A modified c = 1 matrix model with new critical behavior

Abstract

By introducing a dt g (Tr Φ 2 (t)) 2 term into the action of the c = 1 matrix model of two-dimensional quantum gravity, we find a new critical behavior for random surfaces. The planar limit of the path integral generates multiple spherical "bubbles" which touch one another at single points. At a special value of g, the sum over connected surfaces behaves as ∆ 2 log ∆, where ∆ is the cosmological constant (the sum over surfaces of area A goes as A −3 ). For comparison, in the conventional c = 1 model the sum over planar surfaces behaves as ∆ 2 / log ∆.

Matrix model of the (1+1) dimensional dilatonic black hole in the double scaling limit

“…This fits to the general observation by I. Klebanov and collaborators [17,18] that surfaces with critical selftouching interaction (microscopic necks) are described by the non-convenient branch of the Liouville exponent. As a consequence, at the point when the self-touching interaction becomes critical, the string susceptibility exponent γ str jumps to a positive value γ str → γ str γ str − 1 .…”

“…7 The grand canonical ensemble is in fact the simplest realization the "modified c = 1 matrix model" considered in [16,17].…”

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

“…If we restrict to only one of these two vertices, then we get remarkably different behavior in the large N limit: tr Φ 4 leads to a planar limit, while (tr Φ 2 ) 2 leads to bubble diagrams. Mixing the two gives some interesting modifications of scaling exponents of the pure tr Φ 4 theory [27,28,29,30]. For melonic theories, our story so far is less featured, in that the treatment of the two-point and fourpoint functions proceeds almost identically for the myriad theories we can construct.…”