The law of iterated logarithm asserts that if $x_1,x_2,\dots$ are i.i.d $\cal N(0,1)$ random variables and $S_n=x_1+x_2+\cdots+x_n$, then $$\limsup_{n \to \infty} S_n/\sqrt {n \log \log n} = \sqrt 2, $$ almost surely.

##1. Random matrices

Let $(x_{ij})_{1 \le i < \infty, 1 \le j \le i }$ be an array of i.i.d random $\cal N(0,1)$ variables, define $x_{ji}=x_{ij}$ and let $M_n$ be the $n$ by $n$ matrix obtained by restricting the array to $1 \le i \le n, 1 \le j \le n$.

Let $\lambda(n)$ denote the maximum eigenvalue of $M_n$. The variance and limiting distribution of the maximum eigenvalue (for these and related classes of matrices) were discovered by Tracy and Widom.

Question 1: What is the law of iterated logarithm for this scenario? Namely, can one identifies functions $F(n)$ and $G(n)$ such that

$\limsup_{n \to \infty} (\lambda (n)- {\bf E}(\lambda(n))/ F(n) = 1,$ almost surely, and

$\liminf_{n \to \infty} (\lambda (n)- {\bf E}(\lambda(n))/ G(n) = -1$ almost surely.

##2. Random permutations

Choose a sequence of real numbers in the unit interval at random independently. Use the first $n$ numbers in the sequence to describe a random permutation $\pi$ in $S_n$. Look at the length $L_n$ of the maximum increasing sub-sequence of $\pi$. The expectation, variance and limiting distributions of the maximal increasing subsequence are known by the works of Logan-Shepp, Kerov-Vershik, (expectation) Baik, Deift & Johansson (variance and limiting distribution) and many subsequent works.

Question 2: What is the law of iterated logarithm for this scenario?

##3. Motivation.

My initial motivation is to try to understand the law of iterated logarithm itself, and more precisely, to understand why do we have the $\sqrt {\log \log n}$ factor. (I don't ecpect to understand the $\sqrt 2$.)