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SpringerBriefs in Probability and Mathematical Statistics

Lectures on Random Interfaces

Authors: Funaki, Tadahisa

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  • 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
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eBook $39.99
price for USA in USD (gross)
  • ISBN 978-981-10-0849-8
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA in USD
  • ISBN 978-981-10-0848-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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.    

Reviews

“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)
    

Table of contents (5 chapters)

Table of contents (5 chapters)

Buy this book

eBook $39.99
price for USA in USD (gross)
  • ISBN 978-981-10-0849-8
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA in USD
  • ISBN 978-981-10-0848-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Lectures on Random Interfaces
Authors
Series Title
SpringerBriefs in Probability and Mathematical Statistics
Copyright
2016
Publisher
Springer Singapore
Copyright Holder
The Author(s)
eBook ISBN
978-981-10-0849-8
DOI
10.1007/978-981-10-0849-8
Softcover ISBN
978-981-10-0848-1
Series ISSN
2365-4333
Edition Number
1
Number of Pages
XII, 138
Number of Illustrations
35 b/w illustrations, 9 illustrations in colour
Topics