An up-down chain is a Markov chain in which each transition can be decomposed into a growth step followed by a reduction step. These chains often exhibit nice combinatorial, algebraic, and asymptotic properties, but it is generally not understood how this structure arises.

In the first part of this talk, we will shed light on this mystery by presenting a general framework for analyzing up-down chains. Our approach will mainly be algebraic but will lead to scaling limits.

Afterwards, we demonstrate our theory by discussing an example on permutations. This example gives rise to a new family of permutons. Other settings where our framework can be applied include integer partitions, integer compositions, trees, graphs, and words.

Based on joint work with Valentin Féray.