scholarly journals QUARTIC ANHARMONIC OSCILLATOR AND RANDOM MATRIX THEORY

1996 ◽  
Vol 11 (02) ◽  
pp. 119-129 ◽  
Author(s):  
G.M. CICUTA ◽  
S. STRAMAGLIA ◽  
A.G. USHVERIDZE
Keyword(s):  
Orthogonal Polynomials ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Ground State ◽  
Recurrence Relations ◽  
State Energy ◽  
Matrix Theory ◽  
Anharmonic Oscillator ◽  
Hermitian Matrices ◽  
The Relationship

In this letter the relationship between the problem of constructing the ground state energy for the quantum quartic oscillator and the problem of computing mean eigenvalue of large positively definite random hermitian matrices is established. This relationship enables one to present several more or less closed expressions for the oscillator energy. One of such expressions is given in the form of simple recurrence relations derived by means of the method of orthogonal polynomials which is one of the basic tools in the theory of random matrices.

Time-and-band limiting for matrix orthogonal polynomials of Jacobi type

2017 ◽  
Vol 06 (04) ◽  
pp. 1740001 ◽  
Author(s):  
M. Castro ◽  
F. A. Grünbaum
Keyword(s):  
Signal Processing ◽  
Differential Operator ◽  
Orthogonal Polynomials ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Jacobi Polynomials ◽  
Matrix Theory ◽  
Matrix Orthogonal Polynomials

We extend to a situation involving matrix-valued orthogonal polynomials a scalar result that plays an important role in Random Matrix Theory and a few other areas of mathe-matics and signal processing. We consider a case of matrix-valued Jacobi polynomials which arises from the study of representations of [Formula: see text], a group that plays an important role in Random Matrix Theory. We show that in this case an algebraic miracle, namely the existence of a differential operator that commutes with a naturally arising integral one, extends to this matrix-valued situation.

Random matrix theory and integrable systems

2018 ◽  
Author(s):  
Alexander R. Its
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Models ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Integrable Systems ◽  
Matrix Theory ◽  
Kp Hierarchy ◽  
Brownian Motions ◽  
Airy Process ◽  
Virasoro Constraints

This article discusses the interaction between random matrix theory (RMT) and integrable theory, leading to ordinary and partial differential equations (PDEs) for the eigenvalue distribution of random matrix models of size n and the transition probabilities of non-intersecting Brownian motion models, for finite n and for n → ∞. It first provides an overview of the connection between the theory of orthogonal polynomials and the KP-hierarchy in integrable systems before examining matrix models and the Virasoro constraints. It then considers multiple orthogonal polynomials, taking into account non-intersecting Brownian motions on ℝ (Dyson’s Brownian motions), a moment matrix for several weights, Virasoro constraints, and a PDE for non-intersecting Brownian motions. It also analyses critical diffusions, with particular emphasis on the Airy process, the Pearcey process, and Airy process with wanderers. Finally, it describes the Tacnode process, along with kernels and p-reduced KP-hierarchy.

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