I will start with a brief review of large-N tensor models, and of some of their applications in theoretical physics. On the one hand, they provide a general platform to investigate random geometry in an arbitrary number of dimensions, in analogy with the matrix models approach to two-dimensional quantum gravity. Previously known universality classes of random geometries have been identified in this context, with continuous random trees acting as strong attractors. On the other hand, the same combinatorial structure supports a generic family of large-N quantum theories, collectively known as melonic theories. Being largely solvable, they have opened a new window into strongly-coupled quantum theory, and via a 2d version of the AdS/CFT correspondence, into quantum gravity.
In a second and more hands-on part of the talk, I will explain how the melonic large N limit of random tensors transforming under irreducible representations of O(N) can be proven. Such proofs involve graph-theoretic arguments, which I will outline in a pictorial way. Based on arXiv:1712.00249, arXiv:1803.02496 and arXiv:2104.03665.