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Symmetrized random permutations. (English) Zbl 0989.60010

Bleher, Pavel (ed.) et al., Random matrix models and their applications. Based on talks and lectures from the workshop, Berkeley, CA, USA, February 22-26, 1999. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 40, 1-19 (2001).
Summary: Selecting \(N\) random points in a unit square corresponds to selecting a random permutation. Placing symmetry restrictions on the points, we obtain special kinds of permutations: involutions, signed permutations and signed involutions. We are interested in the statistics of the length (in numbers of points) of the longest up/right path in each symmetry type as the number of points increases to infinity. The limiting distribution functions are expressed in terms of a Painlevé II equation. In addition to the Tracy-Widom distributions of random matrix theory, we also obtain two new classes of distribution functions interpolating between the GOE and GSE, and between the GUE and \(\text{GOE}^2\) Tracy-Widom distribution functions. Applications to random vicious walks and site percolation are also discussed.
For the entire collection see [Zbl 0967.00059].

MSC:

60C05 Combinatorial probability
05A05 Permutations, words, matrices
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