Finite Size Scaling in 2d Causal Set Quantum Gravity

Abstract

We study the $N$-dependent behaviour of $2d$ 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 $2d$ causal set action $S$, we find that $\beta ⟨S⟩$ 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 $2d$ causal set quantum gravity for $N\gtrsim 65$.

32 pages, 27 figures (v2 typos and missing reference fixed)

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