Exponential Decay for the Near-Critical Scaling Limit of the Planar Ising Model

Federico Camia, Jianping Jiang, Charles M. Newman

Research output: Contribution to journalArticlepeer-review


We consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a = 1, the mass (inverse correlation length) is of order h8/15 as h ↓ 0; for the Euclidean field, it equals exactly Ch8/15 for some C. Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.

Original languageEnglish (US)
Pages (from-to)1371-1405
Number of pages35
JournalCommunications on Pure and Applied Mathematics
Issue number7
StateAccepted/In press - Jan 1 2020

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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