CRITICAL BEHAVIOR IN c=1 MATRIX MODEL WITH BRANCHING INTERACTIONS

Abstract

In order to understand the phase structure of d>1 strings we investigate the c=1 matrix model with g′(tr M(t)2)2 interaction which is the simplest approximation of the large-N reduced model of odd-dimensional matrix field theory. We find three distinct phases: (i) an ordinary c=1 gravity phase, (ii) a branched polymer phase and (iii) an intermediate phase, and compute the disk and cylinder amplitudes in the phases (i) and (iii). Further we also analyze the model with slightly generalized [Formula: see text][Formula: see text] interaction. As a result the multi-critical versions of the phase (ii) are found.

A modified c = 1 matrix model with new critical behavior

“…After this paper was completed, we learned of an interesting paper by F. Sugino and O. Tsuchiya [10] in which results similar to ours were obtained by means of collective field theory.…”

A scenario for the c > 1 barrier in non-critical bosonic strings

“…in our language for t x < 0, the singular part of the free energy scales as f ∝ t 2 0 / log(t 0 ), and at the multicritical point, i.e. for t x = 0, f ∝ t 2 0 log(t 0 ) [38,39]. This is simply reproduced with the β-functions (18), if we take into account the ρ renormalization.…”

“…Then is becomes a branched polymer critical line, with γ = 1/2, which passes through x = 1/2, g = 0. These features persist for 0 < c ≤ 1 matrix models with touching interactions [36][37][38][39]. When the touching coupling constant x is switched on, the 2-D gravity critical line, characterized by the exponent γ ≤ 0 given by the KPZ formula, ends at a end-point, characterized by the new positive exponent [36,40,41] …”

Higher melonic theories

“…In the infra-red limit,g is large, and to leading order in 1/g we can set G(t) = 0. To solve (28) in the IR, we will adopt the methodology of [20], and so we introduce multiplicative characters defined as π s (t) ≡ |t| s , π s,sgn (t) ≡ |t| s sgn(t) .…”