi-ΛM′ηi,i=1,…,d. This implies that (100)yi=ΛM′ηi.Therefore by the proceeding proof we have large deviations lower bound for the sequence of probability measures {νn}, i.e., for every δ>0,(101)liminfn→∞logνnBy,δ≥-Λ¯M∗y≥-Λ¯∗y>-∞. Note that the process Sn(M) has the same distribution as Sn+V/Mn, where V is the standard multivariate normal random vector. Therefore, in distributional sense (102)Sn=SnM-VMn, and using that (103)Sn∈By,2δ⊃SnM∈By,δ\SnM∈By,δ,VMn∈By,δc we have (104)PSn∈By,2δ≥PSnM∈By,δ-PSnM∈By,δ,VMn∈By,δc. Therefore(105)μnBy,2δ≥νnBy,δ-PVMn∈By,δc=νnBy,δ-PV≥Mnδ≥νnBy,δ+Mδ2n2≥νnBy,δ. Thus (106)liminfn→∞μnBy,2δ≥liminfn→∞νnBy,δ≥-Λ¯∗y=-infx∈By,2δΛ¯∗x.


### 4.2. Proof of Theorem3

Proof.
The proof of this theorem follows from that of theR-valued case upon making appropriate substitutions. For instance, every R, Rd, λiyi, and Λ′(ηi) in the proof of the R-valued case should be replaced with Rl, (Rl)d, 〈λi,yi〉, and ∇Ληi, respectively. In particular, in establishing the exponential tightness of the μn’s we use that for any ρ>0, we define Hρ=[-ρ,ρ]ld. Then Hρc⊂∪i=1d∪j=1l{x∈Rd:|xji|>ρ} and(107)μnHρc≤∑i=1d∑j=1lμnijρ,∞+∑i=1d∑j=1lμnij-∞,-ρ,where xji is the jth coordinate of the ith-component of x=(x1,…,xd)∈(Rl)d and μnij is the law of the empirical mean(108)Sni,j=1ni∑k∈IniXkj. Here Xkj is the jth coordinate of Xk.


### 4.3. Proof of Theorem4

Proof.
The proof follows from the large deviations principle for vector of empirical means ofR|Σ|-valued random vectors restricted to probability vectors in R|Σ|. Recall from (17) that Λ(·) is finite and differentiable convex function. Therefore the map λ↦∑i=1d‍αi〈λi,νi〉-Λ(λi) is concave. Therefore, the supremum of this map is attained at the value of λ such that(109)νi-∇Λλi=0,foreveryi=1,2,…,d.Let ci=∑j=1|Σ|‍eλjiμj. Here λji is the jth component of the vector λi∈R|Σ|. Then for every i=1,…,d and j=1,…|Σ| it follows from (109) that(110)λij=logνjiμj+logci. Substituting this form of λ into ∑i=1d‍αi〈λi,νi〉-Λ(λi) leads to(111)Λ¯∗ν=∑i=1dαi∑j=1Σνjilogνjiμj=∑i=1dαiHνi∣μ.


