Finite size scaling in 2d causal set quantum gravity

We study the $N$-dependent behaviour of $\mathrm{2d}$ causal set quantum gravity. This theory is known to exhibit a phase transition as the analytic continuation parameter $\beta$, 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 $\mathrm{2d}$ causal set action $S$, we find that $\beta \langle S\rangle $ scales like $ N^\nu$ where the scaling exponent $\nu$ 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 $\beta$ 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 $\mathrm{2d}$ causal set quantum gravity for $N \gtrsim 65$.