Lectures on Random Interfaces

1. Front Matter

   Pages i-xii

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2. Scaling Limits for Pinned Gaussian Random Interfaces in the Presence of Two Possible Candidates

   • Tadahisa Funaki

   Pages 1-28

3. Dynamic Young Diagrams

   • Tadahisa Funaki

   Pages 29-79

4. Stochastic Partial Differential Equations

   • Tadahisa Funaki

   Pages 81-92

5. Sharp Interface Limits for a Stochastic Allen-Cahn Equation

   • Tadahisa Funaki

   Pages 93-110

6. KPZ Equation

   • Tadahisa Funaki

   Pages 111-124

7. Back Matter

   Pages 125-138

   PDF

Interfaces are created to separate two distinct phases in a situation in which phase coexistence occurs. This book discusses randomly fluctuating interfaces in several different settings and from several points of view: discrete/continuum, microscopic/macroscopic, and static/dynamic theories. The following four topics in particular are dealt with in the book.
Assuming that the interface is represented as a height function measured from a fixed-reference discretized hyperplane, the system is governed by the Hamiltonian of gradient of the height functions. This is a kind of effective interface model called ∇φ-interface model. The scaling limits are studied for Gaussian (or non-Gaussian) random fields with a pinning effect under a situation in which the rate functional of the corresponding large deviation principle has non-unique minimizers.
Young diagrams determine decreasing interfaces, and their dynamics are introduced. The large-scale behavior of such dynamicsis studied from the points of view of the hydrodynamic limit and non-equilibrium fluctuation theory. Vershik curves are derived in that limit.
A sharp interface limit for the Allen–Cahn equation, that is, a reaction–diffusion equation with bistable reaction term, leads to a mean curvature flow for the interfaces. Its stochastic perturbation, sometimes called a time-dependent Ginzburg–Landau model, stochastic quantization, or dynamic P(φ)-model, is considered. Brief introductions to Brownian motions, martingales, and stochastic integrals are given in an infinite dimensional setting. The regularity property of solutions of stochastic PDEs (SPDEs) of a parabolic type with additive noises is also discussed.
The Kardar–Parisi–Zhang (KPZ) equation , which describes a growing interface with fluctuation, recently has attracted much attention. This is an ill-posed SPDE and requires a renormalization. Especially its invariant measures are studied.    

• Scaling limits for pinned interface model
• Dynamic Young diagrams
• Introduction to stochastic partial differential equations
• Sharp interface limit for stochastic Allen--Cahn equations
• Kardar--Parisi--Zhang equation
• partial differential equations

“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)     

• Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan

  Tadahisa Funaki

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