2d quantum gravity with discrete edge lengths

Abstract

An approximation of the Standard Regge Calculus (SRC) was proposed by the $Z_2$-Regge Model ($Z_2$RM). There the edge lengths of the simplicial complexes are restricted to only two possible values, both always compatible with the triangle inequalities. To examine the effect of discrete edge lengths, we define two models to describe the transition from the $Z_2$RM to the SRC. These models allow to choose the number of possible link lengths to be $n = {4,8,16,32,64,...}$ and differ mainly in the scaling of the quadratic link lengths. The first extension, the $X^1_n$-Model, keeps the edge lengths limited and still behaves rather similar to the "spin-like" $Z_2$RM. The vanishing critical cosmological constant is reproduced by the second extension, the $X^C_n$-Model, which allows for increasing edge lengths. In addition the area expectation values are consistent with the scaling relation of the SRC.