For a β ensemble on with real analytic potential and general , under the assumption that its equilibrium measure is supported on q intervals where , we prove the following rigidity property for its particles.
1. In the bulk of the spectrum, with overwhelming probability, the distance between a particle and its classical position is of order .
2. If k is close to 1 or close to N, that is, near the extreme edges of the spectrum, then with overwhelming probability, the distance between the kth largest particle and its classical position is of order .
Here is an arbitrarily small constant. Our main idea is to decompose the multi-cut β ensemble as a product of probability measures on spaces with lower dimensions and show that each of these measures is very close to a β ensemble in one-cut regime for which the rigidity of particles is known.