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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Gaussian free field ： ウィキペディア英語版 Gaussian free field In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of the Gaussian free field. The discrete version can be defined on any graph, usually a lattice in ''d''-dimensional Euclidean space. The continuum version is defined on R''d'' or on a bounded subdomain of R''d''. It can be thought of as a natural generalization of one-dimensional Brownian motion to ''d'' time (but still one space) dimensions; in particular, the one-dimensional continuum GFF is just the standard one-dimensional Brownian motion or Brownian bridge on an interval. In the theory of random surfaces, it is also called the harmonic crystal. It is also the starting point for many constructions in quantum field theory, where it is called the Euclidean bosonic massless free field. A key property of the 2-dimensional GFF is conformal invariance, which relates it in several ways to the Schramm-Loewner Evolution, see and . Similarly to Brownian motion, which is the scaling limit of a wide range of discrete random walk models (see Donsker's theorem), the continuum GFF is the scaling limit of not only the discrete GFF on lattices, but of many random height function models, such as the height function of uniform random planar domino tilings, see . The planar GFF is also the limit of the fluctuations of the characteristic polynomial of a random matrix model, the Ginibre ensemble, see . The structure of the discrete GFF on any graph is closely related to the behaviour of the simple random walk on the graph. For instance, the discrete GFF plays a key role in the proof by of several conjectures about the cover time of graphs (the expected number of steps it takes for the random walk to visit all the vertices). ==Definition of the discrete GFF== Let ''P''(''x'', ''y'') be the transition kernel of the Markov chain given by a random walk on a finite graph ''G''(''V'', ''E''). Let ''U'' be a fixed non-empty subset of the vertices ''V'', and take the set of all real-valued functions $\backslash varphi$ with some prescribed values on ''U''. We then define a Hamiltonian by : $H(\; \backslash varphi\; )\; =\; \backslash frac\; \backslash sum\_\; P(x,y)\backslash big(\backslash varphi(x)\; -\; \backslash varphi(y)\backslash big)^2.$ Then, the random function with probability density proportional to $\backslash exp(-H(\backslash varphi))$ with respect to the Lebesgue measure on $\backslash R^$ is called the discrete GFF with boundary ''U''. It is not hard to show that the expected value $\backslash mathbb()$ is the discrete harmonic extension of the boundary values from ''U'' (harmonic with respect to the transition kernel ''P''), and the covariances $\backslash mathrm()$ are equal to the discrete Green's function ''G''(''x'', ''y''). So, in one sentence, the discrete GFF is the Gaussian random field on ''V'' with covariance structure given by the Green's function associated to the transition kernel ''P''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gaussian free field」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.