A rectangular additive convolution for polynomials

We define the rectangular additive convolution of polynomials with nonnegative real roots as a generalization of the asymmetric additive convolution introduced by Marcus, Spielman and Srivastava. We then prove a sliding bound on the largest root of this convolution. The main tool used in the analysis is a differential operator derived from the "rectangular Cauchy transform" introduced by Benaych-Georges. The proof is inductive, with the base case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer polynomials which may be of independent interest

Crystallization of random matrix orbits

Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta=1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices can be extrapolated to general values of $\beta>0$ through associated special functions. We show that $\beta\to\infty$ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta$ self-adjoint matrix with fixed eigenvalues is known as $\beta$-corners process. We show that as $\beta\to\infty$ these eigenvalues crystallize on the irregular lattice of all the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field (dGFF) put on top of this lattice, which provides a new explanation of why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.Comment: 25 pages. v2: misprints corrected, to appear in IMR

Generation of spin-polarized currents via cross-relaxation with dynamically pumped paramagnetic impurities

Key to future spintronics and spin-based information processing technologies is the generation, manipulation, and detection of spin polarization in a solid state platform. Here, we theoretically explore an alternative route to spin injection via the use of dynamically polarized nitrogen-vacancy (NV) centers in diamond. We focus on the geometry where carriers and NV centers are confined to proximate, parallel layers and use a 'trap-and-release' model to calculate the spin cross-relaxation probabilities between the charge carriers and neighboring NV centers. We identify near-unity regimes of carrier polarization depending on the NV spin state, applied magnetic field, and carrier g-factor. In particular, we find that unlike holes, electron spins are distinctively robust against spin-lattice relaxation by other, unpolarized paramagnetic centers. Further, the polarization process is only weakly dependent on the carrier hopping dynamics, which makes this approach potentially applicable over a broad range of temperatures.C.A.M. acknowledges support from the National Science Foundation through Grant No. NSF-1314205. M.W.D. acknowledges support from the Australian Research Council through Grant No. DP120102232

Photolysis of Diborane at 1849 Å

The photolysis of diborane at 1849 Å has been studied in a specially constructed, internal‐type mercury‐vapor lamp. The products have been found to be H_2, B_(4)H_(10), B_(5)H_(11), and, at low pressures, a —BH— polymer. Reaction orders at 4°C have been obtained from linear plots of reaction products vs time for a range of diborane pressures from 0.08 to 80 cm, and at two light intensities. Linear relations between products and time existed only at very low conversions (∼1%), which required the development of a low‐temperature separation method for manipulating and analyzing the traces of B_(4)H_(10) and B_(5)H_(11). Because of the reactivity of these compounds, a detailed conditioning procedure was employed for the glass system. A mechanism consistant with the kinetic data and suggested by the kinetic results of thermal and photosensitized decomposition of diborane is postulated: the B_(5)H_(11) is assumed to be formed from a dissociation of B_(2)H_6 into BH_3's, the latter arising from an excited molecule. The B_(4)H_(10) and polymer are assumed to be formed from a dissociation of B_(2)H_6 into B_(2)H_5 and H, followed by radical recombination. There is a significant difference between the kinetics of thermal and photochemical B_(5)H_(11) formation, a result which may be due to the considerable energy excess of the 1849 quantum over that needed for dissociation (∼125‐kcal excess). These kinetic results raise a number of interesting questions, questions which can only be resolved through further investigations of effects due to light intensity, added inert gases, and temperature. The primary quantum yield of the step forming B_(2)H_5 and H is about 10 times higher than that of the one forming BH_3's. A rather rough estimate suggests that the former is of the order of magnitude of unity