## 4.1. Proof of Theorem2

Proof.
The proof comes in two steps. In step one we will establish a large deviations upper bound and the corresponding lower bound is proved in step two. The proof is an adaptation of arguments used to prove the large deviations principle for the empirical mean of i.i.d sequence ofRd-valued random vectors.
Step 1. Let I(x)=Λ∗(x) be the intended rate function of the problem and define the δ-rate function as follows:(67)Iδx=minIx-δ,1δ. The proof of the upper bound will follow if we can show for every δ>0 and every closed subset F of Rd that(68)limn→∞sup1nlogμnF≤δ-infx∈FIδx. We will first of all prove this inequality for compact subsets and extend it to closed subsets with the help of exponential tightness argument. Suppose K is a compact subset of Rd. Then for every δ>0 and q∈K, we can find a λq∈Rd such that(69)∑i=1qαiqiλqi-Λλqi≥Iδq. It follows from the definition of Λ∗ that such a λq exists. For any λq,u∈Rd, define(70)λq,u=u1λq1,…,udλqd. Therefore if n¯=(n1,…,nd), then 1/nλq,n¯=(αn1λq1,…,αndλqd). For each q∈Rd, choose ρq>0 such that ρq|λq|≤δ and define(71)Bq,ρq=x∈Rd:x-q<ρq, where(72)x=x,x,λ,q=∑i=1dλiqi. Note that for any q∈K(73)-infx∈Bq,ρqλq,α,x≤ρqλq,α-λq,α,q≤δ-λq,α,qsince 0≤αi≤1 for each i=1,2,…,d. Therefore it follows from the exponential Chebycheff inequality that (74)1nlogμnBq,ρq≤1nlogEexpλq,n¯,Sn-infx∈Bq,ρqλq,n¯,x=-infx∈Bq,ρq1nλq,n¯,x+1nlogEexp∑i=1dλqi∑j∈IniXj=-infx∈Bq,ρq1nλq,n¯,x+1n∑i=1dniΛλqi≤δ-∑i=1dαniλqiqi-∑i=1dαniΛλqi.The second equality uses that the sequence X1,X2,… is i.i.d. and the second inequality uses that ni/n=αni. Since K is compact, it has a finite covering consisting of N=N(K,δ) open balls Bql,ρql centred at q1,…qN. It follows from the subadditivity property of probability measures and the choice of λql’s that(75)logμnK≤1nlogN+δ-minl=1,…,N∑i=1dαniλqliqli-∑i=1dαniΛλqli. Therefore(76)limsupn→∞logμnK≤δ-limsupn→∞minl=1,…,N∑i=1dαniλqliqli-Λλqli=δ-minl=1,…,N∑i=1dαiλqliqli-Λλqli≤δ-minl=1,…,NIδql≤δ-infq∈KIδqThe equality above uses that the quantity minl=1,…,N∑i=1d‍αniλqliqli-Λ(λqli) is finite, by the finiteness assumption on Λ, which is a concave function of αn and consequently it is continuous of αn.
To extend the proof of the above to all closed subsets ofRd, we need to establish exponential tightness of the measures μn. Let ρ>0 and define Hρ=[-ρ,ρ]d. Then Hρc⊂∪i=1d{x∈Rd:|xi|>ρ} and(77)μnHρc≤∑i=1dμniρ,∞+∑i=1dμni-∞,-ρ,where μni is the law of the ith coordinate of Sn, i.e., the law of Sni=1/ni∑k∈Ini‍Xk. Thus μni=μni. Therefore for any λi≥0(78)μniρ,∞≤E1Sni-ρ≥0≤EeniλiSni-ρ=e-niλiρ∏k∈IniEeλiXk=e-niλiρ-Λλi.Therefore, for every ρ≥x¯, it follows from (64) that(79)μniρ,∞≤e-niΛ∗ρ.Similarly, using (65) we get that(80)μni-∞,-ρ≤e-niΛ∗ρ.It follows from the fact that Λ(λ)<∞, for every λ∈R, and Lemma 2.2.20 of [21] that Λ∗(x)→∞ and |x|→∞. Therefore it follows from (78), (79), and (80) that(81)limρ→∞limsupn→∞1nlogμnHρc=-∞.This then implies that the sequence of measures μn is exponentially tight. Hence the upper bound (76) holds for every closed set F of Rd.
Step 2. Suppose y∈DΛ¯∗={x∈Rd:Λ∗(x)<∞} and for some η∈Rd, yi=Λ′(ηi)=dΛ/dλ(ηi), for any i=1,2,…,d. Define, for any i=1,…,d, a probability measure μ~i on R, as(82)dμ~idμzi=eηizi-Λ′ηi and suppose X1,X2,…,Xn be an independent sequence of random variables with the law of Xj being μ~i if j∈Ini. Let μ~n be the corresponding law for Sn. Then (83)dμ~ndz=∏i=1deniηizi-Ληidμnidzi. This then implies that (84)dμnz=∏i=1deniΛηi-ηizidμ~nz=∏i=1deniΛηi-yiηi+yiηi-ηizidμ~nz.
Therefore(85)1nlogμBy,δ=1nlog∫By,δe∑i=1dniΛηi-yiηi+yiηi-ηizidμ~nz=1nloge∑i=1dniΛηi-yiηi+1nlog∫z∈By,δe∑i=1dniηiyi-zidμn~z=∑i=1dαniΛηi-yiηi+1nlog∫z∈By,δeηn¯,y-zdμ~nz≥∑i=1dαniΛηi-yiηi-1nηn¯δ+1nlogμ~nBy,δ=∑i=1dαniΛηi-yiηi-∑i=1dαni2ηi2δ+1nlogμ~nBy,δ≥∑i=1dαniΛηi-yiηi-ηδ+1nlogμ~nBy,δ,
whereηn¯=(n1η1,…,ndηd). For any i=1,…,d, let Xj be such that j∈Ini. Then(86)Eμ~iXj=1Mηi∫xeηixdμx and by the dominated convergence theorem(87)Eμ¯iXj=Λ′ηi=yi. Therefore by the weak law of large numbers, limn→∞μ~n(By,δ)=1 for all δ>0, since the ith component of S^n converges in probability to yi. Since(88)liminfn→∞∑αniΛηi-yiηi=∑αiΛηi-yiηi we have that(89)liminfn→∞1nlogμnBy,δ≥∑i=1dαiΛηi-yiηi-ηδ, and by the definition of Λ∗(90)Λ∗y≥∑i=1dαi-Ληi+yiηi. Hence(91)liminfn→∞1nlogμnBy,δ≥Λ¯∗y-ηδ. Therefore(92)liminfn→∞1nlogμnBy,δ≥limδ→0liminfn→∞1nlogμnBy,δ≥-Λ¯∗y. Next, suppose that y∈DΛ¯∗, but there is no η∈Rd such that yi=Λ′(ηi) for all i=1,…,d. Here we change the common law μ of the X1,X2,… sequence by adding a small normal random variable so that the perturbed μ will admit logarithmic moment generating function ΛM, for which we can find an η∈Rd that satisfies ΛM′(ηi)=yi, for every i=1,…,d. With this we get a lower bound for the large deviations principle which is later used to deduce the lower for the unperturbed case. Formerly, let Y1,Y2,… be an i.i.d sequence given by(93)Yi=Xi+ViM, where Vi is a sequence of independent and identically distributed standard normal random variables that are independent of the X1,X2,… sequence. The quantity M<∞. Let ν be the common law of the Yi’s and Sn(M)=(Sn,Mi,…,Sn,Md), where(94)Sn,Mi=1ni∑j∈IniYj,fori=1,2,…,d. Then(95)ΛMλ1=logEeλ1Y1=Λλ1+12Mλ12≥Λλ1. Therefore (96)Λ¯∗y=supλ∈Rd∑αiyiλi-Λλi≥supλ∈Rd∑αiyiλi-ΛMλi. Recall that x¯=E(X1)<∞. Hence by the Jensen’s inequality we have that Λ(λ)≥λx¯, for all λ∈R. Therefore the function(97)gλ=∑αiλiyi-ΛMλi is finite, differentiable and it satisfies(98)limρ→∞supλ>ρgλ=-∞. Therefore g(·) attains its supremum over Rd at some η∈Rd, for which(99)0=∂g∂λiηi=αiyi-ΛM′ηi,i=1,…,d. This implies that (100)yi=ΛM′ηi.Therefore by the proceeding proof we have large deviations lower bound for the sequence of probability measures {νn}, i.e., for every δ>0,(101)liminfn→∞logνnBy,δ≥-Λ¯M∗y≥-Λ¯∗y>-∞. Note that the process Sn(M) has the same distribution as Sn+V/Mn, where V is the standard multivariate normal random vector. Therefore, in distributional sense (102)Sn=SnM-VMn, and using that (103)Sn∈By,2δ⊃SnM∈By,δ\SnM∈By,δ,VMn∈By,δc we have (104)PSn∈By,2δ≥PSnM∈By,δ-PSnM∈By,δ,VMn∈By,δc. Therefore(105)μnBy,2δ≥νnBy,δ-PVMn∈By,δc=νnBy,δ-PV≥Mnδ≥νnBy,δ+Mδ2n2≥νnBy,δ. Thus (106)liminfn→∞μnBy,2δ≥liminfn→∞νnBy,δ≥-Λ¯∗y=-infx∈By,2δΛ¯∗x.


