An Erd\H{o}s--Hajnal analogue for permutation classes

Let $\mathcal{C}$ be a permutation class that does not contain all layered permutations or all colayered permutations. We prove that there is a constant $c$ such that every permutation in $\mathcal{C}$ of length $n$ contains a monotone subsequence of length $cn$

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 $\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young diagrams from each pair $(\lambda,\mu)$ which contributes to this decomposition turn out to be conjugate to each other, i.e.~$\mu=\lambda'$. We show that the Young diagram $\lambda$ which corresponds to a randomly selected irreducible component $(\lambda,\lambda')$ has the same distribution as the Young diagram which consists of the boxes with entries $\leq p$ of a random Young tableau of rectangular shape with $m$ rows and $n$ columns. This observation allows treatment of the asymptotic version of this decomposition in the limit as $m,n,p\to\infty$ 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