The paper by E. Brézin, C. Itzykson, G. Parisi, and J. B. Zuber investigates the planar approximation to field theory through the limit of a large internal symmetry group. This approach provides an alternative and powerful method for counting planar diagrams, particularly for cubic and quartic vertices. The authors demonstrate that quantum mechanics in this approximation is equivalent to a free Fermi gas system. The motivation for this work includes the potential to perform reliable computations in the large coupling phase of non-abelian gauge fields in four dimensions and the connection to dual string models. The paper also discusses combinatorial aspects and presents explicit calculations for the ground state energy of a one-dimensional \(gx^4\) anharmonic oscillator, showing that the problem can be restated as finding the ground state energy of a one-dimensional uninteracting Fermi gas. The planar sum is analytic near the origin in the complex coupling constant space, revealing significant changes in the large field region of the Feynman path integral. The authors introduce a field theory where the field is an \(N \times N\) matrix, and the large \(N\) limit selects only planar diagrams, which maximize the number of factors \(N\) associated with closed index loops. The Lagrangian is chosen to describe different types of matrices, and the Feynman rules are derived to illustrate the planar diagrams.The paper by E. Brézin, C. Itzykson, G. Parisi, and J. B. Zuber investigates the planar approximation to field theory through the limit of a large internal symmetry group. This approach provides an alternative and powerful method for counting planar diagrams, particularly for cubic and quartic vertices. The authors demonstrate that quantum mechanics in this approximation is equivalent to a free Fermi gas system. The motivation for this work includes the potential to perform reliable computations in the large coupling phase of non-abelian gauge fields in four dimensions and the connection to dual string models. The paper also discusses combinatorial aspects and presents explicit calculations for the ground state energy of a one-dimensional \(gx^4\) anharmonic oscillator, showing that the problem can be restated as finding the ground state energy of a one-dimensional uninteracting Fermi gas. The planar sum is analytic near the origin in the complex coupling constant space, revealing significant changes in the large field region of the Feynman path integral. The authors introduce a field theory where the field is an \(N \times N\) matrix, and the large \(N\) limit selects only planar diagrams, which maximize the number of factors \(N\) associated with closed index loops. The Lagrangian is chosen to describe different types of matrices, and the Feynman rules are derived to illustrate the planar diagrams.