307 research outputs found
Longest increasing subsequences of random colored permutations
We compute the limit distribution for (centered and scaled) length of the
longest increasing subsequence of random colored permutations. The limit
distribution function is a power of that for usual random permutations computed
recently by Baik, Deift, and Johansson (math.CO/9810105). In two--colored case
our method provides a different proof of a similar result by Tracy and Widom
about longest increasing subsequences of signed permutations (math.CO/9811154).
Our main idea is to reduce the `colored' problem to the case of usual random
permutations using certain combinatorial results and elementary probabilistic
arguments.Comment: AMSTeX, 11 page
Universality of the Distribution Functions of Random Matrix Theory. II
This paper is a brief review of recent developments in random matrix theory.
Two aspects are emphasized: the underlying role of integrable systems and the
occurrence of the distribution functions of random matrix theory in diverse
areas of mathematics and physics.Comment: 17 pages, 3 figure
On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words
We consider the distributions of the lengths of the longest weakly increasing
and strongly decreasing subsequences in words of length N from an alphabet of k
letters. We find Toeplitz determinant representations for the exponential
generating functions (on N) of these distribution functions and show that they
are expressible in terms of solutions of Painlev\'e V equations. We show
further that in the weakly increasing case the generating function gives the
distribution of the smallest eigenvalue in the k x k Laguerre random matrix
ensemble and that the distribution itself has, after centering and normalizing,
an N -> infinity limit which is equal to the distribution function for the
largest eigenvalue in the Gaussian Unitary Ensemble of k x k hermitian matrices
of trace zero.Comment: 30 pages, revised version corrects an error in the statement of
Theorem
A Fredholm Determinant Identity and the Convergence of Moments for Random Young Tableaux
We obtain an identity between Fredholm determinants of two kinds of
operators, one acting on functions on the unit circle and the other acting on
functions on a subset of the integers. This identity is a generalization of an
identity between a Toeplitz determinant and a Fredholm determinant that has
appeared in the random permutation context. Using this identity, we prove, in
particular, convergence of moments for arbitrary rows of a random Young diagram
under Plancherel measure.Comment: 45 pages, ALX-LaTex, 9 figures, Lemma 2 (ii) is changed, |n|->n in
(2.7) and (2.8), new index system, Remark 2.3, 2.13 adde
On the average of the Airy process and its time reversal
We show that the supremum of the average of the Airy process and its time
reversal minus a parabola is distributed as the maximum of two independent GUE
Tracy-Widom random variables. The proof is obtained by considering a directed
last passage percolation model with a rotational symmetry in two different
ways. We also review other known identities between the Airy process and the
Tracy-Widom distributions.Comment: 12 page
Increasing Subsequences and the Classical Groups
We show that the moments of the trace of a random unitary matrix have combinatorial interpretations in terms of longest increasing subsequences of permutations. To be precise, we show that the 2n-th moment of the trace of a random k-dimensional unitary matrix is equal to the number of permutations of length n with no increasing subsequence of length greater than k. We then generalize this to other expectations over the unitary group, as well as expectations over the orthogonal and symplectic groups. In each case, the expectations count objects with restricted "increasing subsequence" length
On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations
The authors consider the length, , of the length of the longest
increasing subsequence of a random permutation of numbers. The main result
in this paper is a proof that the distribution function for , suitably
centered and scaled, converges to the Tracy-Widom distribution [TW1] of the
largest eigenvalue of a random GUE matrix. The authors also prove convergence
of moments. The proof is based on the steepest decent method for
Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 [DZ1] in the
context of integrable systems. The applicability of the Riemann-Hilbert
technique depends, in turn, on the determinantal formula of Gessel [Ge] for the
Poissonization of the distribution function of .Comment: 60 pages, 14 figures, AMS-LaTeX, typo correstions, new reference
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