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Fluctuations of shapes of large areas under paths of random walks
Authors:R Dobrushin  O Hryniv
Institution:(1) Institute for Problems of Information Transmission, Russian Academy of Sciences, Ermolovoi 19, 103051 Moscow, Russia;(2) Institute for Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, Naukova 3ldquobrdquo, 290601 Lviv, Ukraine;(3) Present address: E. Schrödinger International Institute for Mathematical Physics, Pasteurgasse 6/7, A-1090 Vienna, Austria
Abstract:Summary We discuss statistical properties of random walks conditioned by fixing a large area under their paths. We prove the functional central limit theorem (invariance principle) for these conditional distributions. The limiting Gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous Gaussian random process obtained by an integral transformation of the white noise. From the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one-dimensional SOS-model.
Keywords:60F17  60F10  60J15  82B24
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