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In this paper, we prove a polynomial Central Limit Theorem for severalintegrable models, and for the $\beta$-ensembles at high-temperature withpolynomial potential. Furthermore, we connect the mean values, the variancesand the correlations of the moments of the Lax matrices of these integrablesystems with the ones of the $\beta$-ensembles. Moreover, we show that thelocal functions' space-correlations decay exponentially fast for the consideredintegrable systems. For these models, we also established a Berry--Esseen typebound.

CLT for $\beta$ ensembles at high-temperature, and for integrable systems: a transfer operator approach