Phases of simplical quantum gravity in four dimensions estimates for the critical exponents

Abstract

A numerical evaluation of the discrete path integral for pure lattice gravity, with and without higher derivative terms, and using the lattice analog of the DeWitt gravitational measure, shows the existence of a well-behaved ground state for sufficiently strong gravity ( G ⩾ G c). Close to the continuous 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 15 × 16 4 = 1 572864 simplices. With periodic boundary conditions (four-torus) the average curvature approaches zero at the critical point. Curvature fluctuations diverge at this point, while the fluctuations in the local volumes remain bounded. The value of the curvature critical exponent is estimated to be δ = 0.626(25), when the critical point is approached from the smooth phase. In this phase, as well as at the criticalpoint, the fractal dimension is consistent with four, the euclidean value. In the (physically unacceptable) rough, collapsed phase the fractal dimensions is closer to two, in agreement with earlier results which suggested a discontinuity in the fractal dimensions at the critical point. For sufficiently small higher derivative coupling, and in particular for the pure Regge-Einstein action, the transition between the smooth and rough phase becomes first order, suggesting the existence of a multicritical point separating the continuous from the discontinuous phase transition line.