Exact solution of discrete two-dimensional R2 gravity

Abstract

We exactly solve a special matrix model of dually weighted planar graphs describing pure two-dimensional quantum gravity with a R 2 interaction in order to study the intermediate regimes between the gravitating and flat metric. The flat space is modeled by a regular square lattice, while localized curvature is being introduced through defects of the lattice. No "flattening" phase transition is found with respect to the R 2 coupling: the infrared behaviour of the system is that of pure gravity for any finite R 2 coupling. In the limit of infinite coupling, we are able to extract a scaling function interpolating between pure gravity and a phase of a dilute gas of curvature defects on a flat background. We introduce and explain some novel techniques concerning our method of large N character expansions and the calculation of Schur characters on big Young tableaux. December 1995 †

Instantons in Non-Critical Strings From the Two-Matrix Model

“…2 A similar approach to the evaluation of the large N characters and heat kernels is used in [39][40][41][42].…”

A few remarks on colour–flavour transformations, truncations of random unitary matrices, Berezin reproducing kernels and Selberg-type integrals

“…They are a variant of integrals obtained by Berezin in his work on quantization in complex symmetric spaces [5,6]. Berezin proved (26) for integer N and (26) for a range of real N that includes N ≥ n + m. In Section 4 we discuss this link and quote some of Berezin's results. Since identities (17) and (23) are so useful in the context of Schur function expansions, we think it is worth to have a closer look at them.…”

“…Identity (16) and its dual version (21) are rather useful in the context of Schur function expansions. For example, the matrix integrals (25) and (26) are straightforward corollaries of these identities. Consider, for example, the matrix integral in (25).…”

“…This makes Schur function expansions a powerful tool for evaluating integrals over unitary groups as has been already demonstrated in [26,4,36,39,31,32,19]. In this paper, we use the Schur function expansion technique to extend the bosonic version of CFT (2) to the range N ≤ 2m ≤ 2N .…”

Gibbs and quantum discrete spaces

“…Note that this is a particular case (with parameters t q = t) of the interaction , considered in [45] q>2 t n(q,T ) q where n(q, T ) is the number of vertices of degree q.…”