Publisher’s Pick: Proving duality for the two-component asymmetric simple exclusion process.

By Phil Brown on October 10, 2017

In this paper Jeffrey Kuan uses symmetry of quantum groups to prove duality for the two-component asymmetric simple exclusion process. He spoke to Journal of Physics A: Mathematical and Theoretical as about his Publisher’s Pick

Who are you?

Jeffrey Kuan, Postdoc in Mathematics at Columbia University.

Jeffrey Kuan

What prompted you to pursue this field of research?

My original background is in representation theory. I’m fascinated by intersections between different disciplines, so I moved into probability and mathematical physics, with representation theory in mind. Because of this, my postdoc advisor suggested that I look at probabilistic interacting particle systems arising from higher rank Lie Algebras, which he conjectured would be multi-component generalizations. Apparently he was right!

What is this latest paper all about?

This paper uses a quantization of the Lie algebras gl₃ and sp₄ to construct two different random interacting particle systems, each with two types of particles. It turns out that one of them is simply the Asymmetric Simple Exclusion Process with second—class particles, which had been introduced by Liggett about 30 years ago. The construction uses a Hamiltonian which arises from a Casimir element of the quantized Lie algebra. By choosing certain nice elements which by definition commute with the Casimir, this algebraic framework leads to stochastic duality with an explicit duality function. The duality statement can be understood directly (that is, without reference to any algebra), but would be more difficult to discover using only probability.

What do you plan to do next?

I would generalize to different Lie algebras and representations.

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