Non-Markov property of certain eigenvalue processes analogous to Dyson's model

TL;DR

This paper demonstrates that altering the coefficient in Dyson's eigenvalue process for a 2x2 matrix causes the eigenvalue dynamics to lose the Markov property, highlighting the delicate nature of Markovianity in such models.

Contribution

It shows that changing the off-diagonal coefficient in Dyson's 2x2 matrix eigenvalue process breaks the Markov property, revealing a non-Markovian behavior.

Findings

Eigenvalue process becomes non-Markov when coefficient is changed.

Markov property is sensitive to the off-diagonal coefficient.

Original Dyson process with coefficient 1/√2 is Markovian.

Abstract

It is proven that the eigenvalue process of Dyson's random matrix process of size two becomes non-Markov if the common coefficient $1/\sqrt{2}$ in the non-diagonal entries is replaced by a different positive number.