Circular law

=== As a Coulomb gas ===

=== As a Coulomb gas ===

Recall the spectral distribution<math display="block">\rho_N\left(z_1, \ldots, z_N\right)=\frac{1}{Z} \exp \left(-\sum_{k=1}^N\left|z_k\right|^2\right) \prod_{1 \leq j<k \leq N}\left|z_j-z_k\right|^2</math>It can be interpreted as the [[Boltzmann distribution]] for a [[Coulomb gas]], or more specifically a one-component plasma, at temperature 1. The gas contains <math>N</math> identical particles, all placed within the plane <math>\mathbb C</math>, with total energy<math display="block">\sum_{k=1}^N |z_k|^2 - 2\sum_{1 \leq j < k \leq N} \ln |z_j - z_k|</math>The first term indicates that every particle is attracted to the origin by a force of magnitude <math>F_k = 2|z_k|</math>. The second term indicates that every particle pair is repelling each other by a force of magnitude <math>F_{jk} = \frac{2}{|z_j - z_k|} </math>.

Recall the spectral distribution<math display="block">\rho_N\left(z_1, \ldots, z_N\right)=\frac{1}{Z} \exp \left(-\sum_{k=1}^N\left|z_k\right|^2\right) \prod_{1 \leq j<k \leq N}\left|z_j-z_k\right|^2</math>It can be interpreted as the [[Boltzmann distribution]] for a [[Coulomb gas]], or more specifically a one-component plasma, at [[Thermodynamic beta|inverse temperature]] <math>\beta = 2</math>. Note that here <math>\beta </math> is used to mean something different, and may take any value within <math>(0, \infty)</math>.

The gas contains <math>N</math> identical particles, all placed within the plane <math>\mathbb C</math>, with total energy<math display="block">\frac 12 \sum_{k=1}^N |z_k|^2 - \sum_{1 \leq j < k \leq N} \ln |z_j - z_k|</math>The first term indicates that every particle is attracted to the origin by a force of magnitude <math>F_k = |z_k|</math>. The second term indicates that every particle pair is repelling each other by a force of magnitude <math>F_{jk} = \frac{1}{|z_j - z_k|} </math>.

=== Mesoscopic law ===

=== Mesoscopic law ===