Loebl Discrete Mathematics in Statistical Physics

Introductory Lectures
2010
ISBN: 978-3-8348-9329-1
Verlag: Vieweg & Teubner
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 187 Seiten, eBook

Reihe: Advanced Lectures in Mathematics

The book first describes connections between some basic problems and technics of combinatorics and statistical physics. The discrete mathematics and physics terminology are related to each other. Using the established connections, some exciting activities in one field are shown from a perspective of the other field. The purpose of the book is to emphasize these interactions as a strong and successful tool. In fact, this attitude has been a strong trend in both research communities recently.

It also naturally leads to many open problems, some of which seem to be basic. Hopefully, this book will help making these exciting problems attractive to advanced students and researchers.

Prof. Dr. Martin Loebl, Dept. of Mathematics, Charles University, Prague

Zielgruppe

Upper undergraduate

Autoren/Hrsg.

Loebl, Martin

Inhaltsverzeichnis

1;Preface;6
2;Contents;7
3;Chapter 1 Basic concepts;9
3.1;1.1 Sets, functions, structures;9
3.2;1.2 Algorithms and Complexity;11
3.3;1.3 Generating functions;14
3.4;1.4 Principle of inclusion and exclusion;15
4;Chapter 2 Introduction to GraphTheory;20
4.1;2.1 Basic notions of graph theory;20
4.2;2.2 Cycles and Euler’s theorem;25
4.3;2.3 Cycle space and cut space;27
4.4;2.4 Flows in directed graphs;32
4.5;2.5 Connectivity;34
4.6;2.6 Factors, matchings, and dimers;36
4.7;2.7 Graph colorings;43
4.8;2.8 Random graphs and Ramsey theory;44
4.9;2.9 Regularity lemma;46
4.10;2.10 Planar graphs;47
4.11;2.11 Tree-width and excluded minors;54
5;Chapter 3 Trees and electricalnetworks;57
5.1;3.1 Minimum spanning tree and greedy algorithm;57
5.2;3.2 Tree isomorphism;58
5.3;3.3 Tree enumeration;61
5.4;3.4 Electrical networks;63
5.5;3.5 Random walks;68
6;Chapter 4 Matroids;71
6.1;4.1 Examples of matroids;73
6.2;4.2 Greedy algorithm;75
6.3;4.3 Circuits;76
6.4;4.4 Basic operations;77
6.5;4.5 Duality;77
6.6;4.6 Representable matroids;79
6.7;4.7 Matroid intersection;80
6.8;4.8 Matroid union and min-max theorems;80
7;Chapter 5 Geometric representationsof graphs;82
7.1;5.1 Topological spaces;82
7.2;5.2 Planar curves: Gauß codes;87
7.3;5.3 Planar curves: rotation;92
7.4;5.4 Convex embeddings;93
7.5;5.5 Coin representations;96
7.6;5.6 Counting fatgraphs: matrix integrals;98
8;Chapter 6 Game of dualities;106
8.1;6.1 Edwards-Anderson Ising model;106
8.2;6.2 Max-Cut for planar graphs;108
8.3;6.3 Van der Waerden’s theorem;110
8.4;6.4 MacWilliams’ theorem;111
8.5;6.5 Phase transition of 2D Ising;113
8.6;6.6 Critical temperature of the honeycomb lattice;115
8.7;6.7 Transfer matrix method;118
8.8;6.8 The Yang-Baxter equation;121
9;Chapter 7 The zeta function andgraph polynomials;123
9.1;7.1 The Zeta function of a graph;123
9.2;7.2 Chromatic, Tutte and flow polynomials;128
9.3;7.3 Potts, dichromate and ice;132
9.4;7.4 Graph polynomials for embedded graphs;135
9.5;7.5 Some generalizations;138
9.6;7.6 Tutte polynomial of a matroid;142
10;Chapter 8 Knots;144
10.1;8.1 Reidemeister moves;145
10.2;8.2 Skein relation;146
10.3;8.3 The knot complement;147
10.4;8.4 The Alexander-Conway polynomial;149
10.5;8.5 Braids and the braid group;151
10.6;8.6 Knot invariants and vertex models;152
10.7;8.7 Alexander-Conway as a vertex model;153
10.8;8.8 The Kauffman derivation of the Jones polynominal;153
10.9;8.9 Jones polynomial as vertex model;156
10.10;8.10 Vassiliev invariants and weight systems;156
11;Chapter 9 2D Ising and dimer models;159
11.1;9.1 Pfaffians, dimers, permanents;159
11.2;9.2 Products over aperiodic closed walks;164
12;Bibliography;174
13;List of Figures;181
14;Index;183

Basic concepts.- to Graph Theory.- Trees and electrical networks.- Matroids.- Geometric representations of graphs.- Game of dualities.- The zeta function and graph polynomials.- Knots.- 2D Ising and dimer models.

Über Autor(innen)

Prof. Dr. Martin Loebl, Dept. of Mathematics, Charles University, Prague