Author(s):  
Leonid V Bogachev ◽  
Zhonggen Su

We obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the ‘spectrum’ of partitions λ ⊢ n ∈ (under the Plancherel measure), thus settling a long-standing problem posed by Logan & Shepp. Namely, under normalization growing like , the corresponding random process in the bulk is shown to converge, in the sense of finite-dimensional distributions, to a Gaussian process with independent values, while local correlations in the vicinity of each point, measured on various power scales, possess certain self-similarity. The proofs are based on the Poissonization techniques and use Costin–Lebowitz–Soshnikov's central limit theorem for determinantal random point processes. Our results admit a striking reformulation after the rotation of Young diagrams by 45°, whereby the normalization no longer depends on the location in the spectrum. In addition, we explain heuristically the link with an earlier result by Kerov on the convergence to a generalized Gaussian process.


10.37236/929 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Piotr Šniady

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.


10.37236/315 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Grigori Olshanski

Let $M_n$ stand for the Plancherel measure on ${\Bbb Y}_n$, the set of Young diagrams with $n$ boxes. A recent result of R. P. Stanley (arXiv: 0807.0383) says that for certain functions $G$ defined on the set ${\Bbb Y}$ of all Young diagrams, the average of $G$ with respect to $M_n$ depends on $n$ polynomially. We propose two other proofs of this result together with a generalization to the Jack deformation of the Plancherel measure.


2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Valentin Féray ◽  
Maciej Dołęga

International audience We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $\alpha$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $\alpha$. Secondly, we describe asymptotically the shape of random Young diagrams under some deformation of Plancherel measure.


Author(s):  
PETER SPACEK

AbstractIn this article we construct Laurent polynomial Landau–Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau–Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.


1984 ◽  
Vol 266 (4) ◽  
pp. 513-518 ◽  
Author(s):  
L. Baggett
Keyword(s):  

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