Connecting q-Whittaker and periodic Schur measures
Location
Zoom link: https://stanford.zoom.us/j/91274693500?pwd=bmtHZnhTMG1HQ3pTOHYxUWJ2Z1ZjQT09
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The q-Whittaker measure has been playing an important role in integrable probability. It describes the position of a particle in discrete models in the KPZ class such as the q-(Push)TASEP. Though the q-Whittaker measure is not directly associated with a determinantal point process(DPP), the q-Laplace transform of its marginal is written as a Fredholm determinant through a few methods, which allows establishing Tracy-Widom asymptotics.
On the other hand, the periodic Schur measure was first introduced by Borodin in 2007. Its shift mix version is a DPP and its correlation functions can be studied in a standard manner. It is also intimately related to a free fermion at finite temperature.
In our recent works [1,2], we have proved an identity between marginals of the two measures. In [1] the identity was shown by a matching of Fredholm determinants while in [2] it was proved bijectively by generalizing and developing substantially the RSK algorithm, and studying its properties leveraging the theory of affine crystal.
Our identity presents a direct connection between discretized models of the KPZ equation and free fermions at finite temperature, providing a new approach to study KPZ models.
In the first part of the talk, we will give an overview of our new results. Details of the most novel part of our results, namely the bijective proof of the identity, will be explained in the second talk.
References:
[1] Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, Identity between restricted Cauchy sums for the q-Whittaker and skew Schur polynomals, arXiv: 2106.11913
[2] Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, Skew RSK dynamics: Greene invariants, affine crystals and applications to q-Whittaker polynomials, arXiv: 2106.11922