Related papers: Universal correlators for multi-arc complex matrix…

Pattern matching is a powerful tool which is part of many functional programming languages as well as computer algebra systems such as Mathematica. Among the existing systems, Mathematica offers the most expressive pattern matching.…

The loop equations for a chain of hermitian random matrices are computed explicitely, including the 1/N^2 corrections. To leading order, the master loop equation reduces to an algebraic equation, whose solution can be written in terms of…

It was shown roughly thirty years ago that the density correlations of eigenvalues of large random matrices display a universal form, independent of most of the details of the distribution of the random matrix itself. We show that when the…

Real complex systems are not rigidly structured; no clear rules or blueprints exist for their construction. Yet, amidst their apparent randomness, complex structural properties universally emerge. We propose that an important class of…

Properties of Hermitian forms are used to investigate several natural questions from CR Geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the…

Dirichlet integrals and the associated Dirichlet statistical densities are widely used in various areas. Generalizations of Dirichlet integrals and Dirichlet models to matrix-variate cases, when the matrices are real symmetric positive…

Earnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying primarily on Ramanujan's…

{Recently, we found that the correlation between the eigenvalues of random hermitean matrices exhibits universal behavior. Here we study this universal behavior and develop a diagrammatic approach which enables us to extend our previous…

We compute all massive partition functions or characteristic polynomials and their complex eigenvalue correlation functions for non-Hermitean extensions of the symplectic and chiral symplectic ensemble of random matrices. Our results are…

In this paper, we propose to provide a general ensemble learning framework based on deep learning models. Given a group of unit models, the proposed deep ensemble learning framework will effectively combine their learning results via a…

This paper is concerned with complex eigenvalues of truncated unitary quaternion matrices equipped with the Haar measure. The joint eigenvalue probability density function is obtained for truncations of any size. We also obtain the spectral…

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local…

We consider non-gaussian ensembles of random normal matrices with the constraint that the ensembles are invariant under unitary transformations. We show that the level density of eigenvalues exhibits disk to ring transition in the complex…

In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations.…

We establish universality of local eigenvalue correlations in unitary random matrix ensembles (1/Z_n) |\det M|^{2\alpha} e^{-n\tr V(M)} dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal…

It is shown that the eigenvalue problem for the hermitian Wilson-Dirac operator of for a uniform magnetic field in two dimensions can be reduced to one-dimensional problem described by a relativistic analog of the Harper equation. An…

This paper concerns the universality of the two-layer neural network with the $k$-rectified linear unit activation function with $k=1,2,\ldots$ with a suitable norm without any restriction on the shape of the domain. This type of result is…

We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix…

For general large non-Hermitian random matrices $X$ and deterministic normal deformations $A$, we prove that the local eigenvalue statistics of $A+X$ close to the critical edge points of its spectrum are universal. This concludes the proof…

Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or…