ABSTRACT
The two-dimensional Euclidean gravity model we have studied satisfies the Wilsonian criterium for universality: to a large extent it is independent of the details of the short distance regularization. We were not restricted to use triangulations as building blocks, but could use any (finite) combination of polygons as building blocks, as long as the weights of polygons were all positive, and we would obtain the same continuum multi-loop functions when the dimensionless cosmological coupling constant https://www.w3.org/1998/Math/MathML"> μ → μ c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math8_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in such a way that https://www.w3.org/1998/Math/MathML"> μ = μ c + ε 2 Λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math8_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where the link length in the graphs went to zero while the continuum cosmological constant Λ survived. In that limit the average number N of polygons in the graphs also diverged for multi-loop functions with three or more loops and we could talk about a finite continuum limit of the volume https://www.w3.org/1998/Math/MathML"> V ∝ N ε 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math8_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where N denoted the number of polygons. We had https://www.w3.org/1998/Math/MathML"> 〈 V 〉 ∝ Λ − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math8_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> if we did not fix the volume of spacetime, but considered the model with a fixed cosmological constant Λ. Further, by studying the two-point function as a function of the so-called geodesic distance, we identified the correlation length which diverged when we approached the critical point μ c . In this sense the two-point function acted precisely as the two-point function of a spin system and the universality of the results could be understood as a result of the divergent correlation length, in the same way as universality of phase transitions of spin systems can be understood as the result of a divergent correlation length between the spins, which makes many details of the short distance lattice structure and interactions irrelevant for scaling limit.
