On random tridiagonal matrices and the beta log-gas
R.J. Breunissen (TU Delft - Electrical Engineering, Mathematics and Computer Science)
W.G.M. Groenevelt – Mentor (TU Delft - Analysis)
E.M. van Elderen – Graduation committee member (TU Delft - Mathematical Physics)
Gioia Carinci – Graduation committee member (TU Delft - Applied Probability)
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Abstract
In this thesis the beta log-gas probability density function is discussed. It is shown that there is a strong link between this density function and Jacobi matrices. A change of variables exercise shows that the distribution of eigenvalues is exactly like the quadratic beta log-gas. The change of variables gives the normalization constant for the quadratic beta log-gas. Finally, it is made likely that the Jacobi matrix adheres to Wigners semicircle law, and that the beta log-gas is limited by the semicircle distribution.