1,594 research outputs found
Random Words, Toeplitz Determinants and Integrable Systems. I
It is proved that the limiting distribution of the length of the longest
weakly increasing subsequence in an inhomogeneous random word is related to the
distribution function for the eigenvalues of a certain direct sum of Gaussian
unitary ensembles subject to an overall constraint that the eigenvalues lie in
a hyperplane.Comment: 15 pages, no 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 Card Shuffling Analysis of Deformations of the Plancherel Measure of the Symmetric Group
We study deformations of the Plancherel measure of the symmetric group by
lifting them to the symmetric group and using combinatorics of card shuffling.
The existing methods for analyzing deformations of Plancherel measure are not
obviously applicable to the examples in this paper. The main idea of this paper
is to find and analyze a formula for the total variation distance between
iterations of riffle shuffles and iterations of "cut and then riffle shuffle".
Similar results are given for affine shuffles, which allow us to determine
their convergence rate to randomness
On the limiting law of the length of the longest common and increasing subsequences in random words
Let and be two sequences of independent
and identically distributed (iid) random variables taking their values,
uniformly, in a common totally ordered finite alphabet. Let LCI be the
length of the longest common and (weakly) increasing subsequence of and . As grows without bound, and when properly
centered and normalized, LCI is shown to converge, in distribution, towards
a Brownian functional that we identify.Comment: Some corrections from the published version are provided, some typos
are also correcte
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