### 4.9 Coupling to matter fields

Ambjørn et al. [7] considered the influence of both Ising spins and Gaussian scalar fields on bulk
geometric quantities. The phase structure remained essentially unchanged, and no improvement in the
scaling behaviour of was found. Taken together with their results on higher-derivative
gravity [16], one finds a universal linear dependence of the cosmological constant , with slope
.
Coupling to -spin variables located on edges was considered by Ambjørn et
al. [15], who added a Wilson loop term to the action. The matter
sector behaves largely as expected when is varied between the crumpled and the elongated gravity
phase. However, in the common critical region of both sectors, where a priori one might have
expected interesting effects, the critical behaviour seems to agree with that of the pure gravity
system.

More recently, Bilke et al. [52] have reported a non-trivial back-reaction of matter on geometry, when
considering coupling to several non-compact -gauge fields. Their study was in part motivated by a
continuum analysis of the dynamics of the conformal factor of Antoniadis et al. [21]. Including three fields
-fields seems to lead to a total suppression of the branched polymer phase, which is replaced by a
new weak-coupling phase with negative susceptibility exponent and a fractal dimension . These
are clearly interesting results, but should be treated with some caution because of the small lattice sizes
involved ().