32,707 research outputs found
On polynomial integrals over the orthogonal group
We consider integrals of type , with respect to the Haar measure
on the orthogonal group. We establish several remarkable invariance properties
satisfied by such integrals, by using combinatorial methods. We present as well
a general formula for such integrals, as a sum of products of factorials.Comment: 20 page
Investigation of continuous-time quantum walk on root lattice and honeycomb lattice
The continuous-time quantum walk (CTQW) on root lattice (known as
hexagonal lattice for ) and honeycomb one is investigated by using
spectral distribution method. To this aim, some association schemes are
constructed from abelian group and two copies of finite
hexagonal lattices, such that their underlying graphs tend to root lattice
and honeycomb one, as the size of the underlying graphs grows to
infinity. The CTQW on these underlying graphs is investigated by using the
spectral distribution method and stratification of the graphs based on
Terwilliger algebra, where we get the required results for root lattice
and honeycomb one, from large enough underlying graphs. Moreover, by using the
stationary phase method, the long time behavior of CTQW on infinite graphs is
approximated with finite ones. Also it is shown that the Bose-Mesner algebras
of our constructed association schemes (called -variable -polynomial) can
be generated by commuting generators, where raising, flat and lowering
operators (as elements of Terwilliger algebra) are associated with each
generator. A system of -variable orthogonal polynomials which are special
cases of \textit{generalized} Gegenbauer polynomials is constructed, where the
probability amplitudes are given by integrals over these polynomials or their
linear combinations. Finally the suppersymmetric structure of finite honeycomb
lattices is revealed. Keywords: underlying graphs of association schemes,
continuous-time quantum walk, orthogonal polynomials, spectral distribution.
PACs Index: 03.65.UdComment: 41 pages, 4 figure
Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups
The Harish-Chandra correlation functions, i.e. integrals over compact groups
of invariant monomials prod tr{X^{p_1} Omega Y^{q_1} Omega^dagger X^{p_2} ...
with the weight exp tr{X Omega Y Omega^dagger} are computed for the orthogonal
and symplectic groups. We proceed in two steps. First, the integral over the
compact group is recast into a Gaussian integral over strictly upper triangular
complex matrices (with some additional symmetries), supplemented by a summation
over the Weyl group. This result follows from the study of loop equations in an
associated two-matrix integral and may be viewed as the adequate version of
Duistermaat-Heckman's theorem for our correlation function integrals. Secondly,
the Gaussian integration over triangular matrices is carried out and leads to
compact determinantal expressions.Comment: 58 pages; Acknowledgements added; small corrections in appendix A;
minor changes & Note Adde
Matrix models for classical groups and ToeplitzHankel minors with applications to Chern-Simons theory and fermionic models
We study matrix integration over the classical Lie groups
and , using symmetric function theory and the equivalent formulation
in terms of determinants and minors of ToeplitzHankel matrices. We
establish a number of factorizations and expansions for such integrals, also
with insertions of irreducible characters. As a specific example, we compute
both at finite and large the partition functions, Wilson loops and Hopf
links of Chern-Simons theory on with the aforementioned symmetry
groups. The identities found for the general models translate in this context
to relations between observables of the theory. Finally, we use character
expansions to evaluate averages in random matrix ensembles of Chern-Simons
type, describing the spectra of solvable fermionic models with matrix degrees
of freedom.Comment: 32 pages, v2: Several improvements, including a Conclusions and
Outlook section, added. 36 page
Asymptotics of unitary and othogonal matrix integrals
In this paper, we prove that in small parameter regions, arbitrary unitary
matrix integrals converge in the large limit and match their formal
expansion. Secondly we give a combinatorial model for our matrix integral
asymptotics and investigate examples related to free probability and the HCIZ
integral. Our convergence result also leads us to new results of smoothness of
microstates. We finally generalize our approach to integrals over the othogonal
group.Comment: 41 pages, important modifications, new section about orthogonal
integral
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