Geodesic
Asymptotics of the Jordan normal form of a~random nilpotent matrix
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 252-262

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We study the Jordan normal form of an upper triangular matrix constructed from a random acyclic graph or a random poset. Some limit theorems and concentration results for the number and sizes of Jordan blocks are obtained. In particular, we study a linear algebraic analog of Ulam's longest increasing subsequence problem. 
@article{ZNSL_2016_448_a16,
     author = {F. V. Petrov and V. V. Sokolov},
     title = {Asymptotics of the {Jordan} normal form of a~random nilpotent matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {252--262},
     publisher = {mathdoc},
     volume = {448},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a16/}
}
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F. V. Petrov; V. V. Sokolov. Asymptotics of the Jordan normal form of a~random nilpotent matrix. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 252-262. http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a16/