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Register a new userFermion Mapping for Orthogonal and Symplectic Ensembles
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Added by
Matthew B. Hastings
Publication date
1996
fields
Physics
and research's language is
English
Authors
M.B. Hastings
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The circular orthogonal and circular symplectic ensembles are mapped onto free, non-hermitian fermion systems. As an illustration, the two-level form factors are calculated.
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Random matrix theory has proven very successful in the understanding of the spectra of chaotic systems. Depending on symmetry with respect to time reversal and the presence or absence of a spin 1/2 there are three ensembles, the Gaussian orthogonal (GOE), Gaussian unitary (GUE), and Gaussian symplectic (GSE) one. With a further particle-antiparticle symmetry the chiral variants of these ensembles, the chiral orthogonal, unitary, and symplectic ensembles (the BDI, AIII, and CII in Cartans notation) appear. A microwave study of the chiral ensembles is presented using a linear chain of evanescently coupled dielectric cylindrical resonators. In all cases the predicted repulsion behavior between positive and negative eigenvalues for energies close to zero could be verified.
We show, using either Fock space techniques or Macdonald difference operators, that certain symplectic and orthogonal analogues of Okounkovs Schur measure are determinantal with kernels given by explicit double contour integrals. We give two applications: one equates certain Toeplitz+Hankel determinants of random matrix theory with appropriate Fredholm determinants and computes SzegH{o} asymptotics for the former; another finds that the simplest examples of said measures exhibit discrete sine kernel asymptotics in the bulk and Airy 2 to 1 kernel---along with a certain dual---asymptotics at the edge. We believe the edge behavior to be universal.
We investigate orthogonal and symplectic bundles with parabolic structure, over a curve.
Theory of Random Matrix Ensembles have proven to be a useful tool in the study of the statistical distribution of energy or transmission levels of a wide variety of physical systems. We give an overview of certain q-generalizations of the Random Matrix Ensembles, which were first introduced in connection with the statistical description of disordered quantum conductors.
By considering the specialisation $s_{lambda}(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $lambda$ in terms of two properties of the boxes in the diagram for $lambda$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $lambda$-tableaux.
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