Diego Alberici

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

A mean-field monomer-dimer model which includes an attractive interaction among both monomers and dimers is introduced and its exact solution rigorously derived. The Heilmann-Lieb method for the pure hard-core interacting case is used to compute upper and lower bounds for the pressure. The bounds are shown to coincide in the thermodynamic limit for a suitable choice of the monomer density m. The computation of the monomer density is achieved by solving a consistency equation in the phase space (h, J), where h tunes the monomer potential and J the attractive potential. The critical point and exponents are computed and show that the model is in the mean-field ferromagnetic universality class.

Solution of the Monomer–Dimer Model on Locally Tree-Like Graphs. Rigorous Results

We consider the monomer–dimer model on sequences of random graphs locally convergent to trees. We prove that the monomer density converges almost surely, in the thermodynamic limit, to an analytic function of the monomer activity. We characterise this limit as the expectation of the solution of a fixed point distributional equation and we give an explicit expression of the limiting pressure per particle.

A Cluster Expansion Approach to the Heilmann–Lieb Liquid Crystal Model

A monomer-dimer model with a short-range attractive interaction favoring collinear dimers is considered on the lattice

A Mean-Field Monomer–Dimer Model with Randomness: Exact Solution and Rigorous Results

\mathbb Z^2

Two Populations Mean-Field Monomer–Dimer Model

Z2. Although our choice of the chemical potentials results in more horizontal than vertical dimers, the horizontal dimers have no long-range translational order—in agreement with the Heilmann–Lieb conjecture (Heilmann and Lieb in J Stat Phys 20(6):679–693, 1979).

Aggregation models on hypergraphs

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.

Non-Gaussian fluctuations in monomer-dimer models

Independent random monomer activities are considered on a mean-field monomer–dimer model. Under very general conditions on the randomness the model is shown to have a self-averaging pressure density that obeys an exactly solvable variational principle. The dimer density is exactly computed in the thermodynamic limit and shown to be a smooth function.

Finite-size corrections for the attractive mean-field monomer-dimer model

A monomer-dimer model with an attractive interaction that favors a phase separation between monomers and dimers is exactly solved in the mean-field case. With the identification of a suitable variational principle the free energy is computed in the large volume limit using the Heilmann-Lieb pure hard-core ansatz. The monomer density, that turns out to be the order parameter of the model, is shown to have a first-order phase transition along a coexistence curve.