55 research outputs found
Moments of a single entry of circular orthogonal ensembles and Weingarten calculus
Consider a symmetric unitary random matrix
from a circular orthogonal ensemble. In this paper, we study moments of a
single entry . For a diagonal entry we give the explicit
values of the moments, and for an off-diagonal entry we give leading
and subleading terms in the asymptotic expansion with respect to a large matrix
size . Our technique is to apply the Weingarten calculus for a
Haar-distributed unitary matrix.Comment: 17 page
Gaussian diagrammatics from Circular Ensembles of random matrices
We uncover a hidden Gaussian ensemble inside each of the three circular
ensembles of random matrices, which provide novel diagrammatic rules for the
calculation of moments. The matrices involved are generic complex for
, complex symmetric for and complex self-dual for ,
and their dimension must be set to . As an application, we compute
moments of traces of submatrices.Comment: 12 pages, 4 figure
Free particles from Brauer algebras in complex matrix models
The gauge invariant degrees of freedom of matrix models based on an N x N
complex matrix, with U(N) gauge symmetry, contain hidden free particle
structures. These are exhibited using triangular matrix variables via the Schur
decomposition. The Brauer algebra basis for complex matrix models developed
earlier is useful in projecting to a sector which matches the state counting of
N free fermions on a circle. The Brauer algebra projection is characterized by
the vanishing of a scale invariant laplacian constructed from the complex
matrix. The special case of N=2 is studied in detail: the ring of gauge
invariant functions as well as a ring of scale and gauge invariant differential
operators are characterized completely. The orthonormal basis of wavefunctions
in this special case is completely characterized by a set of five commuting
Hamiltonians, which display free particle structures. Applications to the
reduced matrix quantum mechanics coming from radial quantization in N=4 SYM are
described. We propose that the string dual of the complex matrix harmonic
oscillator quantum mechanics has an interpretation in terms of strings and
branes in 2+1 dimensions.Comment: 64 pages, v2: Exposition improved, minor corrections; v3: Typos
corrected, published versio
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