Two Populations Mean-Field Monomer–Dimer Model

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 variational principle. A detailed analysis is made in the limit of one population is much smaller than the other and a ferromagnetic mean-field phase transition is found.

Appendix

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

$$\begin{aligned} J=0 \ ,W=\begin{pmatrix}w_A &{} w_{AB}\\ w_{AB} &{} w_B\end{pmatrix} =\begin{pmatrix}e^{h_A} &{} e^{h_{AB}}\\ e^{h_{AB}} &{} e^{h_B}\end{pmatrix} >0 \ . \end{aligned}$$

(40)

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 [15] in the context of Spin Glass Theory. In this case the partition function (7) admits a representation in terms of Gaussian moments:

$$\begin{aligned} Z_N \,=\, \sum _{\Delta \in \mathscr {D}_N} \left( \frac{w_A}{N}\right) ^{D_A}\left( \frac{w_B}{N}\right) ^{D_B}\left( \frac{w_{AB}}{N}\right) ^{D_{AB}} \,=\, \mathbb E\left[ (1+\xi _A)^{N_A}(1+\xi _B)^{N_B}\right] ,\nonumber \\ \end{aligned}$$

(41)

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) and $\mathbb E$ denotes the expectation operator. The representation (41) is based on the Isserlis–Wick formula, see [5] (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

$$\begin{aligned} Z_N^* =\, \mathbb E\left[ (1+\xi _A)^{N_A}(1+\xi _B)^{N_B}\,\mathbb {1}_Q(\xi ) \right] . \end{aligned}$$

(42)

$Z_N^*$ can be rewritten as an integral over $\xi \in Q$, with integrand function proportional to $\exp (N\,f(\xi ) )$ and

$$\begin{aligned} f(\xi )= -\frac{1}{2}\langle W^{-1}\xi ,\xi \rangle +\alpha \log |1+\xi _A|+(1-\alpha )\log |1+\xi _B| . \end{aligned}$$

Since f approaches its global maximum on $\mathbb R^2$ only for $\xi _A\ge 0,\,\xi _B\ge 0$, standard Laplace type estimates implies that

$$\begin{aligned} \frac{Z_N}{Z_N^*} \rightarrow 1 \quad \text {as }N\rightarrow \infty . \end{aligned}$$

(43)

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 that

$$\begin{aligned} Z_{N_1}^*\,Z_{N_2}^* \,\le \, Z_N^* . \end{aligned}$$

(44)

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 Eq. ():

Corollary 1

Under the hypothesis (40), there exists

$$\begin{aligned} \lim _{N\rightarrow \infty } \frac{1}{N}\log Z_N \,=\, \sup _{N} \frac{1}{N}\log Z_N^* \end{aligned}$$

(45)

Only the proposition 1 remains to be proven.

Proof of the Proposition 1

The strategy for the proof follows the basic ideas introduced in [15] 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

$$\begin{aligned} \gamma =N_1/N\,,\ 1-\gamma =N_2/N , \end{aligned}$$

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

$$\begin{aligned} N_i = \alpha N_i + (1-\alpha ) N_i =: N_{iA}+N_{iB} \,,\quad i=1,2 \end{aligned}$$

Now we introduce two independent centred Gaussian vectors:

$$\begin{aligned} \xi _i = (\xi _{iA}\,,\,\xi _{iB})\,\ \text {with covariance matrix }\frac{1}{N_i}\,W \,,i=1,2 \end{aligned}$$

and we prove the following lemmas.

Lemma 1

$$\begin{aligned} \gamma \,\xi _1+(1-\gamma )\,\xi _2 \,\overset{d}{=}\, \xi \end{aligned}$$

Proof

Since $\xi _1,\xi _2$ are independent centred Gaussian vectors, $\xi ':=\gamma \,\xi _1+(1-\gamma )\,\xi _2$ is a centred Gaussian vector. Its covariance matrix is:

$$\begin{aligned} \gamma ^2\,\frac{W}{N_1}+(1-\gamma )^2\,\frac{W}{N_2} \,=\, \gamma \,\frac{W}{N}+(1-\gamma )\,\frac{W}{N} \,=\, \frac{W}{N} \ , \end{aligned}$$

the same of $\xi$. $\square$

Lemma 2

$$\begin{aligned} (1+x)^\gamma \,(1+y)^{1-\gamma } \le 1+\gamma x+(1-\gamma )y \quad \forall \,x>-1,\,y>-1,\gamma \in (0,1) \end{aligned}$$

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 $f(x,y)\le P(x,y)\,$. $\square$ Finally the proof of proposition 1 follows easily using the independence of $\xi _1,\,\xi _2$, Lemmas 2 and 1. $\square$