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Fredholm Determinant Evaluations of the Ising Model Diagonal Correlations and their λ Generalization
Author(s) -
Witte N. S.,
Forrester P. J.
Publication year - 2012
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2011.00534.x
Subject(s) - fredholm determinant , mathematics , toeplitz matrix , diagonal , ising model , unit circle , generalization , operator (biology) , kernel (algebra) , orthogonal polynomials , pure mathematics , gaussian quadrature , mathematical analysis , boundary value problem , statistical physics , biochemistry , chemistry , physics , geometry , repressor , transcription factor , gene , nyström method
The diagonal spin–spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants—one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel. Either determinant allows for a Neumann expansion possessing a natural λ‐parameter generalization and we prove that both expansions are in fact equal, implying a continuous and a discrete representation of the form factors. Our proof employs an extension of the classic study by Geronimo and Case [1], applying scattering theory to orthogonal polynomial systems on the unit circle, to the bi‐orthogonal situation.