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Integrable lattices: Random matrices and random permutations. (English) Zbl 0987.15014
Bleher, Pavel (ed.) et al., Random matrix models and their applications. Based on talks and lectures from the workshop, Berkeley, CA, USA, February 22-26, 1999. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 40, 321-406 (2001).
Introduction: The purpose of this article is to survey recent interactions between statistical questions and integrable theory. Two types of questions will be tackled here:
(i) Consider a random ensemble of matrices, with certain symmetry conditions to guarantee the reality of the spectrum and subjected to a given statistics. What is the probability that all its eigenvalues belong to a given subset \(E\)? What happens, when the size of the matrices gets very large? The probabilities here are functions of the boundary points \(c_i\) of \(E\).
(ii) What is the statistics of the length of the largest increasing sequence in a random permutation, assuming each permutation is equally probable? Here, one considers generating functions (over the size of the permutations) for the probability distributions, depending on the variable \(x\).
The main emphasis of this article is to show that integrable theory serves as a useful tool for finding equations satisfied by these functions of \(x\), and conversely the probabilities point the way to new integrable systems.
These questions are all related to integrals over spaces of matrices. Such spaces can be classical Lie groups or algebras, symmetric spaces or their tangent spaces. In infinite-dimensional situations, the “\(\infty\)-fold” integrals get replaced by Fredholm determinants.
During the last decade, astonishing discoveries have been made in a variety of directions. A first striking feature is that these probabilities are all related to Painlevé equations or interesting generalizations. In this way, new and unusual distributions have entered the statistical world.
Another feature is that each of these problems is related to some integrable hierarchy. Indeed, by inserting an infinite set of time variables \(t_1,t_2,t_3, \dots\) in the integrals or Fredholm determinants – e.g., by introducing appropriate exponentials \(e^{\sum_1^\infty t_iy^i}\) in the integral – this probability, as a function of \(t_1,t_2,t_3\dots\), satisfies an integrable hierarchy. Korteweg-de Vries, KP, Toda lattice equations are only a few examples of such integrable equations.
Typically integrable systems can be viewed as isospectral deformations of differential or difference operators \({\mathcal L}\). Perhaps, one of the most startling discoveries of integrable theory is that \({\mathcal L}\) can be expressed in terms of a single “\(\tau\)-function” \(\tau(t_1, t_2, \dots)\) (or vector of \(\tau\)-functions), which satisfy an infinite set of nonlinear equations, encapsulated in a single “bilinear identity”. The \(t_i\) account for the commuting flows of this integrable hierarchy. In this way, many interesting classical functions live under the same hat: characters of representations, \(\Theta\)-functions of algebraic geometry, hypergeometric functions, certain integrals over classical Lie algebras or groups, Fredholm determinants, arising in statistical mechanics, in scattering and random matrix theory! They are all special instances of “\(\tau\)-functions”.
The point is that the probabilities or generating functions above, as functions of \(t_1,t_2, \dots\) (after some minor renormalization) are precisely such \(\tau\)-functions for the corresponding integrable hierarchy and thus automatically satisfy a large set of equations.
These probabilities are very special \(\tau\)-functions: they happen to be a solution of yet another hierarchy of (linear) equations in the variables \(t_i\) and the boundary points \(c_i\), namely \(\mathbb{J}_k^{(2)} \tau(t;c)=0\), where the \(\mathbb{J}_k^{(2)}\) form – roughly speaking – a Virasoro-like algebra: \[ [\mathbb{J}_k^{(2)}, \mathbb{J}_l^{(2)}]= (k-l) \mathbb{J}_{k+l}^{(2)}+ \cdots. \] Each integrable hierarchy has a natural “vertex operator”, which automatically leads to a natural Virasoro algebra. Then, eliminating the partial derivatives in \(t\) from the two hierarchies of equations, the integrable and the Virasoro hierarchy, and finally setting \(t=0\), lead to partial differential equations or ordinary differential equations satisfied by the probabilities.
For the entire collection see [Zbl 0967.00059].

MSC:
15B52 Random matrices (algebraic aspects)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K60 Lattice dynamics; integrable lattice equations
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