Eigenvalue Dynamics and the Matrix Chain

TL;DR

This paper develops a method to analyze eigenvalue distributions in matrix chains by transforming equations into boundary value problems, recovering known solutions, and exploring critical phenomena including phase transitions.

Contribution

It introduces a novel approach to map matrix chain equations into boundary value problems and derives eigenvalue distributions using saddle-point approximations, clarifying critical behaviors.

Findings

Recovered all known explicit solutions for matrix eigenvalues.

Derived eigenvalue distributions through saddle-point approximations.

Interpreted critical behavior, including Kosterlitz-Thouless transition, via stationary points.

Abstract

We introduce a general method for transforming the equations of motion following from a Das-Jevicki-Sakita Hamiltonian, with boundary conditions, into a boundary value problem in one-dimensional quantum mechanics. For the particular case of a one-dimensional chain of interacting NxN Hermitean matrices, the corresponding large N boundary value problem is mapped into a linear Fredholm equation with Hilbert-Schmidt type kernel. The equivalence of this kernel, in special cases, to a second order differential operator allows us recover all previously known explicit solutions for the matrix eigenvalues. In the general case, the distribution of eigenvalues is formally derived through a series of saddle-point approximations. The critical behaviour of the system, including a previously observed Kosterlitz-Thouless transition, is interpreted in terms of the stationary points. In particular we show that a previously conjectured infinite series of sub-leading critical points are due to expansion about unstable stationary points and consequently not realized.