© 2016

Lectures on Random Interfaces


  • Shows that the microscopic point of view is useful in choosing a real minimizer of a variational problem that determines an interface shape

  • Is the first book to discuss the stochastic extension of the Sharp interface limit for non-random PDEs

  • Is one of the few books dealing with the KPZ equation, a recent hot topic in probability theory


Table of contents

  1. Front Matter
    Pages i-xii
  2. Tadahisa Funaki
    Pages 29-79
  3. Tadahisa Funaki
    Pages 81-92
  4. Tadahisa Funaki
    Pages 111-124
  5. Back Matter
    Pages 125-138

About this book


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 dynamics is 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

Authors and affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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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)