Quantum metrics with very low action inR+R2gravity

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We have run numerical simulations of Euclidean lattice quantum gravity for metrics which are time-independent and spherically symmetric. The radial variable is discretized as $r=h{L}_{\text{Planck}}$, with $h=0,1,\dots{},N$ and $N$ up to ${10}^{5}$. The Lagrangian is of the form $\sqrt{g}(R+\ensuremath{\alpha}{R}^{2})$ (in units $c=\ensuremath{\hbar}=G=1$) and the action is positive-definite, allowing the use of a standard Metropolis algorithm with update probability $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}S)$. By minimizing the $R+{R}^{2}$ action with respect to conformal modes, Bonanno and Reuter have recently found analytical evidence of a nontrivial ``rippled'' ground state resembling a kinetic condensate of QCD. Our simulations at low but finite temperature ($T={\ensuremath{\beta}}^{\ensuremath{-}1}$) also display strong localized oscillations of the metric, whose total action $S$ remains $\ensuremath{\ll}\ensuremath{\hbar}$ thanks to the indefinite sign of $R$. The average metric $⟨{g}_{rr}⟩$ is significantly different from flat space. The scaling properties of $S$ and $⟨{g}_{rr}⟩$ are investigated in dependence on $N$ and $\ensuremath{\beta}$.