On the Width of Handles in Two-dimensional Quantum Gravity

We discuss the average length l̄ of the shortest non-contractible loop on surfaces in the two-dimensional pure quantum gravity ensemble. The value of γstr and the explicit form of the continuum loop functions indicate that l̄ diverges at the critical point. Scaling arguments suggest that the critical exponent of l̄ is 1 2 . We show that this value of the critical exponent is also obtained for branched polymers with loops where the calculation is straightforward. e-mail: [email protected] In spite of the fact that one can calculate the continuum loop functions of twodimensional quantum gravity more or less explicitly [1] we do not have a very good understanding of what the generic surfaces contributing to the loop functions really look like. In some sense these surfaces are “thick” since they are not like trees or branched polymers and the “minimal baby universe” (minbu) picture of [2] has provided a valuable insight and some quantitative understanding. In this note we discuss the beahavior of a very simple quantity which can be regarded as a measure of how far surfaces are from being branched polymers. This is the length l of the shortest non-contractible loop so we have in mind surfaces of genus g > 0. We consider randomly triangulated surfaces and loops which consist entirely of links on the surface. Each link is defined to have length 1 so the length of a loop is the number of links it contains assuming that each link is traversed only once. It is clear that every homotopy class of loops on a triangulated surface contains a link loop of shortest length. Let S denote the class of surfaces under consideration, let SA be the class of surfaces of area (=number of triangles) A and let |S| denote the area of a surface S. The length of the shortest noncontractible loop on S will be denoted l(S). We denote the average value of l in the canonical ensemble by l̄A and in the grand canonical ensemble by l̄(μ). These averages are defined as l̄A = N(A) −1 ∑ S∈SA l(S) (1) and l̄(μ) = Z(μ)−1 ∑ S∈S l(S) e−μ|S| (2) where N(A) = #SA and Z(μ) = ∑ S∈S e−μ|S| (3) is the partition function in the ensemble under study. There are several reasons why it is natural to expect the average l̄(μ) to be large as μ → μ0. First recall that the number Ng(A) of closed surfaces of genus g made up of A triangles, one of which is marked, grows for large A as Ng(A) ∼ Aγ(g)−2eμ0A (4) where γ(g) = − 2 + 5 2 g and μ0 is independent of g. If the typical handle on a genus g surface were thin so it could be created by identifying two small regions on a surface