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Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model
Author(s) -
Camia Federico,
Jiang Jianping,
Newman Charles M.
Publication year - 2020
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21884
Subject(s) - scaling limit , ising model , scaling , critical point (mathematics) , mathematics , exponential decay , exponential function , lattice (music) , euclidean geometry , conformal symmetry , exponential growth , square lattice , critical phenomena , statistical physics , conformal map , mathematical physics , physics , mathematical analysis , condensed matter physics , quantum mechanics , geometry , phase transition , acoustics
We consider the Ising model at its critical temperature with external magnetic field ha 15/8 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 h 8/15 as h ↓ 0 ; for the Euclidean field, it equals exactly Ch 8/15 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. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.