25 research outputs found
An Erd\H{o}s--Hajnal analogue for permutation classes
Let be a permutation class that does not contain all layered
permutations or all colayered permutations. We prove that there is a constant
such that every permutation in of length contains a
monotone subsequence of length
Rectangular Young tableaux and the Jacobi ensemble
It has been shown by Pittel and Romik that the random surface associated with
a large rectangular Young tableau converges to a deterministic limit. We study
the fluctuations from this limit along the edges of the rectangle. We show that
in the corner, these fluctuations are gaussian wheras, away from the corner and
when the rectangle is a square, the fluctuations are given by the Tracy-Widom
distribution. Our method is based on a connection with the Jacobi ensemble
Skew Howe duality and random rectangular Young tableaux
We consider the decomposition into irreducible components of the external
power regarded as a
-module. Skew Howe duality
implies that the Young diagrams from each pair which
contributes to this decomposition turn out to be conjugate to each other,
i.e.~. We show that the Young diagram which corresponds
to a randomly selected irreducible component has the same
distribution as the Young diagram which consists of the boxes with entries
of a random Young tableau of rectangular shape with rows and
columns. This observation allows treatment of the asymptotic version of this
decomposition in the limit as tend to infinity.Comment: 17 pages. Version 2: change of title, section on bijective proofs
improve
Asymptotics of Plancherel-type random partitions
We present a solution to a problem suggested by Philippe Biane: We prove that
a certain Plancherel-type probability distribution on partitions converges, as
partitions get large, to a new determinantal random point process on the set
{0,1,2,...} of nonnegative integers. This can be viewed as an edge limit
ransition. The limit process is determined by a correlation kernel on
{0,1,2,...} which is expressed through the Hermite polynomials, we call it the
discrete Hermite kernel. The proof is based on a simple argument which derives
convergence of correlation kernels from convergence of unbounded self-adjoint
difference operators.
Our approach can also be applied to a number of other probabilistic models.
As an example, we discuss a bulk limit for one more Plancherel-type model of
random partitions.Comment: AMS TeX, 19 pages. Version 2: minor typos fixe
Limit shapes of evacuation and jeu de taquin paths in random square tableaux
We consider large random square Young tableaux and look for typical (in the
sense of probability) jeu de taquin paths and evacuation paths in the
asymptotic setting. We show that the probability distribution of such paths
converges to a random meridian connecting the opposite corners of the square.Comment: 12 pages. Extended abstract of FPSAC 2020 conference. A full version
of this extended abstract will be available soon as a separate pape