  The Gaussian $\beta$-ensemble is a real $n$-point configuration $\{x_j\}_1^n$
picked randomly with respect to the Boltzmann factor $e^{-\frac\beta 2H_n}$,
$H_n=\sum_{i\ne j}\log\frac 1{|x_i-x_j|}+n\sum_{i=1}^n\tfrac 12x_i^2.$ The
point process $\{x_j\}_1^n$ tends to follow the semicircle law
$\sigma(x)=\tfrac 1{2\pi}\sqrt{(4-x^2)_+}$ in certain average senses.
  A Fekete configuration (minimizer of $H_n$) is spread out in a much more
uniform way in the interval $[-2,2]$ with respect to the regularization
$\sigma_n(x)=\max\{\sigma(x),n^{-\frac 1 3}\}$ of the semicircle law. In
particular, Fekete configurations are "equidistributed" with respect to
$\sigma_n(x)$, in a certain technical sense of Beurling-Landau densities.
  We consider the problem of characterizing sequences $\beta_n$ of inverse
temperatures, which guarantee almost sure equidistribution as $n\to\infty$. We
find that a necessary and sufficient condition is that $\beta_n$ grows at least
logarithmically in $n$: $$\beta_n\gtrsim \log n.$$ We call this growth rate the
perfect freezing regime. We give several further results on the distribution of
particles when $\beta_n\gtrsim\log n$, for example on minimal spacing,
discrepancies, and sampling and interpolation for weighted polynomials.
  The condition $\beta_n\gtrsim\log n$ was introduced by some of the authors in
the context of two-dimensional Coulomb gas ensembles, where it is shown to be
sufficient for equidistribution. Although the technical implementation requires
some considerable modifications, the strategy from dimension two adapts well to
prove sufficiency also for one-dimensional Gaussian ensembles. On a technical
level, we use estimates for weighted polynomials due to Levin, Lubinsky,
Gustavsson and others. The other direction (necessity) involves estimates due
to Ledoux and Rider on the distribution of particles which fall near or outside
the boundary.
