Two populations mean-field monomer-dimer model

TL;DR

This paper analyzes a two-population mean-field monomer-dimer model with interactions, deriving a variational principle for pressure density and identifying a ferromagnetic phase transition in a specific limit.

Contribution

It introduces a novel two-population mean-field monomer-dimer model with both hard-core and attractive interactions, and characterizes its phase transition behavior.

Findings

Pressure density satisfies a three-dimensional variational principle.

A ferromagnetic phase transition is identified in a specific population size limit.

The model extends understanding of monomer-dimer systems with multiple populations.

Abstract

A two populations mean-field monomer-dimer model including both hard-core and attractive interactions between dimers is considered. The pressure density in the thermodynamic limit is proved to satisfy a three-dimensional variational principle. A detailed analysis is made in the limit in which one population is much smaller than the other and a ferromagnetic mean-field phase transition is found.

Equations

$$f_{\alpha }(d_{AB})=h_{AB}+J_{AB}^{AB}{d}_{AB}.$$

$$e=\{i,j\}\in \Delta ,e^{\prime }=\{i^{\prime },j^{\prime }\}\in \Delta \Rightarrow e\cap e^{\prime }=\varnothing$$

$$2D_A+D_{AB}+M_A=N_A,2D_B+D_{AB}+M_B=N_B$$

$$D\,:=\,\left(\begin{array}[]{l}D_{A}\\[2.0pt] D_{B}\\[2.0pt] D_{AB}\end{array}\right)\;.$$

$$|D|:=D_A+D_B+D_{AB}$$

$$H_N(D)=-h\cdot D-\frac{1}{2N}JD\cdot D$$

$$h\,=\,\left(\begin{array}[]{l}h_{A}\\[2.0pt] h_{B}\\[2.0pt] h_{AB}\end{array}\right)\quad J\,=\,\left(\begin{array}[]{lll}J_{A}^{A}&J_{A}^{B}&J_{A}^{AB}\\[2.0pt] J_{B}^{A}&J_{B}^{B}&J_{B}^{AB}\\[2.0pt] J_{AB}^{A}&J_{AB}^{B}&J_{AB}^{AB}\end{array}\right)\;.$$

$$Z_N\equiv Z_N(h,J,\alpha )=\sum _{\Delta \in {\mathcal{D}}_N}{N}^{-|D|}{e}^{-H_N(D)}$$

$$⟨f{⟩}_N:=\frac{1}{Z_N}\sum _{\Delta \in {\mathcal{D}}_N}{N}^{-|D|}{e}^{-H_N(D)}f(\Delta )$$

$$2d_A+d_{AB}\le \alpha ,2d_B+d_{AB}\le 1-\alpha .$$

$$\gamma (x):=\exp (x\log x-x),x\ge 0$$

$$\begin{array}{rl}s(d;\alpha ):=& \log \gamma (\alpha )+\log \gamma (1-\alpha )-\log \gamma (\alpha -2d_A-d_{AB})+\\ & -\log \gamma (1-\alpha -2d_B-d_{AB})-\log \gamma (d_A)-\log \gamma (d_B)+\\ & -\log \gamma (d_{AB})-d_A\log 2-d_B\log 2\end{array}$$

$$ϵ(d;h,J):=-h\cdot d-\frac{1}{2}Jd\cdot d$$

$$\psi (d;h,J,\alpha ):=s(d;\alpha )-ϵ(d;h,J).$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}\log Z_N(h,J,\alpha )=\underset{d\in {\Omega }_{\alpha }}{\max }\psi (d;h,J,\alpha )=:p(h,J,\alpha )$$

$$\{\begin{array}{l}{d}_A=\frac{w_A}{2}{m}_A^2\\ d_B=\frac{w_B}{2}{m}_B^2\\ d_{AB}=w_{AB}{m}_A{m}_B\end{array}$$

