Behavior of the Lattice Gaussian Free Field with Weak Repulsive Potentials

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We consider the d(≥3) - dimensional lattice Gaussian free field on ΛN: = [- N, N] d∩ Zd in the presence of a self-potential of the form U(r) = - bI(| r| ≤ a) , a> 0 , b∈ R. When b> 0 , the potential attracts the field to the level around zero and is called square-well pinning. It is known that the field turns to be localized and massive for every a> 0 and b> 0. In this paper, we consider the situation that the parameter b< 0 and self-potentials are imposed on ΛαN,α∈(0,1). We prove that once we impose this weak repulsive potential from the level [- a, a] , most sites are located on the same side and the field is pushed to the same level when the original Gaussian field is conditioned to be positive everywhere, or negative everywhere with probability 12, respectively. This result can be applied to show the similar path behavior for the disordered pinning model in the delocalized regime.

Original languageEnglish
Article number8
JournalJournal of Statistical Physics
Issue number1
Publication statusPublished - 2021 Jan


  • Capacity
  • Entropic repulsion
  • Lattice Gaussian free field
  • Pinning

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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