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The scaling limit of the energy correlations in non-integrable Ising models. (English) Zbl 1278.82011
Summary: We obtain an explicit expression for the multipoint energy correlations of a non-solvable two-dimensional Ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength \(\lambda\), in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one. Our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory, which generalizes the one originally used by Schultz, Mattis, and Lieb for the nearest neighbor Ising model. The interacting model is then analyzed by a multiscale method first proposed by Pinson and Spencer. If the lattice spacing is finite, then the correlations cannot be computed in closed form: rather, they are expressed in terms of infinite, convergent, power series in \(\lambda\). In the scaling limit, these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable Ising model, up to a finite renormalization of the parameters. Explicit bounds on the speed of convergence to the scaling limit are derived.
©2012 American Institute of Physics

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82D40 Statistical mechanical studies of magnetic materials
82B27 Critical phenomena in equilibrium statistical mechanics
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