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Abstract: Stationary measures of a variety of growth models in the Kardar-Parisi-Zhang class can be described in terms of couples of interacting random walks or Brownian motions, called two-layer Gibbs measures. The Boltzman weights used to define them originate in branching rules satisfied by families of symmetric functions in the Macdonald hierarchy (Schur polynomials, Hall-Littlewood functions, (q)-Whittaker...). This framework applies to various models: interacting particle systems between boundary reservoirs, directed polymer models in a strip, last passage percolation, the stochastic six-vertex model. This talk will present an overview of the method, connecting with other approaches such as the Matrix Product Ansatz and Askey-Wilson processes. Time permitting, we will present recent developments related to random matrices.
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Abstract: Let's randomly choose a permutation of size 𝑁 uniformly and focus on observables such as the length of the longest increasing subsequence, the number of descents, the number of cycles of a given size, etc. The asymptotic behavior of these observables as 𝑁 becomes very large is well understood. In particular, it is easy to show that the joint distribution of traces of powers of permutation matrices is asymptotically of type Poisson. Now, if we consider not just a permutation, but a word formed by several independent uniform permutations, we know from the works of Nica, Puder, et al. that the asymptotic behavior of the trace of the word depends on the algebraic properties of the word considered. In this talk, I will review the uniform case and present a generalization to conjugacy invariant permutations. This talk is based on joint work with Mylène Maïda.
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