4,984 research outputs found
Summing free unitary random matrices
I use quaternion free probability calculus - an extension of free probability
to non-Hermitian matrices (which is introduced in a succinct but self-contained
way) - to derive in the large-size limit the mean densities of the eigenvalues
and singular values of sums of independent unitary random matrices, weighted by
complex numbers. In the case of CUE summands, I write them in terms of two
"master equations," which I then solve and numerically test in four specific
cases. I conjecture a finite-size extension of these results, exploiting the
complementary error function. I prove a central limit theorem, and its first
sub-leading correction, for independent identically-distributed zero-drift
unitary random matrices.Comment: 17 pages, 15 figure
The AdS_5xS^5 superstring worldsheet S-matrix and crossing symmetry
An S-matrix satisying the Yang-Baxter equation with symmetries relevant to
the AdS_5xS^5 superstring has recently been determined up to an unknown scalar
factor. Such scalar factors are typically fixed using crossing relations,
however due to the lack of conventional relativistic invariance, in this case
its determination remained an open problem.
In this paper we propose an algebraic way to implement crossing relations for
the AdS_5xS^5 superstring worldsheet S-matrix. We base our construction on a
Hopf-algebraic formulation of crossing in terms of the antipode and introduce
generalized rapidities living on the universal cover of the parameter space
which is constructed through an auxillary, coupling constant dependent,
elliptic curve. We determine the crossing transformation and write functional
equations for the scalar factor of the S-matrix in the generalized rapidity
plane.Comment: 27 pages, no figures; v2: sign typo fixed in (24), everything else
unchange
Multiplication law and S transform for non-hermitian random matrices
We derive a multiplication law for free non-hermitian random matrices
allowing for an easy reconstruction of the two-dimensional eigenvalue
distribution of the product ensemble from the characteristics of the individual
ensembles. We define the corresponding non-hermitian S transform being a
natural generalization of the Voiculescu S transform. In addition we extend the
classical hermitian S transform approach to deal with the situation when the
random matrix ensemble factors have vanishing mean including the case when both
of them are centered. We use planar diagrammatic techniques to derive these
results.Comment: 25 pages + 11 figure
Spectrum of the Product of Independent Random Gaussian Matrices
We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M
independent NxN Gaussian random matrices in the large-N limit is rotationally
symmetric in the complex plane and is given by a simple expression
rho(z,\bar{z}) = 1/(M\pi\sigma^2} |z|^{-2+2/M} for |z|<\sigma, and is zero for
|z|> \sigma. The parameter \sigma corresponds to the radius of the circular
support and is related to the amplitude of the Gaussian fluctuations. This form
of the eigenvalue density is highly universal. It is identical for products of
Gaussian Hermitian, non-Hermitian, real or complex random matrices. It does not
change even if the matrices in the product are taken from different Gaussian
ensembles. We present a self-contained derivation of this result using a planar
diagrammatic technique for Gaussian matrices. We also give a numerical evidence
suggesting that this result applies also to matrices whose elements are
independent, centered random variables with a finite variance.Comment: 16 pages, 6 figures, minor changes, some references adde
Wrapping interactions at strong coupling -- the giant magnon
We derive generalized Luscher formulas for finite size corrections in a
theory with a general dispersion relation. For the AdS_5xS^5 superstring these
formulas encode leading wrapping interaction effects. We apply the generalized
mu-term formula to calculate finite size corrections to the dispersion relation
of the giant magnon at strong coupling. The result exactly agrees with the
classical string computation of Arutyunov, Frolov and Zamaklar. The agreement
involved a Borel resummation of all even loop-orders of the BES/BHL dressing
factor thus providing a strong consistency check for the choice of the dressing
factor.Comment: 35 pages, 2 figures; v2: comments and references adde
Elevated-temperature impact toughness of Mg–(Gd, Y)–Zr alloy
The Charpy impact results for Mg–10Gd–3Y–0.5Zr and Mg–11Y–5Gd–2Zn–0.5Zr alloys at various temperatures showed that Mg–10Gd–3Y–0.5Zr was more sensitive to temperature. The increase in impact toughness with temperature was related to the blunt crack-tip at high temperatures. The delamination and local melt of matrix were responsible for the brittle-to-ductile transition of GW103 alloy. The branch and bridging of cracks resulting from ordered phases played an import role in the change in fracture mode from cleavage fracture to quasi-cleavage and dimple-fracture for WGZ1152 alloy
Real symmetric random matrices and paths counting
Exact evaluation of is here performed for real symmetric
matrices of arbitrary order , up to some integer , where the matrix
entries are independent identically distributed random variables, with an
arbitrary probability distribution.
These expectations are polynomials in the moments of the matrix entries ;
they provide useful information on the spectral density of the ensemble in the
large limit. They also are a straightforward tool to examine a variety of
rescalings of the entries in the large limit.Comment: 23 pages, 10 figures, revised pape
Adding and multiplying random matrices: a generalization of Voiculescu's formulae
In this paper, we give an elementary proof of the additivity of the
functional inverses of the resolvents of large random matrices, using
recently developed matrix model techniques. This proof also gives a very
natural generalization of these formulae to the case of measures with an
external field. A similar approach yields a relation of the same type for
multiplication of random matrices.Comment: 11 pages, harvmac. revised x 2: refs and minor comments adde
Mapping of quantum well eigenstates with semimagnetic probes
We present results of transmission measurements on CdTe quantum wells with
thin semimagnetic CdMnTe probe layers embedded in various positions along the
growth axis. The presence of the probes allow us to map the probability density
functions by two independent methods: analyzing the exciton energy position and
the exciton Zeeman splitting. We apply both approaches to map the first three
quantum well eigenstates and we find that both of them yield equally accurate
results.Comment: Accepted for publication in Physical Review
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