
Lectures on Random Interfaces
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"The book at hand discusses various aspects of random interfaces, both in static and in dynamic settings, from various points of view. . the book may serve as a good introductory text to several aspects of random interfaces." (Leonid Petrov, Mathematical Reviews, February, 2018)More details
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Content
- Intro
- Preface
- Contents
- 1 Scaling Limits for Pinned Gaussian Random Interfaces in the Presence of Two Possible Candidates
- 1.1 Macroscopic Variational Problems
- 1.1.1 Wulff Shape and Winterbottom Shape
- 1.1.2 Variational Problem with Two Media
- 1.1.3 Variational Problem with a Pinning Effect
- 1.1.4 Examples of Minimizers of J with a Pinning Effect
- 1.2 Microscopic Models
- 1.2.1 Wulff Shape from Microscopic Models
- 1.2.2 The f-Interface Model with a Pinning
- 1.3 Results for d=1, n=1
- 1.4 Outline of the Proof
- 1.5 Results for d=3, n=1
- 1.6 Outline of the Proof
- 1.6.1 Proof of the Lower Bound
- 1.6.2 Proof of the Upper Bound
- 1.6.3 Proof of the Large Deviation Type Estimate
- 2 Dynamic Young Diagrams
- 2.1 Static Theory of 2D Young Diagrams
- 2.1.1 Ensembles of 2D Young Diagrams
- 2.1.2 Scaling Limits, Law of Large Numbers, and Vershik Curves
- 2.1.3 Central Limit Theorem
- 2.1.4 Large Deviation Principle
- 2.1.5 Equivalence of Ensembles under Inhomogeneous Conditioning
- 2.1.6 Related Young Diagrams
- 2.2 Dynamic Theory of 2D Young Diagrams
- 2.2.1 Dynamics of 2D Young Diagrams and Their Gradient Fields
- 2.2.2 Hydrodynamic Limits (LLN)
- 2.2.3 Non-Equilibrium Fluctuations
- 2.2.4 Dynamic Large Deviation Principle
- 2.2.5 Conservative Systems
- 2.3 3D Young Diagrams
- 2.3.1 Static Theory
- 2.3.2 Dynamic Theory
- 3 Stochastic Partial Differential Equations
- 3.1 The TDGL Equation
- 3.2 White Noise, Colored Noise, and Stochastic Integrals
- 3.3 SPDEs of Parabolic Type with Additive Noises
- 3.3.1 Two Definitions of Solutions
- 3.3.2 Regularity of Solutions
- 3.3.3 Invariant Measures
- 4 Sharp Interface Limits for a Stochastic Allen-Cahn Equation
- 4.1 Setting, Quick Overview and Background
- 4.2 Sharp Interface Limits in a Stochastic Case
- 4.2.1 One-Dimensional Case with Space-Time White Noise
- 4.2.2 Higher-Dimensional Case with Noise Asymptotically White in Time
- 4.3 A Brief Survey and Supplements
- 4.3.1 Deterministic Case
- 4.3.2 Stochastic Case
- 5 KPZ Equation
- 5.1 The KPZ Equation, Its Ill-Posedness, and Its Renormalization
- 5.2 Cole-Hopf Solution and the Linear Stochastic Heat Equation
- 5.3 KPZ Approximating Equations
- 5.3.1 Simple Approximation
- 5.3.2 Approximation Adapted to Studying Invariant Measures
- 5.4 Passing to the Limit "3223379 0
- 5.5 Invariant Measures of the Cole-Hopf Solution and SHE
- 5.6 Multi-component Coupled KPZ Equation
- References
- Index
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