Harriet Walsh

Random partitions and random matrices, asymptotic and exact connections

Random integer partitions are discrete analogues of random matrix models, notably arising in the study of random elements of symmetric groups. I’ll discuss two connections between a natural infinite parameter family of laws on partitions, Okounkov's Schur measures, and certain matrix ensembles. On the one hand, the universal asymptotic law of the first part of a Schur random partition coincides with that of the largest eigenvalue of a GUE random Hermitian matrix, a connection which relates a diverse group of discrete physical models to the KPZ universality class. On the other hand, the distribution of first part of a Schur random partition is exactly equal to the normalisation of certain unitary matrix models. This second connection is particularly useful in the “multicritical” case of Schur measures escaping GUE universality, which was considered in joint work with Dan Betea and Jérémie Bouttier (arXiv:2012.01995). If time permits, I’ll discuss connections with enumeration of discrete surfaces (maps), and some related joint work with Guillaume Chapuy and Baptiste Louf (arXiv:2206.11315).

Talk recording: https://mediasite.bris.ac.uk/Mediasite/Play/f7f1e4860fb545768ff419383af8e04d1d