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The Critical Ising Model via Kac-Ward Matrices

ContributorsCimasoni, David
Published inCommunications in Mathematical Physics, vol. 316, no. 1, p. 99-126
Collection
  • Open Access - Licence nationale Springer 
Publication date2012
Abstract

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 22g matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer ∂¯¯¯-torsions of the Riemann surface in which G embeds.

Citation (ISO format)
CIMASONI, David. The Critical Ising Model via Kac-Ward Matrices. In: Communications in Mathematical Physics, 2012, vol. 316, n° 1, p. 99–126. doi: 10.1007/s00220-012-1575-z
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Article (Published version)
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ISSN of the journal1432-0916
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