|Location:||MSRI: Simons Auditorium, Online/Virtual|
KPZ Models And Free Fermion At Finite Temperature
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Recently we discovered a direct connection between solvable models in the Kardar-Parisi-Zhang (KPZ) universality class and free fermionic models at positive temperature. In this talk we will explain this.
In 2000, Johansson showed that the current of the totally asymmetric simple exclusion process (TASEP) corresponds to a marginal of a free fermionic system at zero temperature described by the Schur measure. This was achieved using the Robinson-Schensted-Knuth (RSK) bijection and an immediate consequence is a Fredholm determinant formula for the current distribution which admits further asymptotic analysis.
A similar Fredholm determinant formula was found for KPZ equation in 2010 by a few different groups and also for its various discretized models in the following years. However they are usually found by calculations through non-free fermionic methods such as Bethe ansatz and Macdonald operators.
In this talk, we will present a direct connection between discretized models of the KPZ equation and free fermions at finite temperature. The key ingredient in our theory is a new identity between marginals of the q-Whittaker measure and the periodic Schur measure. The latter measure was first introduced by Borodin (2007) and studied recently in connection with free fermions by Betea and Buttier (2018). The identity is proved in a bijective fashion by generalizing and developing substantially the RSK algorithm, and studying its properties leveraging the theory of affine crystal.
We also discuss applications of this approach to a few KPZ models. In particular we plan to report on a symmetrized variant of the identity and its applications to the KPZ models in half spaces with Pfaffian point process.
The talk is based on collaborations with Takashi Imamura and Matteo Mucciconi.No Notes/Supplements Uploaded