Critical behaviour of loop models on causal triangulations

We introduce a dense and dilute loop model on causal dynamical triangulations. Both models are characterised by geometric coupling constant $g$ parameter $\alpha$ in such way that the purely triangulation is recovered for $\alpha=1$. show can be mapped to solvable planar tree model, whose partition function we compute explicitly use determine critical behaviour of model. The likewise model; however, closed-form expression corresponding not obtainable using standard methods employed case. Instead, derive bounds $g_c$ apply transfer matrix techniques examine small.