Bulk correlation functions in 2D quantum gravity

We compute bulk 3- and 4-point tachyon correlators in the 2d Liouville gravity with non-rational matter central charge c<1, following and comparing two approaches. The continuous CFT approach exploits the action on the tachyons of the ground ring generators deformed by Liouville and matter ``screening charges''. A by-product general formula for the matter 3-point OPE structure constants is derived. We also consider a ``diagonal'' CFT of 2D quantum gravity, in which the degenerate fields are restricted to the diagonal of the semi-infinite Kac table. The discrete formulation of the theory is a generalization of the ADE string theories, in which the target space is the semi-infinite chain of points.Comment: 14 pages, 2 figure

More on the exact solution of the O(n) model on a random lattice and an investigation of the case |n|>2

For $n\in [-2,2]$ the $O(n)$ model on a random lattice has critical points to which a scaling behaviour characteristic of 2D gravity interacting with conformal matter fields with $c\in [-\infty,1]$ can be associated. Previously we have written down an exact solution of this model valid at any point in the coupling constant space and for any $n$. The solution was parametrized in terms of an auxiliary function. Here we determine the auxiliary function explicitly as a combination of $\theta$-functions, thereby completing the solution of the model. Using our solution we investigate, for the simplest version of the model, hitherto unexplored regions of the parameter space. For example we determine in a closed form the eigenvalue density without any assumption of being close to or at a critical point. This gives a generalization of the Wigner semi-circle law to $n\neq 0$. We also study the model for $|n|>2$. Both for $n2$ we find that the model is well defined in a certain region of the coupling constant space. For $n<-2$ we find no new critical points while for $n>2$ we find new critical points at which the string susceptibility exponent $\gamma_{str}$ takes the value $+\frac{1}{2}$.Comment: 27 pages, LaTeX file (uses epsf) + 3 eps figures, formulas involving the string susceptibility corrrected, no change in conclusion

Wilson Loops in Large N QCD on a Sphere

Wilson loop averages of pure gauge QCD at large N on a sphere are calculated by means of Makeenko-Migdal loop equation.Comment: Phys.Lett.B329 (1994) 338 (minor corrections in accordance to published version, several Latex figures are removed and available upon request

The Potts-q random matrix model : loop equations, critical exponents, and rational case

In this article, we study the q-state Potts random matrix models extended to branched polymers, by the equations of motion method. We obtain a set of loop equations valid for any arbitrary value of q. We show that, for q=2-2 \cos {l \over r} \pi (l, r mutually prime integers with l < r), the resolvent satisfies an algebraic equation of degree 2 r -1 if l+r is odd and r-1 if l+r is even. This generalizes the presently-known cases of q=1, 2, 3. We then derive for any 0 \leq q \leq 4 the Potts-q critical exponents and string susceptibility.Comment: 7 pages, submitted to Phys. Letters

Scattering in the adjoint sector of the c = 1 Matrix Model

Closed string tachyon emission from a traveling long string in Liouville string theory is studied. The exact collective field Hamiltonian in the adjoint sector of the c=1 matrix model is computed to capture the interaction between the tip of the long string and the closed string tachyon field. The amplitude for emission of a single tachyon quantum is obtained in a closed form using the chiral formalism.Comment: 22 pages, 2 figure

Phase Structure of the O(n) Model on a Random Lattice for n>2

We show that coarse graining arguments invented for the analysis of multi-spin systems on a randomly triangulated surface apply also to the O(n) model on a random lattice. These arguments imply that if the model has a critical point with diverging string susceptibility, then either \g=+1/2 or there exists a dual critical point with negative string susceptibility exponent, \g', related to \g by \g=\g'/(\g'-1). Exploiting the exact solution of the O(n) model on a random lattice we show that both situations are realized for n>2 and that the possible dual pairs of string susceptibility exponents are given by (\g',\g)=(-1/m,1/(m+1)), m=2,3,.... We also show that at the critical points with positive string susceptibility exponent the average number of loops on the surface diverges while the average length of a single loop stays finite.Comment: 18 pages, LaTeX file, two eps-figure

An Iterative Solution of the Three-colour Problem on a Random Lattice

We study the generalisation of Baxter's three-colour problem to a random lattice. Rephrasing the problem as a matrix model problem we discuss the analyticity structure and the critical behaviour of the resulting matrix model. Based on a set of loop equations we develop an algorithm which enables us to solve the three-colour problem recursivelyComment: 14 pages, LaTeX, misprints corrected, approximation of (6.20) refine