Directed Polymers in Random Environments: École d'Été de Probabilités de Saint-Flour XLVI – 2016
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 questionis: 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.
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2 Thermodynamics and Phase Transition
3 The Martingale Approach and the L2 Region
4 Lattice Versus Tree
5 Semimartingale Approach and Localization Transition
6 The Localized Phase
7 LogGamma Polymer Model
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ˆ ˆ ˆ annealed bound Assume Brownian bridge Brownian motion cocycle compute concentration inequality condition consider convex convex function Corollary defined definition denote density dimension directed polymers disorder regime distribution entropy equal exists exponent favourite endpoint finite fixed fluctuations follows free energy Gaussian environment Gibbs measure independent integrable large deviation lattice Lecture Notes Lemma limit theorem localization log-gamma polymer Markov Markov property martingale notation parameter partition function path phase transition point-to-point polymer measure polymer model Polymers in Random positive probability measure probability space proof of Theorem Proposition prove random environment random polymer random variable random walk Recall region Remark result S-invariant sequence simple random walk ſş stationary Step stochastic strong disorder temperature Theorem 3.3 variational formulas weak disorder