# Mathematics Colloquia and Seminars

### A Thinning'' Analogue of de Finetti's Theorem
I will present a simple generalization of de Finetti's theorem, wherein exchangeability -- i.e. permutation invariance -- is replaced by thinning-invariance''. Thinning'' is a random dynamics acting on n-tuples by deleting one component, at random, and left-justifying the remaining n-1 components, keeping their relative order fixed. This has applications to mean-field theory in 1-d spin systems when both the range of the interaction and the size of the lattice approach infinity together, but where the interaction may be asymmetric. Graduate students are welcome: no prior knowledge of de Finetti's theorem will be assumed.