Rigidity of eigenvalues for $\beta$ ensemble in multi-cut regime

For a $\beta$ ensemble on $\Sigma^{(N)}=\{(x_1,\ldots,x_N)\mathbb R^N|x_1\le\cdots\le x_N\}$ with real analytic potential and general $\beta>0$, under the assumption that its equilibrium measure is supported on $q$ intervals where $q>1$, we prove the following rigidity property for its particles. First, in the bulk of the spectrum, with overwhelming probability, the distance between a particle and its classical position is of order $O(N^{-1+\epsilon})$. Second, if $k$ is close to 1 or close to $N$, i.e., near the extreme edges of the spectrum, then with overwhelming probability, the distance between the $k$-th largest particle and its classical position is of order $O(N^{-\frac{2}{3}+\epsilon}\min(k,N+1-k)^{-\frac{1}{3}})$. Here $\epsilon>0$ is an arbitrarily small constant. Our main idea is to decompose the multi-cut $\beta$ ensemble as a product of probability measures on spaces with lower dimensions and show that each of these measures is very close to a $\beta$ ensemble in one-cut regime for which the rigidity of particles is known.