Title of lectures

Random matrix models and integrable systems

Lecturer:

Craig TRACY, U.C., Davis

Outline:

These lectures begin with an introduction to random matrix theory and their connection with integrable systems of the Painlevé type. Following [1], we give newly simplified proofs of the representation of some of the basic distributions in terms of Fredholm determinants. These ideas will be illustrated in the context of the distribution of the largest eigenvalue in the three ensembles GOE, GUE and GSE. In the edge scaling limit, these distributions are all expressible in terms of a certain Painlevé II transcendent [2-5]. These same distribution functions arise in combinatorics [6,7,8] and random growth models [9]. Some of these last connections will be discussed along with the needed fundational combinatorics, e.g. Ch. 7 in [10].

Lectures will be taken from the references below. Obviously, we will not cover the content of each paper, but rather provide the student with an overview of the issues involved. Detailed proofs will be given in the case of the Airy kernel following [5].

Lectures 1 & 2:
We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painleve II. Our goal in these two lectures is to concentrate on this important example of the connection between random matrix theory and integrable systems, and in so doing to introduce the newcomer to the subject as a whole. These two lectures will follow the preprint solv-int/9901004 on the Los Alamos Archives.

Lectures 3 & 4:
Applications of random matrix theory to combinatorial problems of the Robinson-Schensted-Knuth (RSK) type. We explain the RSK correspondence between random permutations (or random words) and Young tableaux. Using this correspondence we show that certain Toeplitz determinants are the generating functions for the distribution function of the length of the longest increasing subsequence in random permutations/words. The connection of these distributions with those of Lectures 1 & 2 will be explained. These lectures will be of an overview nature. Basic references are work of Baik, Deift, Johansson, Rains and others that can be found on the Mathematics section of the Los Alamos Archives.

References:

[1] Tracy, C.A. and Widom, H., Correlation functions, cluster functions and spacing distributions for random matrices, J. Stat. Phys. 92 (1998), 809-835.
[2] Tracy, C.A. and Widom, H., Level-spacing distributions and the Airy kernel, Commun. Math. Phys. 159 (1994), 151-174.
[3] Tracy, C.A. and Widom, H., Fredholm determinants, differential equations and matrix models, Commun. Math. Phys. 163 (1994), 33-72.
[4] Tracy, C.A. and Widom, H., On orthogonal and symplectic matrix ensembles, Commun. Math. Phys. 177 (1996), 727-754.
[5] Tracy, C.A. and Widom, H., Airy kernel and Painlevé II, LANL archives, solv-int/9901004.
[6] Baik, J., Deift, P., and Johansson, K., On the distribution of the length of the longest increasing subsequence of random permutations, LANL archives, math.CO/9810105.
[7] Baik, J., Deift, P., and Johansson, K., On the distribution of the length of the second row of a Young diagram under Plancherel measure, LANL Archives, math.CO/9901118.
[8] Tracy, C.A. and Widom, H., Random unitary matrices, permutations and Painlevé, LANL archives, math.CO/9811154.
[9] Johansson, K., Shape fluctuations and random matrices, preprint. LANL archives, math.CO/9903134.
[10] Stanley, R., Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999.

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