  We explore models with emergent gravity and metric by means of numerical
simulations. A particular type of two-dimensional non-linear sigma-model is
regularized and discretized on a quadratic lattice. It is characterized by
lattice diffeomorphism invariance which ensures in the continuum limit the
symmetry of general coordinate transformations. We observe a collective order
parameter with properties of a metric, showing Minkowski or euclidean
signature. The correlation functions of the metric reveal an interesting
long-distance behavior with power-like decay. This universal critical behavior
occurs without tuning of parameters and thus constitutes an example of
"self-tuned criticality" for this type of sigma-models. We also find a
non-vanishing expectation value of a "zweibein" related to the "internal"
degrees of freedom of the scalar field, again with long-range correlations. The
metric is well described as a composite of the zweibein. A scalar condensate
breaks euclidean rotation symmetry.
