Publication : t96/112

Abstract:
The $c \leq$ 1 and $c>$ 1 matrix models are analyzed within large $N$ renormalization group, taking into account touching (or branching) interactions. The $c>$ 1 modified matrix model with string exponent $\bar \gamma >0$ is naturally associated with an unstable fixed point, separating the Liouville phase ($\gamma <0$) from the branched polymer phase ($ \gamma =1/2$). It is argued that at $c=1$ this multicritical fixed point and the Liouville fixed point coalesce, and that both fixed points disappear for $c>1$. In this picture, the critical behavior of $c>1$ matrix models is generically that of branched polymers, but only within a scaling region which is exponentially small when $c \rightarrow 1$. Large crossover effects occur for $c-1$ small enough, with a c $ \sim 1$ pseudo scaling which explains numerical results.

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