$$m_A=\alpha -2d_A-d_{AB},m_B=1-\alpha -2d_B-d_{AB},$$

$$w_A=e^{h_A+J_{A}d},w_B=e^{h_B+J_{B}d},w_{AB}=e^{h_{AB}+J_{AB}d}.$$

$$d=g(h+Jd,\alpha ).$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}{⟨D⟩}_N=d^{\ast }.$$

$$Z_N=\sum _{D_A=0}^{N_A/2}\sum _{D_B=0}^{N_B/2}\sum _{D_{AB}=0}^{(N_A-2D_A)\wedge (N_B-2D_B)}{\varphi }_N(D)e^{-H_N(D)}$$

$${\varphi }_N(D):=\frac{N_A!N_B!}{(N_A-2D_A-D_{AB})!(N_B-2D_B-D_{AB})!D_A!D_B!D_{AB}!2^{D_A}{2}^{D_B}}$$

$$\frac{1}{N}\log Z_N=\frac{1}{N}\log {\tilde{Z}}_N+\mathcal{O}\left(\frac{\log N}{N}\right),$$

$$\frac{1}{N}\log {\tilde{Z}}_N=\underset{D\in N{\Omega }_{\alpha }}{\max }\frac{1}{N}\left(\log {\tilde{\varphi }}_N(D)-H_N(D)\right)+\mathcal{O}\left(\frac{\log N}{N}\right)$$

$$\frac{1}{N}\left(\log {\tilde{\varphi }}_N(Nd)-H_N(Nd)\right)=\psi (d;h,J,\alpha ),d\in {\Omega }_{\alpha }.$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}\log Z_N=\underset{d\in {\Omega }_{\alpha }}{\max }\psi (d;h,J,\alpha ).$$

$${\nabla }_{d}s=\left(\log \frac{m_A^2}{2d_A},\log \frac{m_B^2}{2d_B},\log \frac{m_A{m}_B}{d_{AB}}\right)$$

$$-{\nabla }_dϵ=\left(h_A+J_A\cdot d,h_B+J_B\cdot d,h_{AB}+J_{AB}\cdot d\right)$$

$${\nabla }_d\psi (d;h,J,\alpha )=0\iff d\text{ is a solution of eqrefeq: system}.$$

$$h:=h_{AB},J:=J_{AB}^{AB}>0$$

Full text

1 Introduction

Monomer-dimer models have been introduced in theoretical physics during the ’70s to explain the absorption of diatomic molecules on a two-dimensional layer [21]. Fundamental results were obtained by Heilmann and Lieb, who proved the absence of phase transitions [15] when only the hard-core interaction is taken into account, while the presence of an additional interaction coupling dimers can generate critical behaviours [16]. Monomer-dimers models have been source of a renewed interest in the last years in mathematical physics [11, 1, 13, 2], condensed matter physics [19] and in the applications to computer science [22, 17] and social sciences [7, 10]. The presence of an interaction beyond the hard-core one that couples different dimers is fundamental for the applications where phase transitions are observed [7, 10]. Indeed in [4, 5, 3] the authors proved that a mean-field monomer-dimer model exhibits a ferromagnetic phase transition when a sufficiently strong interaction is introduced between pairs of dimers.

In this paper the investigation is extended to the case of a mean-field monomer-dimer model defined over two populations. This multi-species framework has been already introduced in the context of spin models [18, 9, 8, 20] reveling interesting mathematical features. Multi-species monomer-dimer models are suitable to describe the experimental situation treated in [7, 10], where a mean-field type phase transition has been observed in the percentage of mixed marriages between native people and immigrants. The hard-core interaction between dimers naturally represents the monogamy constraint in marriages, while, as pointed out by the authors of [7], an additional imitative interaction between individuals can be at the origin of the observed critical behaviour.

