Random permutations and the discrete Bessel kernel (Q2759650)

Let \(l_n(\sigma)\) denote the length of a longest increasing subsequence in a permutation \(\sigma\in S_n\), and let \(l_n\) denote the same length of a random permutation from \(S_n\). Whether the random variable \(L(\alpha)\) represents the Poissonization of \(l_n\), then the aim of this paper is to prove (again) the following formula NEWLINE\[NEWLINE\lim_{\alpha\to \infty}{\mathbf P}[L(\alpha)\leq 2\sqrt\alpha+ t\cdot\alpha^{1/6}]= F(t),\tag{\(\#\)}NEWLINE\]NEWLINE where \(F(t)\), \(t\in\mathbb{R}\), is referred as the Tracy-Widom distribution. The author outlines the basic steps of the proof for the \((\#)\) relation, the novel approach consisting in the avoidance of the use, within the proof of \((\#)\), of the Gessel's formula and Toeplitz determinants. The behaviour of \(l_n(\sigma)\), for large \(n\), as the largest eigenvalue of a large random Hermitian matrix explains why the largest eigenvalue distribution appears in \((\#)\).NEWLINENEWLINEFor the entire collection see [Zbl 0967.00059].