Explore the fascinating connections between random hyperbolic surfaces and random graphs in this 47-minute conference talk from IPAM's New Interactions Between Probability and Geometry Workshop. Begin with a historical overview before learning how to select uniform random hyperbolic surfaces of genus g and examine their length spectra, particularly focusing on short closed geodesics. Discover the remarkable finding that when the genus g is large, the lengths of these geodesics follow the same distribution as short cycles in large random metric maps, revealing deep mathematical connections between geometric and combinatorial structures. Gain insights into cutting-edge research that bridges probability theory and geometry through this joint work with Simon Barazer and Alessandro Giacchetto, presented by Mingkun Liu from Université de Paris XIII.