Lectures On Random Interfaces
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Lectures on Random Interfaces
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.
Lectures on Probability Theory and Statistics
This volume contains two of the three lectures that were given at the 33rd Probability Summer School in Saint-Flour (July 6-23, 2003). Amir Dembo’s course is devoted to recent studies of the fractal nature of random sets, focusing on some fine properties of the sample path of random walk and Brownian motion. In particular, the cover time for Markov chains, the dimension of discrete limsup random fractals, the multi-scale truncated second moment and the Ciesielski-Taylor identities are explored. Tadahisa Funaki’s course reviews recent developments of the mathematical theory on stochastic interface models, mostly on the so-called \nabla \varphi interface model. The results are formulated as classical limit theorems in probability theory, and the text serves with good applications of basic probability techniques.
The Best Interface Is No Interface
Our love affair with the digital interface is out of control. We’ve embraced it in the boardroom, the bedroom, and the bathroom. Screens have taken over our lives. Most people spend over eight hours a day staring at a screen, and some “technological innovators” are hoping to grab even more of your eyeball time. You have screens in your pocket, in your car, on your appliances, and maybe even on your face. Average smartphone users check their phones 150 times a day, responding to the addictive buzz of Facebook or emails or Twitter. Are you sick? There’s an app for that! Need to pray? There’s an app for that! Dead? Well, there’s an app for that, too! And most apps are intentionally addictive distractions that end up taking our attention away from things like family, friends, sleep, and oncoming traffic. There’s a better way. In this book, innovator Golden Krishna challenges our world of nagging, screen-based bondage, and shows how we can build a technologically advanced world without digital interfaces. In his insightful, raw, and often hilarious criticism, Golden reveals fascinating ways to think beyond screens using three principles that lead to more meaningful innovation. Whether you’re working in technology, or just wary of a gadget-filled future, you’ll be enlighted and entertained while discovering that the best interface is no interface.