PHY560A - Complex systems
The goal of this course on ``Complex Systems'' and advanced statistical physics is to show how the concepts and techniques developed in statistical mechanics and probability theory can be relevant to study the emergence of complexity from collective and stochastic behaviour. The lectures will put the emphasis on some key ideas such as scaling, universality, Brownian motion and criticality and explain how elementary models, that often present isomorphic mathematical structures, can be applied to many different fields. These ideas will be illustrated on examples from random geometry, growth processes, critical phenomena, genetic models, network percolation, information coding and finance.
- UNIVERSALITY AND SCALING: a probabilistic approach.
We study the emergence of universal scaling laws in probability theory: Law of Large Numbers, Central Limit Theorem, the Levy-stable distributions. The limiting distributions of the Maximum and
Minimum of independent identical random variables. Introduction to Random Matrices (Wigner's Law).
- BROWNIAN MOTION, LANGEVIN EQUATION AND PATH INTEGRALS.
After reviewing the Brownian Motion (from the physics point of view), simple Stochastic Differential Equations (SDE) are introduced and analyzed. The Wiener Measure and Path Integrals, one of the most important tool for analysis and contemporary physics, are defined. We then apply these techniques to a simple financial model and to derive Arrhenius'Law.
- RANDOM GEOMETRY AND FRACTALS.
Fractals are geometric objects that can have non-integral dimensions and display naturally scaling properties. Adding randomness allows one to define a whole class of stochastic processes that generalize the Brownian Motion and can be used to describe temporal series appearing various contexts (geological data, finance...).
- STOCHASTIC DYNAMICS.
The goal of this more mathematical chapter is to study the reversibility properties of Markov Processes and to derive the diffusion equations associated with an SDE. Applications to the growth of Random Landscapes will be given.
- INTRODUCTION TO INFORMATION THEORY.
Basic concepts of Information and Communication Theory: entropy, coding, transmission, models of noisy channels, mutual information and capacity. Various interpretations of Shannon's entropy are
given and Shannon's noisy-channel coding theorem is proved.
- DISORDERED SYSTEMS.
Statistical mechanics of disordered systems develops a variety of original concepts and techniques, ranging from combinatorial optimization to engineering, with applications in solid state physics (quantum conduction, localization), biology (protein folding, neural networks) and population dynamics. We shall discuss the basic properties of spin glasses (frustration, complex landscape and aging) and present some specific theoretical tools of the field.
- CRITICAL PHENOMENA and THE RENORMALISATION GROUP.
After reviewing various second order critical phenomena (liquid-gas, ferromagnetic-paramagnetic), we explain how various universality classes can be defined through critical exponents that characterize scaling behaviour. The basic ideas of Ken Wilson's renormalisation group will be presented, in real space, and we shall explain how the critical exponents can be related to independent scaling dimensions, that correspond to relevant fields (two in general). This theory will be illustrated on the 2d Ising model on a triangular lattice).
- NETWORKS.
Network theory has provided a new paradigm to represent and analyze data. Its roots lie in graph theory. Elementary combinatorial properties of graphs (Cayley theorem, Pr\ufer sequence) will be discussed. We shall then consider Branching tree processes (Galton's model) and Random Recursive Trees. This will allow us to study scale-free networks and their connectivity properties. Application: the percolation transition in the Erdos-Renyi graph.
- MODELS FOR FINANCE.
Phenomenology of financial markets; the Bachelier model. Statistics of prices and intermittency. Options and derivatives: the Black-Scholes strategy. Analysis of the discrete time Cox, Ross and Rubinstein model and its continuous limit: the Black, Scholes and Merton Pricing formula.
Preriquisite : PHY433 - Physique statistique
Language : English
Credits ECTS : 5
- Teaching coordinator: Mallick Kirone