### Combinatorial identities from an inhomogeneous Ising chain

Probability Seminar

26th January 2024, 3:30 pm – 4:30 pm

Fry Building, 2.04

In recent years we have seen that several intriguing combinatorial identities, old and new, can be obtained by considering correspondences between certain interacting particle systems. Among these results, using a comparison between ASEP and the zero-range process, Balázs and Bowen were able to obtain a new proof of the Jacobi triple product identity. A somewhat natural extension is to allow particles to interact with each other in a more general way. With this in mind we consider an Ising chain with inhomogeneous interactions and its mapping to a particle system with similarities to the zero-range process. This allows us to obtain new, and non-obvious, combinatorial identities relating to generating functions of certain types of partitions. Using the connection to the Ising model we are also able to obtain long-range reversible dynamics for a system of interacting particles on a half-infinite chain.

This is joint work with Jess Jay.

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