Integrability in Random Two-Matrix Models under Finite-Rank Perturbations

Checinski T. Integrability in Random Two-Matrix Models under Finite-Rank Perturbations. Bielefeld: Universität Bielefeld; 2019.In Quantum Chromodynamics low energy spectral properties of the Dirac operator can be described by random matrix ensembles. In time-series analysis strong statistical fluctuations coincide with eigenvalue statistics of random matrices. These two completely different fields share the same type of random matrix ensembles: chiral symmetric random matrices. The analysis of two random-matrix models of this type is presented: the product of two coupled Wishart matrices and the sum of two independent Wishart matrices. Here, we expose the integrability of these models and compute quantities being of interest in Quantum Chromodynamics and in time- series analysis, respectively

Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and P\'ech\'e, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer $G$-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors.Comment: 39 pages, 4 figures; v2: 43 pages, presentation of Thm 1.4 improved, alternative proof of Prop 3.1 and reference added; v3: final typo corrections, to appear in AIHP Probabilite et Statistiqu

Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances

Akemann G, Checinski T, Kieburg M. Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 2016;49(31): 315201.We compute the spectral statistics of the sum H of two independent complex Wishart matrices, each of which is correlated with a different covariance matrix. Random matrix theory enjoys many applications including sums and products of random matrices. Typically ensembles with correlations among the matrix elements are much more difficult to solve. Using a combination of supersymmetry, superbosonisation and bi-orthogonal functions we are able to determine all spectral k-point density correlation functions of H for arbitrary matrix size N. In the half-degenerate case, when one of the covariance matrices is proportional to the identity, the recent results by Kumar for the joint eigenvalue distribution of H serve as our starting point. In this case the ensemble has a bi-orthogonal structure and we explicitly determine its kernel, providing its exact solution for finite N. The kernel follows from computing the expectation value of a single characteristic polynomial. In the general nondegenerate case the generating function for the k-point resolvent is determined from a supersymmetric evaluation of the expectation value of k ratios of characteristic polynomials. Numerical simulations illustrate our findings for the spectral density at finite N and we also give indications how to do the asymptotic large-N analysis

Field- and temperature-modulated spin diode effect in a GMR nanowire with dipolar coupling

The magnetization dynamics of single Co/Cu/Co spin valves, embedded in electrodeposited nanowires of 30 nm avarage diameter, was observed using the spin-diode effect. The electrically-detected magnetic resonances were compared when using modulation of either the magnetic field or a laser irradiation. The effect of temperature modulation was accounted for by introducing the temperature dependence of the saturation magnetization and anisotropy, as well as thermal spin-transfer torque (TSTT). The predictions of the model are compared with experimental data. Two forms of modular ion give rise to qualitative differences in the spectra that are accounted for by the model only if both temperature-modulated magnetization and TSTT are introduced in the model. On the contrary, the temperature modulation of the magnetic anisotropy has a smaller contribution