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Symmetric functions and random partitions. (English) Zbl 1017.05103

Fomin, S. (ed.), Symmetric functions 2001: Surveys of developments and perspectives. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, June 25-July 6, 2001. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 74, 223-252 (2002).
This is a paper, based on three lectures delivered by the author at the Newton Institute in Cambridge in autumn 2001. Its subject matter is the very active subject of random partitions and their asymptotics. The material has links with eigenvalues of random Hermitian matrices (which in turn links to mathematical physics, the Riemann zeta function, etc.) and with longest increasing or decreasing subsequences of random permutations, amongst other topics. The material of the paper is technical: this review only attempts to give the basic ideas and some flavour of the central results.
The starting point is a partition \(\lambda\) of \(N\): that is, a non-increasing sequence \(\lambda_{1}\geq \lambda_{2}\geq \cdots\geq 0\) of non-negative integers with \(\sum_{i}\lambda_{i}=N\) and \(\lambda_{i}=0\) for all large enough \(i\). Think of the partition as a tableau, whose bottom left-hand corner is at the origin, with \(\lambda_{1}\) unit squares in the bottom row, \(\lambda_{2}\) in the second bottom row and so on. (All rows start at the \(x\)-coordinate 0 and move right.) For example, if \(N=9\) and \(\lambda=(4,2,2,1)\) we have a first row with 4 squares in it, then the second row has two squares above the first two squares (from left to right) of the first row, the third row is a copy of the second row, and the fourth row is just one square above the first (from left to right) square of the third row. The boundary of this tableau consists of the lines from \((4,0)\) to \((4,1)\), \((4,1)\) to \((2,1)\), \((2,1)\) to \((2,3)\), \((2,3)\) to \((1,3)\), \((1,3)\) to \((1,4)\) and \((1,4)\) to \((0,4)\) and \((0,4)\) to \((0,\infty)\); the boundary of a general tableau is defined analogously. Our measure on partitions is the Plancherel measure: the probability of getting \(\lambda\) is \((\dim \lambda)^{2}/N!\) where \(\dim\lambda\) is the dimension of the corresponding irreducible representation of the symmetric group \(S_{N}\). (Many other interpretations of \(\dim \lambda\) exist.)
Translate the diagram through 45 degrees anticlockwise, and rescale both axes by multiplying by \(1/\sqrt{N}\): thus the resulting shape has area 1. The boundary of the tableau is thus transformed into a piecewise linear function, some parts having positive gradient (‘ups’) and others negative gradient (‘downs’). (In our example, the ‘downs’, were the rotated images of the lines \((4,0)\) to \((4,1)\), \((2,1)\) to \((2,3)\), \((1,3)\) to \((1,4)\) and \((0,4)\) to \((0,\infty)\).) The idea is to investigate so-called correlation functions, which are the probability of observing (say) several consecutive ‘ups’or several consecutive ‘downs’. It turns out that the probability of observing downs starting at each of a (suitable) finite set of points \(X=\{x_{1},\dots, x_{n}\}\subseteq \mathbb{Z} +\frac{1}{2}\) converges (as \(N\rightarrow \infty\)) to the determinant of a certain square matrix whose \((i,j)\) entry is a certain ‘kernel’ function \(K(x_{i}-x_{j},\phi_{i})\) for certain functions \(\phi_{i}\).
A very similar formula arises in the theory of eigenvalues of random Hermitian matrices. The similarity transpires to be (much) more than purely formal. Indeed, in the \(N\rightarrow\infty\) limit, \[ \frac{\lambda_{i}-2\sqrt{N}}{N^{1/6}} \] converge in joint distribution to a certain so-called Airy ensemble (related to solutions of the Airy differential equation). This had been conjectured by Baik, Deift and Johansson as an analogue of a now famous result of theirs for random matrices. The correlation functions of the Airy ensemble are given by a determintal formula similar to the one discussed in the previous paragraph, albeit for a different kernel function.
In fact even more is true: there are exact formulae for the correlation functions in the random partitions case, from which these asymptotic results can be derived, and this is the subject matter of section 2. What is proved is that the Poissonization of the Plancherel measure has correlation functions which are determinants of a suitable kernel function related to Bessel functions, which can be expressed as a double integral. The proof is a substantial calculation with a quantum-mechanical flavour, symmetric functions playing a key role. Once one has this exact formula, one can use the method of steepest descent on the double integral to recover the asymptotics – this is the subject of section 3.
For the entire collection see [Zbl 0997.00015].

MSC:

05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
60C05 Combinatorial probability
15B52 Random matrices (algebraic aspects)