Acta Mathematica

The energy density in the planar Ising model

Clément Hongler and Stanislav Smirnov

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We study the critical Ising model on the square lattice in bounded simply connected domains with + and free boundary conditions. We relate the energy density of the model to a discrete fermionic correlator and compute its scaling limit by discrete complex analysis methods. As a consequence, we obtain a simple exact formula for the scaling limit of the energy field one-point function in terms of the hyperbolic metric. This confirms the predictions originating in physics, but also provides a higher precision.

Article information

Acta Math., Volume 211, Number 2 (2013), 191-225.

Received: 20 July 2011
Revised: 15 September 2012
First available in Project Euclid: 31 January 2017

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Ising model energy density discrete analytic function fermions conformal invariance hyperbolic geometry conformal field theory

2013 © Institut Mittag-Leffler


Hongler, Clément; Smirnov, Stanislav. The energy density in the planar Ising model. Acta Math. 211 (2013), no. 2, 191--225. doi:10.1007/s11511-013-0102-1.

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