Scaling Limit of Discrete Gaussian Free Fie

Nandan, Shubhamoy (2018) Scaling Limit of Discrete Gaussian Free Fie. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

In this article, we review some well-known results in the literature of random interfaces, on scaling limit of Ginzburg Landao ∇φ interfaces defined on the integer lattice Zd, to be specific. In dimension d = 1, we obtain Brownian bridge as the scaling limit of the model when considered under the effect of a uniformly strictly convex, C² potential. For dimension d ≥ 2, we consider the interface under the effect of quadratic potential which is the well-known discrete Gaussian free field. We prove that, under appropriate scaling, the discrete Gaussian free field converges to the continuous Gaussian free field. For dimension d ≥ 3, this convergence is obtained in C∞c (Rd)', the dual space of compactly supported smooth functions on Rd. While in dimension d = 2, we get the convergence in negatively indexed Sobolev spaces H₀-s 0 for s > 1.5.

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