Gaussian curvature on random planar maps and Liouville quantum gravity

Abstract We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by this, we study asymptotics for the discrete curvature of $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> -mated CRT maps. More precisely, we prove that the discrete curvature integrated against a $$C_c^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> test function is of order $$\epsilon ^{o(1)},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which is consistent with our scaling limit conjecture. On the other hand, we prove the total discrete curvature on a fixed space-filling SLE segment scaled by $$\epsilon ^{\frac{1}{4}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:msup> </mml:math> converges in distribution to an explicit random variable.