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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.



Chapter 1. Introduction

The model we consider all through these notes is easy to define as a random walk in a random potential.

Francis Comets

Chapter 2. Thermodynamics and Phase Transition

An important quantity for this model is the logarithmic moment generating function λ of ω(n, x),

Francis Comets

Chapter 3. The Martingale Approach and the L 2 Region

Martingale theory is well-known as a powerful tool to study random sequences. In this section, we start to use it in our context. First of all, it is efficient for proving that equality can hold in (2.13). The simplest sufficient condition for equality is based on a second moment computation. The parameter region where second moment computations are reliable, will be called the L2 region.

Francis Comets

Chapter 4. Lattice Versus Tree

In this chapter we deal with polymer models on different oriented graphs and compare them with the lattice case. As revealed by Derrida and Spohn in, many interesting questions can be answered on the regular tree. Later on, refined tree-like structures including Derrida’s m-tree has been introduced, yielding further comparisons. There, correlations are simpler compared to the lattice case, since the medium along two paths becomes independent as soon as they visit different sites. In the sense of simplifying the correlation structure, these models play the role of mean-field models.

Francis Comets

Chapter 5. Semimartingale Approach and Localization Transition

The next step in our martingale analysis is to consider lnW n as a semimartingale and to write its Doob’s decomposition. Viewed as a “conditional second moment” method, this new approach is the natural continuation of the techniques from Chap. 3 However this technique was introduced much later. One concrete output is to point out polymer localization, and to relate this phenomenon to strong disorder.

Francis Comets

Chapter 6. The Localized Phase

In this chapter we start to analyse the behavior of the polymer in its localized regime. We will make precise the image of the corridors where the polymer wants to be. We will see that localization happens at all temperature in dimension d = 1 and 2. We illustrate the phenomenon by simulation experiments. Finally, a precise picture will be achieved when the environment has heavy tails. However, we leave some matter on the localized phase for the forthcoming Sects. 7.4 and 9.7

Francis Comets

Chapter 7. Log-Gamma Polymer Model

Recently, significant efforts have been focused on planar polymer models—i.e. (1 + 1)-dimensional—which are integrable. In the line of specific first passage percolation models and interacting particle systems, a few explicitly solvable models were discovered, allowing for detailed descriptions of new scaling limits and statistics characteristic of the KPZ universality class.

Francis Comets

Chapter 8. Kardar-Parisi-Zhang Equation and Universality

Polymer models belong to Kardar-Parisi-Zhang (KPZ) universality class, which is an extended family of models (kinetically roughened surfaces) which all share some non-Gaussian scaling limits and statistics, characterized by a few exponents different from the usual ones. Some other familiar members are some interacting particle systems (exclusion processes SSEP, ASEP, TASEP, q-TASEP), stochastic growth (random deposition, ballistic aggregation, polynuclear growth, Eden and Richardson model), first/last passage percolation, and some stochastic PDE’s (stochastic Burgers equation, stochastic reaction-diffusion equations, stochastic Hamilton-Jacobi equations). An elementary introduction [84] aiming at non-specialists mathematicians covers some of the above models, and can be completed by a more physics-oriented, but still pedestrian, account [155].

Francis Comets

Chapter 9. Variational Formulas

In this chapter we closely follow two recent papers [118] and [199] by Georgiou, Rassoul-Agha, Seppäläinen and Yilmaz.

Francis Comets


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