Integrable Probability

M2 Mathématiques avancées, parcours probabilités, Lyon 2018-2019/2019-2020

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Material

The course is based on chapters 1 and 2 of the book The Surprising Mathematics of Longest Increasing Subsequences by Dan Romik. It is available online here.

In the first part based on chapter 1, we stayed close to the book. For chapter 2 we are slightly diverging from it: here are some preliminary lecture notes (work in progress).

Exercises

Here are some exercises (in French) related to the material presented in the lectures.

Related papers

Here are some papers related to the topics of the course.

• D. Aldous and P. Diaconis, Hammersley's interacting particle process and longest increasing subsequences, Probab. Theory Related Fields 103 (1995), no. 2, 199–213.
• K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), no. 2, 437–476, arXiv:math/9903134 [math.CO]. See also Chapters 4 and 5 of Romik's book.
• A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003), no. 3, 581–603, arXiv:math/0107056 [math.CO].
• M. S. Kammoun, Monotonous subsequences and the descent process of invariant random permutations, Electron. J. Probab., Volume 23 (2018), paper no. 118, arXiv:1805.05253 [math.PR].
• A.-L. Basdevant and L. Gerin, Longest increasing paths with gaps, arXiv:1805.09136 [math.PR].
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