In this work we consider a mean-field model built on two populations $A$ and $B$ (e.g., the immigrants population and the local one) which takes into account both the imitative and the hard-core interactions. Dimers can be divided into three classes: type $A$ if they link two individuals in $A$, type $B$ if they link two individuals in $B$ and type $AB$ if they link a mixed couple. The relative size of the two populations is fixed $N_{A}/N_{B}=\alpha/(1-\alpha)$. The energy contribution of dimers is tuned by a three dimensional vector $h=(h_{A},h_{B},h_{AB})\in\mathbb{R}^{3}$ where $h_{A}$ tunes the activity of a dimer of type $A$ and so on. Individuals have also a certain propensity to imitate or counter-imitate the behaviour of the other individuals which is encoded in an additional contribution to the energy tuned by a $3\times 3$ real matrix $J$. For example the entry $J_{AB}^{AB}$ couples dimers of type $AB$ with other dimers of the same type. The main result we obtain is a representation of the pressure density in the thermodynamic limit in terms of a variational problem in $\mathbb{R}^{3}$ for all the values of the parameters $h$ and $J$ (see Theorem 1 in section 2 for the precise statement). This result is applied in the case where the only non-zero parameters contributing to the energy are $h_{AB}$ and $J_{AB}^{AB}$. As a consequence the relevant degree of freedom of the model is the density of mixed dimers $d_{AB}$ and the above variational problem leads to a consistency equation of the type

[TABLE]

Its analytical properties are investigated in details for small $\alpha$: the mean-field critical exponent $1/2$ is rigorously found, consistently with the experimental situation described in [7, 10].

The paper is structured as follows. In section 2 we introduce the statistical mechanics model with the basic definitions and we prove the main result: the thermodynamic limit of the pressure density is expressed as a three-dimensional variational problem, where the order parameters are the dimer densities $d_{A},\,d_{B}$ internal to each population and the mixed dimer density $d_{AB}$.

In section 3 we focus on three non-zero parameters, $\alpha,\,h_{AB},\,J_{AB}^{AB}$, and we study in detail the critical behaviour of the system when one population is much larger than the other ($\alpha\to 0$), finding a phase transition with standard mean-field exponents.

Finally in the Appendix we give an alternative proof for the existence of thermodynamic limit of the pressure density in the case $J=0$, $h_{A}+h_{B}\geq 2h_{AB}$. This proof, which easily applies also to the standard single population case, uses a convexity inequality and is based on the Gaussian representation for the partition function [6].

2 Model and main result

Consider a system composed by $N$ sites divided into two populations of sizes $N_{A}$ and $N_{B}$ respectively, $N_{A}+N_{B}=N$. We assume that the ratios $\alpha=N_{A}/N$ and $1-\alpha=N_{B}/N$ are fixed when the total size $N$ of the system varies. A monomer-dimer configuration can be identified with a set $\Delta$ of edges that satisfies a hard-core condition:

[TABLE]

Given the configuration $\Delta$ (see Figure 1), the edges in $\Delta$ are called dimers and they can be partitioned into three families: denote by $D_{A}$ the number of dimers having both endpoints in $A$, by $D_{B}$ the number of dimers having both endpoints in $B$ and by $D_{AB}$ the number of dimers having one endpoint in $A$ and the other one in $B$. Monomers, namely sites free of dimers, can be partitioned into two families: denote by $M_{A},\,M_{B}$ the number of monomers in $A,\,B$ respectively. Observe that

[TABLE]

We denote by $\mathscr{D}_{N}$ the set of all possible monomer-dimer configurations on $N$ sites. For a given configuration $\Delta\,\in\,\mathscr{D}_{N}$, $D$ denotes the vector of the cardinalities of the three families of dimers

[TABLE]

while

[TABLE]

represents the total number of dimers. The Hamiltonian function is defined as

[TABLE]

where $\cdot$ denotes the standard scalar product in $\mathbb{R}^{3}$, the dimer vector field $h$ tunes the activity of dimers while the coupling matrix $J$ tunes the interaction between sites according to the types of dimers they host:

[TABLE]

The partition function of the model is

[TABLE]

where the term $N^{-|D|}$ is necessary to ensure a well defined thermodynamic limit of the model. Given $f\!:\mathscr{D}_{N}\to\mathbb{R}$ we call expected value of $f$ with respect to the Gibbs measure the quantity

[TABLE]

where $H_{N}$ is the Hamiltonian function (5).

