This volume contains most of the lectures presented at the meeting held in Carry-le nd Rouet from the 2 to the 4th June 1980 and entitled "Numerical Methods in the Study of Critical Phenomena". Scientific subjects are becoming increasingly differentiated, and the number of journals and meetings devoted to them is continually increasing. Thus it has become very difficult for the non-specialist to approach subjects with which he is not familiar. Hence the purpose of our meeting was to bring together scientists from different disciplines to study a common subject and to stimulate discussion' between participants. We hope this goal was reached. The lectures are grouped in five chapters and, inside the first and the second chapter, under two headings. In each group they are classified in alphabetical order by author. We are pleased to publish these Proceedings in a series whose multidisciplinary character has been emphasized from the beginning. We are indebted to all who provided us with their help, particularly to Mrs. A. Litman of the Centre International de Rencontres Mathematiques at Luminy (C.I.R.M.) whose kindness and efficiency are well known; from the practical point of view, the meetings were organized within the scientific framework of the G.I.S. No.19 (C.N.R.S.), with the participation of the University of Grenoble.
Table of Contents1. Mathematical Methods.- 1.1 Study of Singularities.- Padé-Hermite Approximants.- Somme Comments About the Numerical Utilization of Factorial Series.- to Real Quasianalytic Classes and Continuation Problems.- 1.2 Critical Phenomena in Dynamical Systems.- Groups Transformations and Critical Asymptotics Applications to Non-Linear Differential and Partial Derivative Equations.- Antecedent Invariant Curves of an Endomorphism. Influence Domain of a Stable Cycle Coexisting with an Isolated Stable Invariant Curve.- Topological Entropy As a Measure of Dynamic Chaos in Endomorphisms.- Topological Entropy of Markov Processes for a C0-Endomorphism of the Interval.- Sequential Iteration of Threshold Functions.- Some Properties of Second Order Dynamic Systems with Parametric Resonances.- 2. Applications in Physics.- 2.1 Critical Phenomena in Solid-State Physics.- On the Bifurcation of Certain Kam Tori in the Standard Mapping.- MO Stochasticity Criterion.- Singularities in Saw Numerical Simulations.- Monte Carlo Measurement of the Single Vortex Free Energy in the Kosterlitz-Thouless Theory.- Algebraic Method for the Computation of the Partition Functions of Spin Glasses and Numerical Study of the Distributions of Zeros.- Percolation and Gelation by Additive Polymerization.- Ground State Structure of the Random Frustration Model in Two Dimensions.- Line Defects and the Glass Transition.- Universality in Size-Effects in 2D Percolation.- 2.2 Use of Renormalisation Techniques.- The Phenomenological Renormalization Method.- Computation of the Yang-Lee Edge Singularity in Ising Models.- Real-Space Renormalization-Group Method for Quantum Systems: Application to Quantum Frustration in Two Dimensions.- Yang-Lee Edge Singularity by Real Space Renormalization Group.- 3. Applications in Biology.- Numerical Determination of a Periodical Solution of Discontinuous Type, near a Singular Point, for a Neurophysiological Model.- On the Relation Between the Logical Structure of Systems and Their Ability to Generate Multiple Steady States or Sustained Oscillations.- Critical Delays in Logical Asynchronous Models.- 4. Applications in Chemistry.- A Simulation Technique for Studying Critical Properties of Chemical Dissipative Systems.- Critical Paths and Passes: Application to Quantum Chemistry.- 5. Non-Physical Applications of Statistical Mechanics.- Telephone Network: Statistical Mechanics and Non-Random Connecting Procedures.- The Thermodynamic Formalism in Population Biology.- Asymptotic Inference for Markov Random Fields on Zd.- List of Contributors.