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The Gaussian $\beta$-ensemble is a real $n$-point configuration $\{x_j\}_1^n$picked randomly with respect to the Boltzmann factor $e^{-\frac\beta 2H_n}$,$H_n=\sum_{i\ne j}\log\frac 1{|x_i-x_j|}+n\sum_{i=1}^n\tfrac 12x_i^2.$ Thepoint 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 moreuniform 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. Inparticular, 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 inversetemperatures, which guarantee almost sure equidistribution as $n\to\infty$. Wefind that a necessary and sufficient condition is that $\beta_n$ grows at leastlogarithmically in $n$: $$\beta_n\gtrsim \log n.$$ We call this growth rate theperfect freezing regime. We give several further results on the distribution ofparticles 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 inthe context of two-dimensional Coulomb gas ensembles, where it is shown to besufficient for equidistribution. Although the technical implementation requiressome considerable modifications, the strategy from dimension two adapts well toprove sufficiency also for one-dimensional Gaussian ensembles. On a technicallevel, we use estimates for weighted polynomials due to Levin, Lubinsky,Gustavsson and others. The other direction (necessity) involves estimates dueto Ledoux and Rider on the distribution of particles which fall near or outsidethe boundary.

Gaussian beta ensembles: the perfect freezing transition and its characterization in terms of Beurling-Landau densities