Let us introduce the definitions needed to state our main result. Denote by $\Omega_{\alpha}$ the set of $d=(d_{A},d_{B},d_{AB})^{T}\in(\mathbb{R}_{+})^{3}$ such that

[TABLE]

The above constraints on the vector $d$ reflect the hard-core relations (2). Set

[TABLE]

and define the following functions

[TABLE]

[TABLE]

[TABLE]

The functions $\psi,s,\epsilon$ represent respectively the variational pressure, entropy and energy densities.

Theorem 1**.**

For all $\alpha\in(0,1)$, $h\in\mathbb{R}^{3}$ and $J\in\mathbb{R}^{3\times 3}$, there exists

[TABLE]

The function $\psi(d;h,J,\alpha)$ attains its maximum in at least one point $d^{*}=d^{*}(h,J,\alpha)\in\Omega_{\alpha}$ which solves the following fixed point system:

[TABLE]

where we denote

[TABLE]

[TABLE]

At $J=0$ the system (15) has a unique solution $d^{*}=g(h,\alpha)\in\Omega_{\alpha}$ which is an analytic function of the parameters $h,\alpha$. Clearly at any $J$ the system (15) rewrites as

[TABLE]

Provided that $d^{*}$ is differentiable, $\nabla_{\!h\,}p=d^{*}$ hence there exists

[TABLE]

Proof.

The number of configurations $\Delta\in\mathscr{D}_{N}$ with given cardinalities $D_{A}$, $D_{B}$, $D_{AB}$ can be computed by a standard combinatorial argument. Therefore the partition function rewrites as

[TABLE]

with

[TABLE]

In order to simplify the computations, we approximate the factorial by the continuous function $\gamma$ defined in (10). We denote by $\tilde{\phi}_{N}$ the function obtained from $\phi_{N}$ by substituting any factorial $n!$ with $\gamma(n)$, then we denote by $\tilde{Z}_{N}$ the partition function obtained from $Z_{N}$ by substituting $\phi_{N}$ with $\tilde{\phi}_{N}$. The Stirling approximation and elementary computations give the following properties of $\gamma$:

• i.

$1\lor\sqrt{2\pi n}\,\leq\,n!/\gamma(n)\,\leq\,1\lor e^{\frac{1}{12}}\sqrt{2\pi n}\quad\forall n\in\mathbb{N}$

• ii.

$\frac{d}{dx}\log\gamma(x)=\log x\,,\quad\log\gamma(x)$ is convex

• iii.

$\frac{1}{N}\log\gamma(Nx)=\log\gamma(x)+x\,\log N$

By i. it follows that

[TABLE]

by a standard argument

[TABLE]

and using iii. a direct computation shows that for every $N\in\mathbb{N}$

[TABLE]

Therefore there exists

[TABLE]

Using ii. one can easily compute

[TABLE]

[TABLE]

therefore

[TABLE]

The first derivatives of $p(h,J,\alpha)=\psi(d^{*}(h,J,\alpha);h,J,\alpha)$ can be easily computed since $\nabla_{\!d\,}\psi(d^{*};h,J,\alpha)=0$.

3 The limit $\alpha\,\to\,0$

In this section we choose a particular framework that simplifies the mathematical treatment of the problem and allows a detailed analysis of the thermodynamic properties of the system. The most peculiar parameters of the model are $h_{AB}$ and $J_{AB}^{AB}$, describing respectively the $AB$-dimer field and the interaction between couples of $AB$-dimers, indeed they have no correspondence in a bipopulated Ising model [18]. Moreover we focus on the case where one population is much smaller than the other ($\alpha\to 0$), since it is interesting for the social applications [7]. Thus in this section we set $h_{A}=h_{B}=0$, $J_{A}^{A}=J_{A}^{B}=J_{B}^{A}=J_{A}^{AB}=J_{AB}^{A}=J_{B}^{AB}=J_{AB}^{B}=0$ and we consider only the remaining coefficients $h_{AB}$ and $J_{AB}^{AB}$. From now on, with a slight abuse of notation, we will denote

