  How a system initially at infinite temperature responds when suddenly placed
at finite temperatures is a way to check the existence of phase transitions. It
has been shown in [R. da Silva, IJMPC 2023] that phase transitions are
imprinted in the spectra of matrices built from time evolutions of
magnetization of spin models. In this paper, we show that this method works
very accurately in determining the critical temperature in the mean-field Ising
model. We show that for Glauber or Metropolis dynamics, the average eigenvalue
has a minimum at the critical temperature, which is corroborated by an
inflection at eigenvalue dispersion at this same point. Such transition is
governed by a gap in the density of eigenvalues similar to short-range spin
systems. We conclude that the thermodynamics of this mean-field system can be
described by the fluctuations in the spectra of Wishart matrices which suggests
a direct relationship between thermodynamic fluctuations and spectral
fluctuations.
