In this work I reported recent results in the field of Statistical
Mechanics of Equilibrium, and in particular in Spin Glass models and Monomer Dimer models .
We start giving the mathematical background and the general formalism
for Spin (Disordered) Models with some of their applications to physical and mathematical problems.
Next we move on general aspects of
the theory of spin glasses, in particular to the Sherrington-Kirkpatrick model which is of fundamental interest for the
work. In Chapter 3, we introduce the Multi-species Sherrington-Kirkpatrick
model (MSK), we prove the existence of the thermodynamical limit and the
Guerra's Bound for the quenched pressure together with a detailed analysis of
the annealed and the replica symmetric regime. The result is a multidimensional
generalization of the Parisi's theory. Finally we brie
y illustrate the strategy of
the Panchenko's proof of the lower bound.
In Chapter 4 we discuss the Aizenmann-Contucci and the Ghirlanda-Guerra identities for a wide class of Spin Glass models. As an example of
application, we discuss the role of these identities in the proof of the lower bound.
In Chapter 5 we introduce the basic mathematical formalism of Monomer
Dimer models. We introduce a Gaussian representation of the partition function
that will be fundamental in the rest of the work.
In Chapter 6, we introduce an interacting Monomer-Dimer model.
Its exact solution is derived and a detailed study of its analytical properties and
related physical quantities is performed.
In Chapter 7, we introduce a quenched randomness in the
Monomer Dimer model
and show that, under suitable conditions the
pressure is a self averaging quantity. The main result is that, if we consider
randomness only in the monomer activity, the model is exactly solvable.
| Identifer | oai:union.ndltd.org:unibo.it/oai:amsdottorato.cib.unibo.it:6963 |
| Date | 04 June 2015 |
| Creators | Mingione, Emanuele <1980> |
| Contributors | Contucci, Pierluigi |
| Publisher | Alma Mater Studiorum - Università di Bologna |
| Source Sets | Università di Bologna |
| Language | English |
| Detected Language | English |
| Type | Doctoral Thesis, PeerReviewed |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
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