Vikus van Rensburg

I am a Ph.D. candidate in Mathematics at the University of Florida, working under the supervision of Mike Jury. I earned a B.S. in Mathematics from Oregon State University in 2023 and began my graduate studies at the University of Florida the same year. I completed my M.S. in Mathematics in 2025 and subsequently advanced to Ph.D. candidacy.

My research interests lie in random matrix theory and free probability. My current work focuses on the development of a free strong Szegő limit theorem, where the second-order Szegő asymptotics are described through fluctuation moments of independent random unitary matrices. More broadly, I am interested in the interplay between free probability, operator theory, and random matrix theory.

Originally from Pretoria, South Africa, I was born and raised there before moving to the United States in 2020 to pursue my studies in mathematics.

Eigenvalue development of a linear pencil evaluated at Haar unitaries — This animation shows the linear pencil \[ L_X(U_1,U_2)=I+X_1\otimes U_1+X_2\otimes U_2 \] evaluated at two independent \(d\times d\) Haar-distributed random unitaries. The figure illustrates the development of the eigenvalues as the unitary dimension \(d\) increases.

\[ X_1= \begin{pmatrix} 0.218011 & 0.218011+1.74409i\\ 0 & 0 \end{pmatrix}, \qquad X_2= \begin{pmatrix} 0 & 0.218011+1.74409i\\ 0.218011 & 0.436022 \end{pmatrix}. \]