On the phase structure of commuting matrix models

Abstract

We perform a systematic study of commutative SO(p) invariant matrix models with quadratic and quartic potentials in the large N limit. We find that the physics of these systems depends crucially on the number of matrices with a critical rôle played by p = 4. For p ≤ 4 the system undergoes a phase transition accompanied by a topology change transition. For p > 4 the system is always in the topologically trivial phase and the eigenvalue distribution is a Dirac delta function spherical shell. We verify our analytic work with Monte Carlo simulations.

Lectures on Matrix Field Theory

“…Alternative Derivation: In deriving the two-cut solution, we may follow the more compact formalism of [204]. We rewrite the saddle point equation (5.174), together with (5.289) and introducing a symmetric density of eigenvalues ρ(λ), as…”

“…The one-cut solution to (5.223) corresponds to an unbounded function y(z), at c 1 = µ, given explicitly by [204] y(z) = 1 2π…”

Commuting quantum matrix models

“…The corresponding saddle point equation is given by: 22) comparing this to equation (2.6) in ref. [18] we conclude that all of the results of [18] for gaussian potential are valid here provided we multiply the radius of the distribution by a…”

“…[15]- [18]. The corresponding saddle point equation is given by: 22) comparing this to equation (2.6) in ref.…”

“…Therefore, we obtain that for 2 ≤ p ≤ 4 the radius of the distribution is given by [18]: . Note that at ξ = 1 the radius of the distribution collapses to zero.…”