[TABLE]

and the mixed dimer density

[TABLE]

In this framework the degrees of freedom of the variational problem (14) reduces from three to one, since $d_{A},d_{B}$ are explicit functions of $d_{AB}\equiv d$ as can be easily observed by looking to the consistency equation (15). Precisely, by setting $x_{\alpha}(d):=m_{A}=\sqrt{2d_{A}}\,$, $y_{\alpha}(d):=m_{B}=\sqrt{2d_{B}}$ one can easily see that $x_{\alpha}(d),\,y_{\alpha}(d)$ are the positive solutions of the following quadratic equations respectively

[TABLE]

namely

[TABLE]

Then one can easily prove from Theorem 1 that

[TABLE]

where $\psi_{1}$ coincides with the function $\psi$ defined by equation (13) evaluated at

[TABLE]

Any solution ${d}^{*}={d}^{*}(h,J,\alpha)$ of the one-dimensional variational problem (29) satisfies the fixed point equation

[TABLE]

It is convenient to set $f_{\alpha}(d):=\log d-\log x_{\alpha}(d)-\log y_{\alpha}(d)$ and rewrite equation (31) as $f_{\alpha}(d)=h+Jd\,$. Fix $\alpha\in(0,1)$. $f_{\alpha}$ is the inverse function of a sigmoid function111It is easy to check that $f_{\alpha}(d)\to-\infty$ as $d\searrow 0$, $f_{\alpha}(d)\to\infty$ as $d\nearrow\alpha$, $f_{\alpha}^{\prime}>0$, $f_{\alpha}^{\prime\prime}$ vanishes exactly once.. Therefore the point $(d_{c},h_{c},J_{c})$ such that $f^{\prime\prime}_{\alpha}(d_{c})=0$, $f^{\prime}_{\alpha}(d_{c})=J_{c}$, $f_{\alpha}(d_{c})=h_{c}+J_{c}\,d_{c}$ is the critical point of the system, where the density ${d}^{*}$ branches from one to two values (see Figure 2).

For small values of $\alpha$, the following estimates for the critical point can be obtained by expanding $f_{\alpha}(d)$ as $\alpha\to 0$:

[TABLE]

[TABLE]

[TABLE]

Fixing $\alpha$ close to zero and moving the parameters $(h,J)$ towards their critical values, along the half line h-h_{c}(\alpha)=-d_{c}(\alpha)\,\big{(}J-J_{c}(\alpha)\big{)}, $J\geq J_{c}$, the mixed dimer density $d^{*}(h,J,\alpha)$ exhibits the following critical behaviour:

[TABLE]

with $C(\alpha)=\sqrt{\frac{3}{16}\alpha^{3}+\mathcal{O}(\alpha^{6})}$. This fact can be proven using the Taylor expansion of $f_{\alpha}(d)$ around $d=d_{c}(\alpha)$ up to the third order.

Remark 1*.*

It is remarkable that our model is in good agreement with the experimental results in [7] where the authors find that the fraction of mixed marriage over total number of marriages

[TABLE]

undergoes a mean-field like phase transition for small values of $\alpha$. More precisely they obtain that a function of the type

[TABLE]

is a very good fit for the experimental values of $d_{mix}$ versus $\alpha$.

The critical behaviour (37) can be predicted by the model presented in this section, with coupling $J=\alpha\,(1-\alpha)\,J^{\prime}$, $J^{\prime}\gg 1$. Indeed, for fixed $J^{\prime}\gg 1$, the critical point of the system is given by $(d_{c},h_{c},\alpha_{c})$, where

[TABLE]

[TABLE]

[TABLE]

and the critical behaviour of $d_{mix}$ as $\alpha\to\alpha_{c}$, $h=h_{c}-d_{c}\,(\alpha-\alpha_{c})$, is the following:

[TABLE]

where

[TABLE]

Remark 2*.*

Equation (41) is a consequence of the fact that at the critical point the lowest order non vanishing derivative of the variational pressure $\psi_{1}$ in (29) is the fourth one. This fact suggests that the fluctuations of the order parameter at the critical point follows the standard mean field theory [12, 3]. From the above considerations we expect the fluctuations scale as $N^{3/4}$ and converge to a quartic exponential distribution agreement with the experimental results in [10].

