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Advances in Disordered Systems, Random Processes and Some Applications.

Author: Pierluigi Contucci; Cristian Giardinà
Publisher: Cambridge : Cambridge University Press, 2016.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
This book offers a unified perspective on the study of complex systems, with contributions written by leading scientists from various disciplines.
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Contucci, Pierluigi.
Advances in Disordered Systems, Random Processes and Some Applications.
Cambridge : Cambridge University Press, ©2016
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Pierluigi Contucci; Cristian Giardinà
ISBN: 9781316868508 1316868508 9781316866887 1316866882 9781107124103 1107124107
OCLC Number: 969637057
Description: 1 online resource (384 pages)
Contents: Cover --
Half-title --
Title page --
Copyright information --
Table of contents --
List of Contributors --
Preface --
1 Topological Field Theory of Data: Mining Data Beyond Complex Networks --
1.1 A Philosophical Introduction --
1.2 The Reference Landscape --
1.3 The Challenge --
1.4 Step One: Topological Data Analysis --
1.5 Step Two: from Data Topology to Data Field --
1.6 The Topological Field Theory of Data --
1.7 The Formal Language Theory Facet --
1.8 Language, Structure and Behavior, Automata --
1.9 Emergence of Patterns --
1.10 Conclusions --
1.11 Acknowledgments --
References --
2 A Random Walk in Diffusion Phenomena and Statistical Mechanics --
2.1 Introduction --
2.2 Different Approaches to Diffusion --
2.2.1 Macroscopic Perspective (Fick's Law) --
2.2.2 Physical Perspective (Einstein's Theory) --
2.2.3 Mathematical Perspective (Wiener Process) --
2.2.4 Microscopic Perspective (Random Walks) --
2.2.4.1 The Continuum Limit of Random Walks --
2.2.5 Continuous-time Random Walks --
2.2.6 Mathematical Tools: Characteristic Functions --
2.2.7 Mathematical Tools: Generating Functions and Tauberian Theorems --
2.2.8 Anomalous Diffusion --
2.2.9 Mathematical Tools: an Algebraic Approach --
2.2.10 Resistance Networks --
2.2.11 The Pólya Problem and the Probability of Return to the Origin --
2.3 Random Walks on Graphs --
2.3.1 A Short Introduction to Graphs --
2.3.2 Definitions --
2.3.3 Elements of Spectral Graph Theory --
2.3.3.1 Spectral Gap --
2.3.4 The Random Walk Problem on Graphs --
2.3.4.1 Generating Functions --
2.3.5 Random Walks on Finite Graphs --
2.3.6 Infinite Graphs --
2.3.7 Again on the Type Problem --
2.3.8 The (Local) Spectral Dimension --
2.3.8.1 An Alternative Approach to Local Spectral Dimension --
2.3.9 Averages in Infinite Graphs --
2.3.10 Average Properties of Random Walkers. 2.3.10.1 Alternative Approaches to Average Spectral Dimension --
2.3.10.2 Pure and Mixed Transience on Average --
2.3.11 Non-uniqueness of the Thermodynamic Limit in Inhomogeneous Graphs --
2.3.12 The Two-particle Problem --
2.3.13 Other First-passage Quantities --
2.3.14 Quantum Walks --
2.3.14.1 Definitions --
2.3.14.2 Average Displacement --
2.3.14.3 Return Probability --
2.3.14.4 Systems with Absorption --
2.4 A Brief Introduction to Statistical Mechanics Models on Graphs (and Their Relation to Random Walk) --
2.4.1 A Survey of Models and Results --
2.4.1.1 The Oscillating Network --
2.4.1.2 The Gaussian Model --
2.4.1.3 The Spherical Model and O(n) Models --
2.4.2 Generalization of the Mermin-Wagner Theorem --
2.5 Statistical Mechanics of Simple Mean Field Models through Analogies with Mechanical and Diffusive Dynamical Systems --
2.5.1 Statistical Mechanics in a Nutshell --
2.5.2 Getting Familiar with the Curie-Weiss Model --
2.5.3 Statistical Mechanics via Mechanical Analogy --
2.5.3.1 Shock Waves and Spontaneous Symmetry Breaking --
2.5.3.2 Conservation Laws: Noether Invariants in Mechanics and Self-averaging in Statistical Mechanics --
2.5.4 Statistical Mechanics via Analogy with a Diffusion Problem --
2.5.5 Generalized Models and Techniques for Mean Field Many-body Problems --
2.5.6 Further Problems Solved through the Mechanical Analogy --
References --
3 Legendre Structures in Statistical Mechanics for Ordered and Disordered Systems --
3.1 Introduction --
3.2 The General Legendre Structure for Statistical Mechanics Systems: the Entropic Principle --
3.3 Ordered Mean Field Models --
3.4 Direct Methods versus Convexity and Interpolation Methods --
3.5 The Entropic Principle in the Random Energy Model --
3.6 Functional Order Parameter and the Inverted Variational Principle --
3.7 The Inverted Principle in the Random Energy Model. 3.8 The Squared Interaction in the Random Energy Model --
