Seminar Gibbs measures on graded graphs

The seminar is devoted to the study of Gibbs (or central) measures on graded graphs. The general theory has appeared in works of Vershik, Kerov, and Olshanski and continues to develop nowadays. On top of its intrinsic beauty, it has connections with classical probability questions, combinatorics, dynamical systems, representation theory, and even biology.

We will start with earlier (and more elementary) examples related to de Finetti's theorem and Kingman's partition structures, and their variations.

If this will go well, we will discuss more complicated settings: the Gibbs measures on the Young graph and the Gelfand-Tsetlin graph.

Friday 14:00 -- 16:00, room 0.007 , Endenicher Allee 60

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Schedule

5 April -- Introduction ( by Alexey)

12 April -- no seminar

19 April --Holiday

26 April -- no seminar

3 May -- Pascal's graph and de Finetti's theorem ( Alexey)

10 May -- Kingman partition structures: motivation, definition, examples

17 May -- Poisson-Dirichlet process

24 May -- Kingman partition structures: classification

31 May -- Symmetric functions and branching graphs.

7 June -- general de Finetti's theorem

14 June -- Holiday

21 June -- Coherent measures on Young's graph: motivation, examples.

28 June -- no seminar

5 July --- TBA

12 July -- no seminar

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Possible topics

0) Introduction (by Alexey)

1) de Finetti's theorem and Pascal graph. ( https://projecteuclid.org/euclid.aop/1176995566 , .... )

2) Kingman's partition structures: motivation, definition, examples. ( Sections 2 and 3 of https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1978.0089 )

3) Kingman's partition structures: classification ( https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-18.2.374 )

4) Poisson-Dirichlet process.

5) Symmetric functions and branching. Kingman's branching graph.

6) Young's graph. Classification of Gibbs measures.

7) RSK algorithm. Sampling of Gibbs measures on Young's graph.

8) Gelfand-Tsetlin graph. Classification of Gibbs measures.

9) Gibbs measures and total positivity.