Emanuele Mingione

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.

Interpolating the Sherrington-Kirkpatrick replica trick

Interpolation techniques have become, in the past decades, a powerful approach to describe several properties of spin glasses within a simple mathematical framework. Intrinsically, for their construction, these schemes were naturally implemented in the cavity field technique, or its variants such as stochastic stability and random overlap structures. However the first and most famous approach to mean field statistical mechanics with quenched disorder is the replica trick. Among the models where these methods have been used (namely, dealing with frustration and complexity), probably the best known is the Sherrington–Kirkpatrick spin glass. In this paper we apply the interpolation scheme to the original replica trick framework and test it directly on the cited paradigmatic model. Although the problem, at a mathematical level, has been deeply investigated by Talagrand, it is still rich in information from a theoretical physics perspective; in fact, by treating the number of replicas n ∈ (0, 1] as an interpolating parameter (far from its original interpretation) the proof of the attendant commutativity of the zero replica and the infinite volume limits can be easily obtained. Further, within this perspective, we can naturally think of n as a quenching temperature close to that introduced in off-equilibrium approaches to gain some new insight into our understanding of the off-equilibrium features encountered in equilibrium statistical mechanics of spin glasses.

Factorization Properties in d -Dimensional Spin Glasses. Rigorous Results and Some Perspectives

In this paper we show that d-dimensional Gaussian spin glass models are strongly stochastically stable, fulfill the Ghirlanda-Guerra identities in distribution and the ultrametricity property.

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

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.

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

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.

The exact solution of a mean-field monomer-dimer model with attractive potential

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.

Two Populations Mean-Field Monomer–Dimer Model

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.

Aggregation models on hypergraphs

Following a newly introduced approach by Rasetti and Merelli we investigate the possibility to extract topological information about the space where interacting systems are modelled. From the statistical datum of their observable quantities, like the correlation functions, we show how to reconstruct the activities of their constitutive parts which embed the topological information. The procedure is implemented on a class of polymer models on hypergraphs with hard-core interactions. We show that the model fulfils a set of iterative relations for the partition function that generalise those introduced by Heilmann and Lieb for the monomer-dimer case. After translating those relations into structural identities for the correlation functions we use them to test the precision and the robustness of the inverse problem. Finally the possible presence of a further interaction of peer-to-peer type is considered and a criterion to discover it is identified.

Non-Gaussian fluctuations in monomer-dimer models

A hard-core monomer-dimer mean-field model is considered with the addition of an attraction potential between similar particles. We find that in the curve where the monomer and dimer phases coexist, the equilibrium state, due to the lack of gauge symmetry, turns out to be a superposition with unequal weights. We show, moreover, that at the critical point the number of monomers has non-Gaussian, quartic exponential, fluctuations.