3.9 The Legendre Structure in Mean Field Spin Glass Models --
3.10 Outlook and Perspectives --
References --
4 Extrema of Log-correlated Random Variables: Principles and Examples --
4.1 Introduction --
4.1.1 Statistics of Extremes --
4.1.2 Log-correlated Fields --
4.1.3 Relations to Statistical Physics --
4.2 Examples and General Properties --
4.2.1 Branching Random Walk --
4.2.2 2D Gaussian Free Field --
4.3 Order of the Maximum --
4.3.1 Leading Order of the Maximum --
4.3.2 Subleading Order of the Maximum --
4.4 Universality Classes of Log-correlated Fields --
4.4.1 Maximum of the Riemann Zeta Function on an Interval --
4.4.2 Maximum of the Characteristic Polynomialof Random Unitary Matrices --
References --
5 Scaling Limits, Brownian Loops, Conformal Fields --
5.1 Introduction --
5.1.1 Critical Scaling Limits --
5.1.2 Near-critical Scaling Limits --
5.1.3 Random Walk Loop Soups and Brownian Loop Soups --
5.1.4 Conformal Correlation Functions in the Brownian Loop Soup --
5.2 Loop Soups --
5.2.1 Random Walk Loop Soups --
5.2.2 Boundary Correlations in the Discrete Gaussian Free Field --
5.2.3 Brownian Loop Soups --
5.2.4 Some Properties of the Massive Brownian Loop Soup --
5.3 Scaling Limits --
5.3.1 The Critical Case --
5.3.2 The Near-critical Case --
5.4 Conformal Correlation Functions in the Brownian Loop Soup --
5.4.1 Motivation and Summary of Results --
5.4.2 Correlation Functions of the Layering and Winding Models --
5.4.2.1 Correlation Functions of the Layering Model --
5.4.2.2 The 1-point Function in the Layering Model --
5.4.2.3 The 1-point Function in the Winding Model --
5.4.2.4 The 2-point Function in the Layering Model --
5.4.3 Conformal Covariance of the n-point Functions --
5.4.3.1 The Layering Model in Finite Domains --
5.4.3.2 The Winding Model in Finite Domains. 5.4.3.3 The Layering Model in the Plane --
Appendix A: Occupation Field and Gaussian Free Field --
Appendix B: The Brownian Loop Measure --
References --
6 The Brownian Web, the Brownian Net, and their Universality --
6.1 Introduction --
6.2 The Brownian Web --
6.2.1 The Space of Compact Sets of Paths --
6.2.2 Construction and Characterization of the Brownian Web --
6.2.3 The Brownian Web and its Dual --
6.2.4 The Coalescing Point Set --
6.2.5 Special Points of the Brownian Web --
6.3 The Brownian Net --
6.3.1 The Left-right Brownian Web and its Dual --
6.3.2 The Hopping Construction of the Brownian Net --
6.3.3 The Wedge Construction of the Brownian Net --
6.3.4 The Mesh Construction of the Brownian Net --
6.3.5 The Branching-coalescing Point Set --
6.3.6 Special Points of the Brownian Net --
6.4 Coupling the Brownian Web and Net --
6.4.1 Relevant Separation Points of the Brownian Net --
6.4.2 Finite Graph Representation of the Brownian Net --
6.5 Scaling Limits of Random Walks in i.i.d. Space-time Environment --
6.5.1 Stochastic Flows of Kernels and the Howitt-Warren Flows --
6.5.2 The Space-time Random Environment for the Howitt-Warren Flows --
6.5.3 Properties of the Howitt-Warren Flow --
6.6 Convergence to the Brownian Web and Net --
6.6.1 General Convergence Criteria for the Brownian Web --
6.6.2 Convergence Criteria for Non-crossing Paths --
6.6.3 Convergence of Coalescing Simple Random Walks to the Brownian Web --
6.6.4 Convergence of General Coalescing Random Walks to the Brownian Web --
6.6.5 Convergence to the Brownian Net --
6.7 Survey on Related Results --
6.7.1 Alternative Topologies --
6.7.1.1 Weak Flow Topology --
6.7.1.2 Tube Topology --
6.7.1.3 Marked Metric Measure Spaces --
6.7.2 Other Models which Converge to the Brownian Web and Net --
6.7.2.1 Voter Model and Spatial Fleming-Viot Processes --
6.7.2.2 Biased Voter Model. 6.7.2.3 True Self-avoiding Walks and True Self-repelling Motion --
6.7.2.4 Planar Aggregation Models --
6.7.2.5 Drainage Networks and Directed Forests --
6.7.2.6 Supercritical Oriented Percolation --
6.7.3 Brownian Web, Critical Percolation, and Noise --
6.7.3.1 Brownian Web, Black Noise, and Noise Sensitivity --
6.7.3.2 Dynamical Brownian Web --
6.8 Open Questions --
6.8.1 Voter Model Perturbations and Brownian Net with Killing --
6.8.1.1 Brownian Net with Killing --
6.8.2 Fractal Structure of the Brownian Net --
6.8.3 Miscellaneous Open Questions --
References --
Index.

Abstract:

This book offers a unified perspective on the study of complex systems with contributions written by leading scientists from various disciplines, including mathematics, physics, computer science,  Read more...

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