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

Abstract

The c ≤ 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 exponentγ > 0 is naturally associated with an unstable fixed point, separating the Liouville phase (γ < 0) from the branched polymer phase (γ = 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 → 1. Large crossover effects occur for c − 1 small enough, with a c ∼ 1 pseudo scaling which explains numerical results. * CNRS 1

Beyond the c = 1 barrier in two-dimensional quantum gravity

“…This behavior is, in our opinion, not compatible with the existence of a phase transition, and we conclude that there is only a branched polymer phase. This strongly supports the conjecture in [2] about the nature of the c = 1 "barrier".…”

“…2 Touching random surfaces A conjecture for the observed c > 1 behavior, was put forward in [2]: "For c > 1 the dynamical triangulation model is always in a branched polymer phase. But finite size effects are exponentially enhanced as c → 1 + , due to the influence of the c = 1 fixed point (which becomes complex for c > 1)."…”

Compact support probability distributions in random matrix theory

“…By comparing eq(4.4) with the analogous saddle point equation for the "canonical" quartic probability distribution (4.3) it is obvious that they have the same eigenvalue density, in the large-n limit, for both phases of the model, provided the effective coupling eq. (4.6) are precisely identified with those of the "canonical" distribution 14) which is a border of existence for the model. If g ′ 4 is fixed positive, the one-cut solution holds for any real g ′ 2 such that 15) which is the line of phase transition to the symmetric two-cut solution : …”

“…These type of models, where the exponent of the Boltzmann weight is a sum of different powers of traces of even powers of the random matrix was analyzed in several matrix models in zero and one dimension [8] - [14]. The additional "trace-squared" terms were interpreted to provide touching interactions to the dynamical triangulated surfaces defined by the matrix potential Tr V (M ).…”

Kosterlitz-Thouless phase transitions on discretized random surfaces

“…On the contrary, very little is known about c > 1 strings, but their physical properties are probably quite different [4].…”