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We consider the GUE minor process, where a sequence of GUE matrices is drawnfrom the corner of a doubly infinite array of i.i.d. standard normal variablessubject to the symmetry constraint. From each matrix, we take its largesteigenvalue, appropriately rescaled to converge to the standard Tracy-Widomdistribution. We show the analogue of the law of iterated logarithm for thissequence, i.e. we divide the normalized n-th eigenvalue by a logarithmic factorand show the limsup of this sequence is a constant almost surely. We also givealmost sure bounds for the appropriately scaled liminf.

Extremal eigenvalue fluctuations in the GUE minor process and the law of fractional logarithm