This paper is concerned with certain connections between the ensemble of n×n unitary matrices - specifically the characteristic function of the random variable tr(U) - and combinatorics - specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups - and the appearance of Painlevé functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painlevé V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painlevé II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups.
Communications in Mathematical Physics
- Pub Date:
- Mathematics - Combinatorics;
- Mathematics - Probability;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- 21 pages, 1 figure. Revised paper simplifies the statement of Theorem 1 and adds some additional references