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We study the asymptotic expansion in $n$ for the partition function of$\beta$ matrix models with real analytic potentials in the multi-cut regime upto the $O(n^{-1})$ terms. As a result, we find the limit of the generatingfunctional of linear eigenvalue statistics and the expressions for theexpectation and the variance of linear eigenvalue statistics, which in thegeneral case contain the quasi periodic in $n$ terms.

Fluctuations of linear eigenvalue statistics of $\beta$ matrix models in the multi-cut regime