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 $S$ of arbitrary order $n$, up to some integer $p$, 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 $n$ limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large $n$ 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 $N$ 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