Semicircle Law for Generalized Curie–Weiss Matrix Ensembles at Subcritical Temperature

AbstractHochstättler et al. (J Theor Probab 29:1047–1068, 2016) showed that the semicircle law holds for generalized Curie–Weiss matrix ensembles at or above the critical temperature. We extend their result to the case of subcritical temperatures for which the correlations between the matrix entries are stronger. Nevertheless, one may use the concept of approximately uncorrelated ensembles that was first introduced in Hochstättler et al. (2016). In order to do so, one needs to remove the average magnetization of the entries by an appropriate modification of the ensemble that turns out to be of rank 1, thus not changing the limiting spectral measure.

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