Directed polymers in random environments. École d'Été de Probabilités de Saint-Flour XLVI -- 2016
type: Item
instance of: scholarly article
title: Directed polymers in random environments. École d'Été de Probabilités de Saint-Flour XLVI -- 2016
zbMATH Open document ID: 1392.60002
DOI: 10.1007/978-3-319-50487-2
author: Francis Comets
published in: Lecture Notes in Mathematics
publication date: 2017-02-14T00:00:00Z
review text: 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 …
reviewed by: Marius Iosifescu
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 82D60
Mathematics Subject Classification ID: 82B26
zbMATH DE Number: 6684616
zbMATH Keywords: polymers in random environments
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-50487-2
OpenAlex ID: W2585339115
description: scientific article; zbMATH DE number 6684616