Geodesic
Deducing a variational principle with minimal \textit{a priori} assumptions
Andrew Krieger1 ; Georg Menz1 ; Martin Tassy2
1University of California, Los Angeles
2Dartmouth College
The electronic journal of combinatorics, Tome 27 (2020) no. 4
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We study the well-known variational and large deviation principle for graph homomorphisms from $\mathbb{Z}^m$ to $\mathbb{Z}$. We provide a robust method to deduce those principles under minimal a priori assumptions. The only ingredient specific to the model is a discrete Kirszbraun theorem i.e. an extension theorem for graph homomorphisms. All other ingredients are of a general nature not specific to the model. They include elementary combinatorics, the compactness of Lipschitz functions, and a simplicial Rademacher theorem. Compared to the literature, our proof does not need any other preliminary results like e.g. concentration or strict convexity of the local surface tension. Therefore, the method is very robust and extends to more complex and subtle models, as e.g. the homogenization of limit shapes or graph-homomorphisms to a regular tree.
DOI : 10.37236/9121
Classification : 82B41, 82B20, 60G60, 60F10, 05C60, 05C05
Mots-clés : large deviation principle, graph homomorphisms, random fields

Andrew Krieger  1   ; Georg Menz  1   ; Martin Tassy  2

1 University of California, Los Angeles
2 Dartmouth College 
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Andrew Krieger; Georg Menz; Martin Tassy. Deducing a variational principle with minimal \textit{a priori} assumptions. The electronic journal of combinatorics, Tome 27 (2020) no. 4. doi: 10.37236/9121

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