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BIRS Workshop Lecture Videos

Invariant measures for KdV and Toda-type discrete integrable systems Croydon, David


This talk is based on joint work with Makiko Sasada (University of Tokyo) and Satoshi Tsujimoto (Kyoto University). I will give a brief introduction to four discrete integrable systems, which are derived from the KdV and Toda lattice equations, and discuss some arguments that are useful in identifying invariant measures for them. As a first key input, I will describe how it is possible to construct global solutions for each of the systems of interest using variants of Pitman's transformation. Secondly, I will present a "detailed balance" criterion for identifying i.i.d.-type invariant measures, and will relate this to approaches used to study various stochastic integrable systems, such as last passage percolation, random polymers, and higher spin vertex models. In many of the examples I discuss, solutions to the detailed balance criterion are given by well-known characterizations of certain standard distributions, including the exponential, geometric, gamma and generalized inverse Gaussian distributions. Our work leads to a number of natural conjectures about the characterization of some other standard distributions.

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