Dr. D. Croydon
Invariant measures for KdV and Toda-type discrete integrable systems
I will describe how to construct infinite versions of four
well-studied discrete integrable models (namely the ultra-discrete KdV equation,
the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation),
and discuss some arguments that are useful in identifying invariant measures for them.
In each case, the current configuration is represented as a certain path encoding,
and the dynamics of the system can be understood in terms of a version of Pitman’s
transformation (i.e. reflection in the past maximum), which is well-known amongst probabilists.
For identifying spatially homogeneous and independent random configurations that are invariant
under the dynamics in particular, I will present a ‘detailed balance’ criterion,
which is also relevant for various stochastic integrable systems.
This talk is based on joint work with Makiko Sasada (University of Tokyo) and Satoshi Tsujimoto (Kyoto University).
Tuesday 9 March 2021, 1PM Sydney Time