Directed polymers in random environments. École d'Été de Probabilités de Saint-Flour XLVI -- 2016
Contents: 1. Introduction 1.1 Polymer models; 1.2 Particle in a random potential; 1.3 History, experiments and related models; 1.4 Highlights. 2. Thermodynamics and phase transition. 2.1 Preliminaries; 2. 2 Free energy; 2.3 Upper bounds; 2.4 Monotonicity; 2.5 Phase transition. 3. The martingale approach and the L2 region. 3.1 A martingale associated to the partition function; 3.2 The second moment method and the L2 region; 3.3 Diffusive behavior in the L2 region; 3.4 Local limit theorem in the L2 region; 3.5 Analytic function method: energy and entropy in the L2 region; 3.6 Concentration revisited; 3.7 Rate of martingale convergence. 4. Lattice versus tree. 4.1 A mean field approximation; 4.2 Majorizing lattice polymers by \(m\)-tree polymers; 4.3 Phase diagram on the tree; 4.4 Free energy is strictly convex for the polymer on the lattice; 4.5 Conclusions and related models. 5. Semimartingale approach and localization transition. 5.1 Semimartingale decomposition; 5.2 Weak disorder and diffusive regime; 5.3 Bounds on the critical temperature by size-biasing; 5.4 Localization versus delocalization 6. The localized phase 6.1 Path localization; 6.2 Low dimensions; 6.3 Simulations of the localized phase; 6.4 Localization for heavy-tails environment 7. Log-gamma polymer model. 7.1 Log-gamma model and its stationary version; 7.2 Free energy and fluctuations for the stationary model; 7.3 Model without boundaries; 7.4 Localization in the log-gamma polymer model with boundaries; 7.5 Final remarks and complements 8. Kardar-Parisi-Zhang equation and universality. 8.1 Kardar-Parisi-Zhang equation; 8.2 Hopf-Cole solution; 8.3 The continuum random polymer; 8.4 Intermediate disorder regime for lattice polymers 8.5 Fluctuations and universality 9. Variational formulas. 9.1 Free energy revised: A more detailed account; 9.2 Large deviation principle for the polymer endpoint; 9.3 Cocycle variational formula for the point-to-Level case; 9.4 Tilt-velocity duality; 9.5 Busemann functions and minimizing cocycles; 9.6 Gibbs' variational formulas: energy/entropy balance; 9.7 Variational formula with a functional order parameter. A. Toolbox for random motion and Gibbs measures. A.1 Superadditive lemmas; A.2 Concentration inequalities for martingales with bounded increments; A.3 Gaussian laws; A.4 Correlation inequalities; A.5 Local limit theorem for the simple random walk; A.6 Perron-Frobenius eigenvalue of a nonnegative matrix; A.7 Linear differential equations: Duhamel form References; Index. Publisher's description: ``Analyzing the phase transition from diffusive to localized behavior in a model of directed polymers in a random environment, this volume places particular emphasis on the localization phenomenon. The main questions: What does the path of a random walk look like if rewards and penalties are spatially randomly distributed? This model, which provides a simplified version of stretched elastic chains pinned by random impurities, has attracted much research activity, but it (and its relatives) still holds many secrets, especially in high dimensions. It has non-Gaussian scaling limits and it belongs to the so-called KPZ universality class when the space is one-dimensional. Adopting a Gibbsian approach, using general and powerful tools from probability theory, the discrete model is studied in full generality. Presenting the state-of-the art from different perspectives, and written in the form of a first course on the subject, this monograph is aimed at researchers in probability or statistical physics, but is also accessible to masters and Ph.D. students.'' The reference list contains 245 well chosen titles. I conclude this review by warmly recommending the book to any mathematically interested person.
scientific article; zbMATH DE number 6684616