Abstract
We consider homogenization of random surfaces and study the variational principle for graph homomorphisms from subsets of into , where the underlying uniform measure is perturbed by a random potential. Motivated by the theories of random walks in random potentials, we assume that random potential is stationary, ergodic, and bounded in . We show that the variational principle holds in probability and that the entropy functional homogenizes, i.e. is independent of the values taken by the random potential. The main ingredients in the argument are the existence of the quenched surface tension, the equivalence of the quenched and the annealed surface tension, and robustness of the surface tension under change in boundary data. These ingredients are deduced by a combination of a superadditive ergodic theorem and combinatorial results, especially the Kirszbraun theorem.
Funding Statement
This work has been partially supported by NSF award DMS-1954343.
Dedication
Dedicated to the memory of Thomas Ligget
Acknowledgments
The authors want to thank Tim Austin, Nathanaël Berestycki, Marek Biskup, Antoine Gloria, Michelle Ledoux, Thomas Liggett, Igor Pak, Larent Saloff-Coste, and Tianyi Zheng for the many discussions on this topic.
Citation
Andrew Krieger. Georg Menz. Martin Tassy. "Homogenization of the variational principle for discrete random maps." Electron. J. Probab. 30 1 - 52, 2025. https://doi.org/10.1214/24-EJP1236
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