Simplicial quantum gravity in three dimensions: Analytical and numerical results

Abstract

The weak-field expansion and the nonperturbative ground state of three-dimensional simplicial quantum gravity are discussed. The correspondence between lattice and continuum operators is shown in the context of the lattice weak-field expansion, around a simplicial network built of rigid hypercubes, and the lattice translational zero modes are exhibited. A numerical evaluation of the discrete path integral for pure lattice gravity (with and without higher-derivative terms) shows the existence of a well-behaved ground state for sufficiently strong gravity ($G>{G}_{c}$). At the critical point, separating the smooth from the rough phase of gravity, the critical exponents are estimated using a variety of methods on lattices with up to 7\ifmmode\times\else\texttimes\fi{}${64}^{3}$=1 835 008 edges. As in four dimensions, the average curvature approaches zero at the critical point. Curvature fluctuations diverge at this point, while the fluctuations in the local volumes remain bounded.