The discrete Gaussian free field on a compact manifold |
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| Institution: | 1. Department of Mathematics, University of Utah, United States;2. Department of Mathematics, University of Mississippi, United States;1. School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China;2. Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China;3. Department of Statistics and Actuarial Science, University of Waterloo, Canada;4. KLAS MOE and School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China;5. Department of Statistics and Applied Probability, National University of Singapore, Singapore;6. Department of Statistics and Actuarial Science, The University of Hong Kong, Hongkong, China |
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| Abstract: | In this article we define the discrete Gaussian free field (DGFF) on a compact manifold. Since there is no canonical grid approximation of a manifold, we construct a random graph that suitably replaces the square lattice in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field (GFF). Furthermore using Voronoi tessellations we can interpret the DGFF as element of a Sobolev space and show convergence to the GFF in law with respect to the strong Sobolev topology. |
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