Giacomin G. Random Polymer Models

London: Imperial College Press, 2007. — 258 p.This work deals with a well-defined class of probabilistic models. The focus is on polymers. More precisely I should say that the focus is on the equilibrium statistical mechanics of a class of polymers: dynamical and non-equilibrium phenomena are not treated, reflecting the fact that these directions are at the moment under-developed, at least from the viewpoint of the so called rigorous results. Moreover, only a subset of the world of polymer models is taken into consideration. Chapter 1 is a collection of models, with partial solutions, that lead to the general framework given in Section
1.8. Chapter 2 and Chapter 3 deal with non-disordered models and the Renewal Theory approach is developed in detail: some of the arguments are relatively heavy. Chapter 4 deals with the problem of the existence of the free energy for disordered models. An elementary proof is worked out in detail and some other proofs are sketched. Chapter 5 and Chapter 6 focus, respectively, on the phase diagram of the pinning model and of the copolymer models, which are in a sense the two basic models (by combining them one finds the general framework). Chapter 7 and Chapter 8 deal with the properties of polymer paths. These are sharply different in the localized regime (Ch. 7) and in the delocalized one (Ch. 8). Chapter 9 deals with computational approaches to analyze these systems. Appendix A is rather long and aims at making this work as self-contained as possible. Appendix B collects some important technical estimates, and Appendix C is a very quick reminder that directed random walks in (1 + 1)-dimension are also effective models of interfaces and that some of our polymer models can in fact be reinterpreted in a different light.Random Polymer Models and their Applications
The Homogeneous Pinning Model
Weakly Inhomogeneous Models
The Free Energy of Disordered Polymer Chains
Disordered Pinning Models: The Phase Diagram
Disordered Copolymers and Selective Interfaces: The Phase Diagram
The Localized Phase of Disordered Polymers
The Delocalized Phase of Disordered Polymers
Numerical Algorithms and Computations
Mathematical Tools
Some Technical Estimates
Effective Interface Models