The distributions of the entries of Young tableaux

Retrieved November 2, 2007 from http://xxx.lanl.gov/find/grp_math/1/au:+Morse_J/0/1/0/all/0/1.Let T be a standard Young tableau of shape λ ⊢ k. We show that the probability that a randomly chosen Young tableau of n cells contains T as a subtableau is, in the limit n → ∞, equal to f_/k!, where f_ is the number of all tableaux of shape λ. In other words, the probability that a large tableau contains T is equal to the number of tableaux whose shape is that of T , divided by k!. We give several applications, to the probabilities that a set of prescribed entries will appear in a set of prescribed cells of a tableau, and to the probabilities that subtableaux of given shapes will occur. Our argument rests on a notion of quasirandomness of families of permutations, and we give sufficient conditions for this to hold

The oriented swap process and last passage percolation

We present new probabilistic and combinatorial identities relating three random processes: the oriented swap process on $n$ particles, the corner growth process, and the last passage percolation model. We prove one of the probabilistic identities, relating a random vector of last passage percolation times to its dual, using the duality between the Robinson-Schensted-Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of 'last swap times' in the oriented swap process, is conjectural. We give a computer-assisted proof of this identity for $n\le 6$ after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence. The conjectural identity provides precise finite-$n$ and asymptotic predictions on the distribution of the absorbing time of the oriented swap process, thus conditionally solving an open problem posed by Angel, Holroyd and Romik.Comment: 36 pages, 6 figures. Full version of the FPSAC 2020 extended abstract arXiv:2003.0333

Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation

We study the combinatorial structure of the irreducible characters of the classical groups ${\rm GL}_{n}(\mathbb{C})$, ${\rm SO}_{2n+1}(\mathbb{C})$, ${\rm Sp}_{2n}(\mathbb{C})$, ${\rm SO}_{2n}(\mathbb{C})$ and the "non-classical" odd symplectic group ${\rm Sp}_{2n+1}(\mathbb{C})$, finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of ${\rm Sp}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$ and between characters of ${\rm SO}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$. We identify the first family as a one-parameter specialization of Koornwinder polynomials, for which we thus provide a novel combinatorial structure; on the other hand, the second family appears to be new. We next develop a method of Gelfand-Tsetlin pattern decomposition to establish identities between all these polynomials that, in the case of characters, can be viewed as describing the decomposition of irreducible representations of the groups when restricted to certain subgroups. Through these formulas we connect orthogonal and symplectic characters, and more generally the interpolating polynomials, to LPP models with various symmetries, thus going beyond the link with classical Schur polynomials originally found by Baik and Rains [BR01a]. Taking the scaling limit of the LPP models, we finally provide an explanation of why the Tracy-Widom GOE and GSE distributions from random matrix theory admit formulations in terms of both Fredholm determinants and Fredholm Pfaffians.Comment: 60 pages, 11 figures. Typos corrected and a few remarks adde

Poisson limit theorems for the Robinson-Schensted correspondence and for the multi-line Hammersley process

We consider Robinson-Schensted-Knuth algorithm applied to a random input and study the growth of the bottom rows of the corresponding Young diagrams. We prove multidimensional Poisson limit theorem for the resulting Plancherel growth process. In this way we extend the result of Aldous and Diaconis to more than just one row. This result can be interpreted as convergence of the multi-line Hammersley process to its stationary distribution which is given by a collection of independent Poisson point processes.Comment: version 2: 41 pages, the proofs are now more detaile

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