Random Polymer Models

by Giacomin, Giambattista

• ISBN13: 9781860947865
• ISBN10: 1860947867
• Edition: 1st
• Format: Hardcover
• Copyright: 2007-03-01
• Publisher: Imperial College Pr

Summary

This volume introduces readers to the world of disordered systems and to some of the remarkable probabilistic techniques developed in the field. The author explores in depth a class of directed polymer models to which much attention has been devoted in the last 25 years, in particular in the fields of physical and biological sciences. The models treated have been widely used in studying, for example, the phenomena of polymer pinning on a defect line, the behavior of copolymers in proximity to an interface between selective solvents and the DNA denaturation transition. In spite of the apparent heterogeneity of this list, in mathematical terms, a unified vision emerges. One is in fact dealing with the natural statistical mechanics systems built on classical renewal sequences by introducing one-body potentials. This volume is also a self-contained mathematical account of the state of the art for this class of statistical mechanics models.

Table of Contents

Preface	p. vii
Random Polymer Models and their Applications	p. 1
Random Polymers and (De)Localization Phenomena	p. 1
A First Model: Pinning on a Defect Line	p. 6
Pinning a walk on a defect line	p. 6
Pinning more general processes on a defect line	p. 13
Entropic Repulsion and Wetting Phenomena	p. 17
Walls versus penetrable substrates	p. 20
The Denaturation Transition: Poland-Scheraga Models	p. 21
Force Induced Unzipping	p. 26
Inhomogeneous Charge Distributions: Copolymers and Pinning	p. 29
Periodic pinning models	p. 30
Copolymers, periodic copolymers and selective interfaces	p. 32
Copolymers with adsorption	p. 36
Fully Inhomogeneous Charge Distributions and Disordered Polymer Models	p. 37
Annealed models	p. 41
On the Return Time Viewpoint	p. 42
On Related Classes of Models	p. 43
More general return times	p. 43
Rewards and penalties on general structures	p. 45
Bibliographic Complements	p. 46
The Homogeneous Pinning Model	p. 49
The Free Energy	p. 49
Renewal Theory and Sharp Estimates	p. 52
Path Properties: The Infinite Volume Limit	p. 56
Path Properties: The Scaling Limit	p. 57
Bibliographic Complements	p. 63
Weakly Inhomogeneous Models	p. 69
A Formula for the Free Energy: Reduction to a Finite Dimension Problem	p. 69
Sharp Estimates on the Partition Function	p. 75
The Limit Path Trajectories	p. 77
The infinite volume limit	p. 78
The scaling limit	p. 80
First Order Transitions, Non-Uniqueness and Phase Diagrams	p. 82
Bibliographic Complements	p. 87
The Free Energy of Disordered Polymer Chains	p. 89
Preliminary Observations	p. 89
Existence of the Free Energy and Self-Averaging	p. 90
Periodic versus Disordered Charge Sequences	p. 94
Alternative Approaches to the Existence of the Free Energy	p. 96
Concentration inequalities and self-averaging	p. 96
The Super-Additive Ergodic Theorem approach	p. 97
Bibliographic Complements	p. 98
Disordered Pinning Models: The Phase Diagram	p. 101
Preliminary Observations on the Free Energy	p. 101
An Improved Lower Bound on the Free Energy	p. 102
On the Upper Bound: Annealing and Constrained Annealing	p. 109
The Effect of the Disorder on the Order of the Transition: A Smoothing Inequality	p. 111
What is One Expecting: The Renormalization Group Viewpoint and the Harris Criterion	p. 116
A Fully Inhomogeneous Non-Disordered Model	p. 118
Bibliographic Complements	p. 122
Disordered Copolymers and Selective Interfaces: The Phase Diagram	p. 127
Copolymers and Homogeneous Localization	p. 130
Rare Stretch Strategy for Localization Estimates	p. 135
The Weak Coupling Limit: Brownian Motion Model and Universality	p. 140
On more general copolymer models and the Brownian scaling	p. 142
Copolymers with adsorption	p. 144
Bibliographic Complements	p. 146
The Localized Phase of Disordered Polymers	p. 151
A First Tightness Estimate	p. 151
Quenched versus Quenched Averaged Estimates	p. 154
Quenched Averaged Estimates and the Infinite Volume Polymer Measure	p. 160
Bibliographic Complements	p. 164
The Delocalized Phase of Disordered Polymers	p. 167
Large Deviations and Delocalization	p. 168
Beyond Large Deviations: O(log N) Results	p. 170
The Strongly Delocalized Regime	p. 174
Bibliographic Complements	p. 177
Numerical Algorithms and Computations	p. 181
Computing the Partition Function	p. 182
Random walk models	p. 182
General renewal models	p. 183
The Fixman-Freire Algorithm	p. 184
A Statistical Test for Localization	p. 185
Applications: A Few Examples	p. 188
Bibliographic Complements	p. 191
Mathematical Tools	p. 193
Limit Theorems in Probability	p. 193
Some remarks on convexity	p. 196
Entropy	p. 196
Concentration Properties	p. 198
Slow Variation and Laplace Transforms	p. 200
Renewal Theory for N Valued Random Variables	p. 202
The renewal theorem	p. 204
The renewal mass function: Transient case	p. 206
The renewal mass function: Null recurrent case	p. 207
Scaling limits of renewal processes	p. 208
Random Walks	p. 212
The Super-Additive Ergodic Theorem	p. 214
Perron-Frobenius Theory	p. 216
Some Technical Estimates	p. 219
Homogeneous Localization Strategy: The Entropy Density	p. 219
The Weak Localization Limit	p. 221
Localization and Loss of Memory	p. 222
Effective Interface Models	p. 229
Ising Model, Effective Interface Models and Random Walks	p. 229
Bibliography	p. 233
Index	p. 241
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