## 4.2. Proof of Theorem3

Proof.
The proof of this theorem follows from that of theR-valued case upon making appropriate substitutions. For instance, every R, Rd, λiyi, and Λ′(ηi) in the proof of the R-valued case should be replaced with Rl, (Rl)d, 〈λi,yi〉, and ∇Ληi, respectively. In particular, in establishing the exponential tightness of the μn’s we use that for any ρ>0, we define Hρ=[-ρ,ρ]ld. Then Hρc⊂∪i=1d∪j=1l{x∈Rd:|xji|>ρ} and(107)μnHρc≤∑i=1d∑j=1lμnijρ,∞+∑i=1d∑j=1lμnij-∞,-ρ,where xji is the jth coordinate of the ith-component of x=(x1,…,xd)∈(Rl)d and μnij is the law of the empirical mean(108)Sni,j=1ni∑k∈IniXkj. Here Xkj is the jth coordinate of Xk.


## 4.3. Proof of Theorem4

Proof.
The proof follows from the large deviations principle for vector of empirical means ofR|Σ|-valued random vectors restricted to probability vectors in R|Σ|. Recall from (17) that Λ(·) is finite and differentiable convex function. Therefore the map λ↦∑i=1d‍αi〈λi,νi〉-Λ(λi) is concave. Therefore, the supremum of this map is attained at the value of λ such that(109)νi-∇Λλi=0,foreveryi=1,2,…,d.Let ci=∑j=1|Σ|‍eλjiμj. Here λji is the jth component of the vector λi∈R|Σ|. Then for every i=1,…,d and j=1,…|Σ| it follows from (109) that(110)λij=logνjiμj+logci. Substituting this form of λ into ∑i=1d‍αi〈λi,νi〉-Λ(λi) leads to(111)Λ¯∗ν=∑i=1dαi∑j=1Σνjilogνjiμj=∑i=1dαiHνi∣μ.


