Semiclassical Approach to Finite-N Matrix Models

Abstract

We reformulate the zero-dimensional hermitean one-matrix model as a (nonlocal) collective field theory, for finite~$N$. The Jacobian arising by changing variables from matrix eigenvalues to their density distribution is treated {\it exactly\/}. The semiclassical loop expansion turns out {\it not\/} to coincide with the (topological) $\frac{1}{N}$~expansion, because the classical background has a non-trivial $N$-dependence. We derive a simple integral equation for the classical eigenvalue density, which displays strong non-perturbative behavior around $N=\infty$. This leads to IR singularities in the large-$N$ expansion, but UV divergencies appear as well, despite remarkable cancellations among the Feynman diagrams. We evaluate the free energy at the two-loop level and discuss its regularization. A simple example serves to illustrate the problems and admits explicit comparison with orthogonal polynomial results.

27 pages / 3 figures (ps file fixed)

Cited by in corpus (5)

• Breakdown of universality in multi-cut matrix models
• Matrix Models for the Black Hole Information Paradox
• Soliton solutions of Calogero model in harmonic potential
• Notes on Mayer Expansions and Matrix Models
• Symmetric Sextic Freud Weight