Mathematics

The course will present an introduction to the theory of determinantal point processes (DPP) and its use for the solution to the problem of the length of the longest increasing sub- sequence in a large random permutation. A celebrated result, belonging to what is now called ”integrable probability” and first proved by Baik-Deift-Johansson in 1999, asserts that the fluctuation of this length around its average is asymptotically distributed according to the Tracy-Widom distribution, similarly to the largest eigenvalue of a random Hermitian Gauss- ian matrix. The proof of this result will be the common thread of the course that will cover the following topics: definition and properties of DPP, trace class operators and Fredholm determinants, Macchi-Soshnikov/Shirai-Takahashi existence Theorem for DPP; orthogonal polynomial ensembles from random matrix theory; Robinson-Schensted correspondence and poissonized Plancherel measure; symmetric functions, Okounkov’s theory of Schur measures, fermionic Fock space and Boson-Fermion correspondance; Wiener-Helson classification of shift invariant subspaces; asymptotic analysis of the discrete Bessel kernel and proof of the Baik-Deift-Johansson Theorem. The course requires basic knowledge in probability theory, functional analysis and complex analysis, while a substantial part of its content consists of algebraic combinatorics, for which no prerequite is needed.