Random Fibonacci words via clone Schur functions

• Jeanne Scott (University of Minnesota)

The Young lattice of integer partitions is well known for its role in representation theory and the study of symmetric functions. It is simultaneously the Bratelli diagram of the symmetric groups, expressing how irreducible representations of $S_n$ restrict to $S_{n-1}$, and also responsible for how Schur functions multiply --- as distilled in the celebrated Littlewood-Richardson rule. In 1988 R. Stanley introduced a similar lattice called the Young-Fibonacci (YF) lattice, which consists of Fibonacci words: i.e. binary words made of the digits 1 and 2 and ranked according to the sum of their digits. In 1994 Okada showed that the YF-lattice is the Bratelli diagram of a tower of semi-simple algebras and also responsible for the YF-analogue of a Littlewood-Richardson rule governing multiplication of "clone" Schur functions. In joint work with L. Petrov, we introduce a system of coherent measures on the YF-lattice using certain positive biserial specializations of Okada's clone Schur functions. We characterize these specializations using the theory of totally positive tridiagonal matrices and describe the asymptotic behaviour of random Fibonacci words sampled with respect to the associated coherent measures. Our results have connections with Stieltjes moment sequences, orthogonal polynomials from the (q-)Askey scheme, and residual allocation (stick-breaking) models.

Multipoint correlation functions in the 1+1 dimensional quantum Sinh-Gordon model

• Alex Simon (ENS de Lyon)

Since their first heuristic constructions, the quantum field theories have been able to provide the physicists with numerous, albeit non-rigorous, results. For a few particular theories, mostly free and perturbative theories, rigorous constructions have been made in the last decades. However, so far, some observed phenomena such as the universality of the behaviour of the correlation functions in the ultraviolet regime are still out of reach for these constructions. In this talk, we use the bootstrap program to construct a mathematically rigorous and explicit theory for the 1+1 dimensional quantum Sinh-Gordon model which is one of the simplest integrable models. This could then hopefully provide a starting point for tackling the universality questions. As we will see, the study of the multipoint correlation functions leads us to the study of N-fold integrals whose asymptotic behaviour is crucial to find. This talk is based on a joint work with K. Kozlowski.