Limit Theorems for Monomer–Dimer Mean-Field Models with Attractive Potential

Abstract

The number of monomers, in a monomer-dimer mean-field model with an attractive potential, fluctuates according to the central limit theorem when the parameters are outside the critical curve. At the critical point the model belongs to the same universality class of the mean-field ferromagnet. Along the critical curve the monomer and dimer phases coexist.

Inverse problem for the mean-field monomer-dimer model with attractive interaction

“…Beside such interaction though, as first noticed by Peierls [5], the attractive component of the Van der Waals potentials might influence the phase structure of the model and the thermodynamic behaviour of the material. In the mean field setting analysed here the monomer-dimer model displays the phenomenon of 1 phase coexistence among the two types of particles [6,7,8]. This makes the inverse problem particularly challenging since in the presence of phase coexistence the non uniqueness of its solution requires a special attention in identifying the right set of configurations.…”

Mean-Field Monomer-Dimer Models. A Review

“…In this section we study the distributional limit of the random variable number of monomers with respect to the Gibbs measure on the complete graph [3,4]. We show that a law of large numbers holds outside the coexistence curve γ, whereas on γ the limiting distribution is a convex combination of two Dirac deltas representing the two phases (theorems 5.1, 5.2).…”

“…The law of large numbers holds outside the coexistence curve γ, on γ instead it breaks down in a convex combination of two Dirac deltas. Precisely it holds Theorem 5.2 (see [3]). ii) On the coexistence curve (h, J) ∈ γ, denoting by m 1 , m 2 the two global maximum points ofp(m), it holds…”

“…In [3] we follow the Gaussian convolution method introduced by Ellis and Newman for the mean-field Ising model (Curie-Weiss) in [21][22][23] in order to deal with the imitative potential. An additional difficulty stems from the fact that even in the absence of imitation the system keeps an interacting nature due to the presence of the hard-core interaction: we use the Gaussian representation 2.7 to decouple the hard-core interaction.…”

“…At the critical point its breakdown results in a different scaling N 3/4 and in a different limiting distribution Ce −cx 4 dx . Precisely Theorem 5.4 (see [3]). where λ c = ∂ 4 p ∂m 4 (m c ) < 0, m c ≡ m * (h c , J c ) and C −1 = R exp( λc 24 x 4 )dx .…”