ABSTRACT
In the last section we saw how universal scaling limits describing aspects of two-dimensional quantum gravity could be obtained. While we found critical points, critical surfaces and approached them in various ways, which provided a wonderful realization of the Wilsonian point of view, where the continuum quantum theory is related to the approach to critical surfaces, somehow the most important and intuitive part of this picture was missing. The primary intuitive reason for the Wilsonian universality is the existence of a correlation length which diverges when we approach the critical surface. It is this divergence of a correlation length which makes the underlying lattice structure irrelevant and allows us to define a continuum theory with no reference to the lattice. But where is this correlation length when we consider two-dimensional quantum gravity? A priori it is not so clear how to define a correlation length in a theory of quantum gravity. In the path integral we have to integrate over all geometries, but at the same time a correlation length, being a “length”, has to refer to a geometry. Still, we will show in this section that one can define a two-point function on the triangulations with a correlation length https://www.w3.org/1998/Math/MathML"> ξ ( μ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math7_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> which diverges as https://www.w3.org/1998/Math/MathML"> | μ − μ c | − ν https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math7_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> when we approach the critical point μ c , and where the scaling exponent https://www.w3.org/1998/Math/MathML"> ν = 1 / 4 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math7_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> determines the Hausdorff dimension ( https://www.w3.org/1998/Math/MathML"> d H = 4 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math7_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ) and where the susceptibility exponent γ calculated from this two-point function precisely is the https://www.w3.org/1998/Math/MathML"> γ = − 1 / 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003320562/573084c4-5e38-4573-b4ce-5412e9a1c018/content/math7_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> already determined in the former chapter.
