Gauge theories induced by scalars in the fundamental representation of the $\mathrm{U}{(N}_{c}{)}_{\mathrm{gauge}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(N}_{f}{)}_{\mathrm{global}}$ group are investigated in the large ${N}_{c}$ and ${N}_{f}$ limits. A master field is defined from bilinears of the scalar field following an Eguchi-Kawai type reduction of spacetime. The density function for the master field satisfies an integral equation that can be solved exactly in two dimensions $(D=2)$ and in a convergent series of approximations at $D>2$. While at $D=2$ the system is in the same phase at all $\ensuremath{\epsilon}{=N}_{c}{/N}_{f},$ it undergoes a phase transition at a critical value, ${\ensuremath{\epsilon}}_{c}(D)$, for $D>2$.