R. Dobrushin, O. Hryniv
Fluctuations of Shapes of Large Areas Under Paths of Random Walks
The paper is published: Probab. Theory Relat. Fields 105 (1996) 423-458

MSC:

60F17 Functional limit theorems; invariance principles

60F10 Large deviations

60J15 Random walks

82B24 Interface problems

Abstract: We discuss statistical properties of random walks conditioned by
fixing a large area under their paths. We prove the functional
central limit theorem for these conditional distributions. The
limiting Gaussian measure coincides with the conditional
probability distribution of a
certain time-nonhomogeneous Gaussian random process obtained by
an integral transformation of 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: Random walks, large deviations, invariance principle, conditional limit theorems, Solid-On-Solid model, Wulff construction
Notes: second version