C2 Regularity of the Surface Tension for the ∇ϕ Interface Model

Scott Armstrong, Wei Wu

Research output: Contribution to journalArticlepeer-review


We consider the ∇ϕ interface model with a uniformly convex interaction potential possessing Hölder continuous second derivatives. Combining ideas of Naddaf and Spencer with methods from quantitative homogenization, we show that the surface tension (or free energy) associated to the model is at least C2,β for some β > 0. We also prove a fluctuation-dissipation relation by identifying its Hessian with the covariance matrix characterizing the scaling limit of the model. Finally, we obtain a quantitative rate of convergence for the Hessian of the finite-volume surface tension to that of its infinite-volume limit.

Original languageEnglish (US)
Pages (from-to)349-421
Number of pages73
JournalCommunications on Pure and Applied Mathematics
Issue number2
StatePublished - Feb 2022

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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