Finite size scaling in 2d causal set quantum gravity

We study the dependence on size, N, of 2d causal set quantum gravity. This theory is known to exhibit a phase transition as the analytic continuation parameter β, akin to an inverse temperature, is varied. Using a scaling analysis we find that the asymptotic regime is reached at relatively small values of N. Focussing on the $2d$ causal set action S, we find that $\beta ⟨S⟩$ scales like $N^{\nu }$ where the scaling exponent ν takes different values on either side of the phase transition. For $\beta >{\beta }_c$ we find that $\nu =2$ which is consistent with our analytic predictions for a non-continuum phase in the large β regime. For $\beta <{\beta }_c$ we find that $\nu =0$, consistent with a continuum phase of constant negative curvature thus suggesting a dynamically generated cosmological constant. Moreover, we find strong evidence that the phase transition is first order. Our results strongly suggest that the asymptotic regime is reached in $2d$ causal set quantum gravity for $N\gtrsim 65$. (paper)