Arbitrary Rotation Invariant Random Matrix Ensembles and Supersymmetry

We generalize the supersymmetry method in Random Matrix Theory to arbitrary rotation invariant ensembles. Our exact approach further extends a previous contribution in which we constructed a supersymmetric representation for the class of norm-dependent Random Matrix Ensembles. Here, we derive a supersymmetric formulation under very general circumstances. A projector is identified that provides the mapping of the probability density from ordinary to superspace. Furthermore, it is demonstrated that setting up the theory in Fourier superspace has considerable advantages. General and exact expressions for the correlation functions are given. We also show how the use of hyperbolic symmetry can be circumvented in the present context in which the non-linear sigma model is not used. We construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the Green functions.Comment: 36 page

Level Repulsion in Constrained Gaussian Random-Matrix Ensembles

Introducing sets of constraints, we define new classes of random-matrix ensembles, the constrained Gaussian unitary (CGUE) and the deformed Gaussian unitary (DGUE) ensembles. The latter interpolate between the GUE and the CGUE. We derive a sufficient condition for GUE-type level repulsion to persist in the presence of constraints. For special classes of constraints, we extend this approach to the orthogonal and to the symplectic ensembles. A generalized Fourier theorem relates the spectral properties of the constraining ensembles with those of the constrained ones. We find that in the DGUEs, level repulsion always prevails at a sufficiently short distance and may be lifted only in the limit of strictly enforced constraints.Comment: 20 pages, no figures. New section adde

On bulk singularities in the random normal matrix model

We extend the method of rescaled Ward identities of Ameur-Kang-Makarov to study the distribution of eigenvalues close to a bulk singularity, i.e. a point in the interior of the droplet where the density of the classical equilibrium measure vanishes. We prove results to the effect that a certain "dominant part" of the Taylor expansion determines the microscopic properties near a bulk singularity. A description of the distribution is given in terms of a special entire function, which depends on the nature of the singularity (a Mittag-Leffler function in the case of a rotationally symmetric singularity).Comment: This version clarifies on the proof of Theorem

Critical Conductance of a Mesoscopic System: Interplay of the Spectral and Eigenfunction Correlations at the Metal-Insulator Transition

We study the system-size dependence of the averaged critical conductance $g(L)$ at the Anderson transition. We have: (i) related the correction $\delta g(L)=g(\infty)-g(L)\propto L^{-y}$ to the spectral correlations; (ii) expressed $\delta g(L)$ in terms of the quantum return probability; (iii) argued that $y=\eta$ -- the critical exponent of eigenfunction correlations. Experimental implications are discussed.Comment: minor changes, to be published in PR

Learning 3D Human Pose from Structure and Motion

3D human pose estimation from a single image is a challenging problem, especially for in-the-wild settings due to the lack of 3D annotated data. We propose two anatomically inspired loss functions and use them with a weakly-supervised learning framework to jointly learn from large-scale in-the-wild 2D and indoor/synthetic 3D data. We also present a simple temporal network that exploits temporal and structural cues present in predicted pose sequences to temporally harmonize the pose estimations. We carefully analyze the proposed contributions through loss surface visualizations and sensitivity analysis to facilitate deeper understanding of their working mechanism. Our complete pipeline improves the state-of-the-art by 11.8% and 12% on Human3.6M and MPI-INF-3DHP, respectively, and runs at 30 FPS on a commodity graphics card.Comment: ECCV 2018. Project page: https://www.cse.iitb.ac.in/~rdabral/3DPose

Smooth analysis of the condition number and the least singular value

Let \a be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of \a and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when \a is gaussian. Our investigation reveals an interesting fact that the "core" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n}$. This does not occur in Spielman-Teng studies when \a is gaussian. Consequently, our general estimate involves the norm $\|M\|$. In the special case when $\|M\|$ is relatively small, this estimate is nearly optimal and extends or refines existing results.Comment: 20 pages. An erratum to the published version has been adde

Spectral fluctuation properties of constrained unitary ensembles of Gaussian-distributed random matrices

We investigate the spectral fluctuation properties of constrained ensembles of random matrices (defined by the condition that a number N(Q) of matrix elements vanish identically; that condition is imposed in unitarily invariant form) in the limit of large matrix dimension. We show that as long as N(Q) is smaller than a critical value (at which the quadratic level repulsion of the Gaussian unitary ensemble of random matrices may be destroyed) all spectral fluctuation measures have the same form as for the Gaussian unitary ensemble.Comment: 15 page

Magnetic switching in granular FePt layers promoted by near-field laser enhancement

Light-matter interaction at the nanoscale in magnetic materials is a topic of intense research in view of potential applications in next-generation high-density magnetic recording. Laser-assisted switching provides a pathway for overcoming the material constraints of high-anisotropy and high-packing density media, though much about the dynamics of the switching process remains unexplored. We use ultrafast small-angle x-ray scattering at an x-ray free-electron laser to probe the magnetic switching dynamics of FePt nanoparticles embedded in a carbon matrix following excitation by an optical femtosecond laser pulse. We observe that the combination of laser excitation and applied static magnetic field, one order of magnitude smaller than the coercive field, can overcome the magnetic anisotropy barrier between "up" and "down" magnetization, enabling magnetization switching. This magnetic switching is found to be inhomogeneous throughout the material, with some individual FePt nanoparticles neither switching nor demagnetizing. The origin of this behavior is identified as the near-field modification of the incident laser radiation around FePt nanoparticles. The fraction of not-switching nanoparticles is influenced by the heat flow between FePt and a heat-sink layer