## 5. Proof of Theorem5

Proof.
The proof comes in three steps. In step one we proof a lower for (23). Using condition (21), we proof an upper bound for (23) in step two. Step three shows that condition (22) implies condition (21) and that completes the proof.
Step One. Fix x∈X and δ>0. Due to the equi-continuity of the sequence ϕn we have that the functions ϕn are lower semicontinuous. Thus, there exists a neighbourhood G of x such that(112)infy∈Gϕny≥ϕnx-δ,foralln=1,2,3,…. We then have that(113)EenϕnSn≥EenϕnSn1Sn∈G≥enϕnx-δμnG. We get from here that(114)liminfn→∞1nlogEenϕnSn≥liminfn→∞ϕnx-δ+liminfn→∞1nlogμnG≥ϕx-δ-infy∈GIy≥ϕx-Ix-δ. Since x∈X and δ>0 were arbitrary chosen, we have that(115)liminfn→∞1nlogEenϕnSn≥supx∈Xϕx-Ix.
Step Two. Suppose that the functions are ϕn and uniformly bounded; i.e., there is a constant M, such that supx∈Mϕn(x)≤M. Then condition (21) holds. Let α<∞ and δ>0 and define ΨI(α)={x∈X:I(x)≤α}. Then ΨI is a compact level set since I is a good rate function. It follows from the lower semicontinuity property of I, the upper semicontinuity property of ϕn, and the fact that X is a regular topological space that for every x∈ΨI(α), there is a neighbourhood Ax of x such that(116)infy∈Ax¯Iy≥Ix-δ,supy∈Ax¯ϕny≤ϕnx+δ,foralln=1,2,3,….Since ΨI(α) is compact, there are finite number of points x1,x2,…,xN∈ΨI(α) such that the neighbourhoods Ax1,Ax2,…,AxN cover ΨI(α). Therefore(117)EenϕnSn≤∑i=1NEenϕnSn1Sn∈Axi+enMμn⋃i=1NAxic≤∑i=1Nenϕnxi+δμnAxi¯+enMμn⋃i=1NAxic. The last inequality follows from (116). Note that ⋃i=1NAxic⊂ΨI(α)c and (118)limsupn→∞1nlogEenϕnSn≤maxmaxi=1Nlimsupn→∞ϕnxi+δ-infy∈A¯xiIy,M-infy∈⋃i=1NAxicIy≤maxmaxi=1Nϕxi-Ixi+2δ,M-α≤maxsupxi∈Xϕxi-Ixi,M-α+2δ. Since the ϕn’s are bounded above, so is the limiting ϕ. The upper bound of interest follows upon taking the limits δ→0 and α→∞.
If theϕn’s are not uniformly bounded above, we set ϕnM(x)=ϕn(x)∧M≤ϕn(x). Then ϕnM’s are uniformly bounded above and using the arguments for the above proof we get that(119)limsupn→∞1nlogEenϕnSn≤supx∈Xϕx-Ix∨limsupn→∞1nlogEenϕnSn1ϕnSn≥M. The result for this case follows from the tail condition (21) as we let M→∞.
Step Three. In this step we show that condition (22) implies the tail condition (21). In this regard we let ϵ>0, γ>1, and define Xn=expn[ϕn(Sn)-M]. Then(120)e-nMEenϕnSn1ϕnSn≥M=EXn1Xn≥1≤EXnγ≤e-nγMEenγϕnSn. It follows from here that(121)limsupn→∞1nlogEenϕnSn1ϕnSn≥M≤M1-γ+limsupn→∞1nlogEenγϕnSn. Therefore condition (22) implies condition (21) upon taking M→∞ limit.


## 6. Conclusion

This paper has developed large deviations machinery for the empirical means and measures for partitions of independent and identically distributed sequence of random variables. The large deviations result is further applied to derive the limiting free energy for multipopulation Curie-Weiss and Potts’ models. The method proposed here can be applied to multipopulation versions of spin models with continuous spins such as theO(n) model [11], the spherical model [12, 13], the liquid crystal model [11, 14, 15], and the Kuramoto model [16]. The multipopulation Potts’ model may have applications in discrete choice context with more than two alternatives to choose from. This serves as a natural extension to the Ising cases considered in [8–10, 22, 23].The knowledge gained from the study of the minimizers of the associated minimization problem that leads to the limiting pressure will offer insight to the scaling limit behaviour of the empirical measures associated with the multipopulation Potts’ model. This will be a natural extension of the work in [24].



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