A mean-field monomer-dimer model with attractive interaction: Exact solution and rigorous results

Abstract

A mean-field monomer-dimer model which includes an attractive interaction among both monomers and dimers is introduced and its exact solution rigorously derived.

Mean-Field Monomer-Dimer Models. A Review

“…The hard-core interaction is not sufficient to cause a phase transition, but adding also the imitative interaction the system can have phase transitions [6,15,16,31]: in sections 4 we will study this phase transition on the complete graph. The location of the zeros of the partition function in the complex plane in presence of imitation is an open problem.…”

“…· GN ,x will be the corresponding Gibbs expected value. The pressure density is 6) and the monomer density is…”

“…This model has been initially studied in [6,7], where the behaviour of the pressure and monomer densities in the thermodynamic limit is analysed. where for all t ∈ R Furthermore the function ψ(m) attains its maximum in (at least) one point m * = m * (h, J) ∈ (0, 1), which is a solution of the the consistency equation In order to prove Theorem 4.1, first we need to deal with the case J = 0, then the limit (4.6) with J > 0 follows by a convexity argument introduced by Guerra [28] for the Curie-Weiss model.…”

“…which will be fundamental in the study of the fluctuations of M N (Section 5). The function m * (see [6]) is single-valued and smooth on the plane (h, J) with the exception of an implicitly defined open curve h = γ(J) and its endpoint (h c , J c ). Instead on γ there are two global maximum points m 1 < m 2 that correspond to the dimer phase and the monomer phase respectively.…”

Dilution of Ferromagnets via a Random Graph‐Based Strategy

“…The study of diluted ferromagnets [1][2][3][4][5][6] dates back to several years ago, following two main paths sometimes overlapping, i.e. the statistical mechanics approach to lattices, and the graph theory approach to networks [7,8].…”