Fractional exponential decay in the forbidden region for Toeplitz operators

We prove several results of concentration for eigenfunctions in Toeplitz quantization. With mild assumptions on the regularity, we prove that eigenfunctions are $O(exp(-c{N}^{\delta }))$ away from the corresponding level set of the symbol, where N is the inverse semiclassical parameter and $0 < \delta < 1$ depends on the regularity. As an application, we prove a precise bound for the free energy of spin systems at high temperatures, sharpening a result of Lieb.