Acknowledgment: We thank Pierluigi Contucci for bringing the problem to our attention and we acknowledge financial support by GNFM-INdAM Progetto Giovani 2017.

Appendix

Here we give a directed proof of the existence of the thermodynamic limit for the pressure density in the particular case

[TABLE]

where $W>0$ means that the matrix $W$ is positive definite. This proof is independent from Theorem 1 and the strategy follows a basic idea introduced in [14] in the context of Spin Glass Theory. In this case the partition function (7) admits a representation in terms of Gaussian moments:

[TABLE]

where $\xi=(\xi_{A},\xi_{B})$ is a centred Gaussian vector of covariance matrix $\frac{1}{N}W$ (the hypothesis of positive definiteness is crucial). The representation (43) is based on the Isserlis-Wick formula, see [6] (Proposition 2.2) for the proof.

Now consider the set $Q=\{\xi\in\mathbb{R}^{2}\,:\,1+\xi_{A}>0,\,1+\xi_{B}>0\}\,$ and define a modified partition function

[TABLE]

$Z_{N}^{*}$ rewrites as an integral over $\xi\in Q$ with integrand function proportional to $\exp(N\,f(\xi))$ where

[TABLE]

Since $f$ approaches its global maximum on $\mathbb{R}^{2}$ only for $\xi_{A}\geq 0,\,\xi_{B}\geq 0$, standard Laplace type estimates implies that

[TABLE]

Hence we can restrict our attention to the sequence $\log Z_{N}^{*}$, $N\in\mathbb{N}$. We claim that

Proposition 1**.**

For every $N_{1},N_{2},N\in\mathbb{N}$ such that $N=N_{1}+N_{2}$, it holds

[TABLE]

Then the sequence $\log Z_{N}^{*}$ is super-additive and the “monotonic” convergence of the pressure density will follow immediately by Fekete’s lemma and equation (45):

Corollary 1**.**

Under the hypothesis (42), there exists

[TABLE]

Only the proposition 1 remains to be proven.

Proof of the proposition 1.

The strategy for the proof follows the basic ideas introduced in [14] for mean field spin models. For a fixed $N$ consider two integers $N_{1},N_{2}$, such that $N=N_{1}+N_{2}$ and set

[TABLE]

We decompose each of the two parts of the system $N_{1},N_{2}$ in two populations $A,B$ according to the fixed ratio $\alpha$, namely according to the relation

[TABLE]

Now we introduce two independent centred Gaussian vectors:

[TABLE]

and we prove the following lemmas.

Lemma 1**.**

[TABLE]

Proof.

Since $\xi_{1},\xi_{2}$ are independent centred Gaussian vectors, $\xi^{\prime}:=\gamma\,\xi_{1}+(1-\gamma)\,\xi_{2}$ is a centred Gaussian vector. Its covariance matrix is:

[TABLE]

the same of $\xi$. ∎

Lemma 2**.**

[TABLE]

Proof.

Consider the function $f(x,y)=(1+x)^{\gamma}\,(1+y)^{1-\gamma}$ and its Taylor polynomial of first order at $(0,0)$, $P(x,y)=1+\gamma x+(1-\gamma)y\,$. The Hessian matrix of $f$ is negative defined for $x>-1,\,y>-1$ (it has zero determinant and negative trace), hence $f(x,y)\leq P(x,y)\,$. ∎

Finally the proof of proposition 1 follows easily using the independence of $\xi_{1},\,\xi_{2}$, lemma 2 and lemma 1 . ∎