Exact Solution of Induced Lattice Gauge Theory at Large $N$

Abstract

We find the exact solution of a recently proposed model of the lattice gauge theory induced by heavy scalar field in adjoint representation at $ N= \infty $ for arbitrary dimension $D$. The nonlinear integral equation for the gauge invariant density of eigenvalues of the vacuum average of the scalar field is derived. In the continuum limit, the density grows as $ \phi^{\alpha} $ where $ \alpha = 1 + \frac{1}{\pi}\arccos\frac{D}{3D-2} $.