The Paquette-Zeitouni law of fractional logarithms for the GUE minor process and the Plancherel growth process

Abstract

It is well-known that the largest eigenvalue of an $n\times n$ GUE matrix and the length of a longest increasing subsequence in a uniform random permutation of length $n$, both converge weakly to the GUE Tracy-Widom distribution as $n\to \infty$. We consider the sequences of the largest eigenvalues of the $n\times n$ principal minor of an infinite GUE matrix, and the the lengths of longest increasing subsequences of a growing sequence of random permutations (which, by the RSK bijection corresponds to the top row of the Young diagrams growing according to the Plancherel growth process), and establish laws of fractional logarithms for these. That is, we show that, under a further scaling of $(\log n)^{2/3}$ and $(\log n)^{1/3}$, the $\limsup$ and $\liminf$ respectively of these scaled quantities converge almost surely to explicit non-zero and finite constants. Our results provide complete solutions to two questions raised by Kalai in 2013. We affirm a conjecture of Paquette and Zeitouni (Ann. Probab., 2017), and give a new proof of $\limsup$, due to Paquette and Zeitouni (Ann. Probab., 2017), who provided a partial solution in the case of GUE minor process.