It is well-known that matrix models admit topological expansions as the size goes to infinity, that the asymptotic expansion to all-order can be determined from loop equations (Virasoro constraints) and take the form of Eynard-Orantin topological recursion. I will show how to a similar approach can be used (not only formally, but also rigorously) for random matrix models with discrete eigenvalues or models of random partitions. Loop equations are replaced by "non-perturbative Dyson-Schwinger equations" similar to those Nekrasov derived in 4d N = 2 supersymmetric gauge theories, there is a topological recursion but it is different from Eynard-Orantin one beyond the leading order. I will discuss application to random lozenge tilings on surfaces and the Kenyon-Okounkov conjecture (fluctuations are described by free bosonic field).
Based on https://arxiv.org/abs/2601.16377 with Vadim Gorin and Alice Guionnet.