Combinatorial Relations from the BKP Integrable Hierarchy for Kontsevich-Type Matrix Models & the Combinatorial Structure of Correlation Functions in the Quartic Kontsevich Model

This thesis studies a special type of matrix model, namely the Kontsevich-type matrix model with arbitrary potential, where a special focus lies on the purely quartic potential. We start with a general overview of matrix models and their applications, especially in the context of free probability, topological recursion and integrable systems. The measure of the original Kontsevich model, developed by Maxim Kontsevich to prove the Witten conjecture, is a Gaussian measure deformed by a cubic potential. We introduce the Kontsevich-type matrix model as a generalization of its original with an arbitrary potential. Correlation functions in this model, which we also call n1 + ⋅ ⋅ ⋅ + nB -point functions, are graded by the genus g and number of boundary components B as they are naturally embedded into Riemann surfaces with the same topology. Recently, it was discovered that a deformation of the partition function of this model with an infinite set of formal parameters is a tau-function of the BKP integrable hierarchy, which then led to the discovery of an infinite amount of linear relations between the model’s moments for an even potential. We proceed with an introduction of the necessary combinatorial tools in order to further examine these linear relations as well as the correlation functions of this model. This includes partitioned permutations, which first appeared in the context of higher-order freeness in free probability theory, as well as Catalan tuples and their possible visualizations, which were first introduced during the study of the combinatorial structure of n1 -point functions in the quartic Kontsevich model. One main result of this thesis includes the expression of the moments of the matrix model and therefore the linear relations between them in terms of its correlation functions. We further show that only n1 -point functions and n1 + n2 -point functions with n1 , n2 odd contribute to the leading order of the genus expansion in these relations. For the quartic Kontsevich model, even more can be said: We show that the linear relations between its moments assemble to an infinite amount of non-linear differential relations between the 1+1-point function and the 2-point function due to the recursion relations that are already known for its correlation functions. These recursion relations state that we can express the correlation functions with boundaries of any length recursively by correlation functions with boundaries of either length 1 or 2 only. While the combinatorial structure of the recursion relation for n1 -point functions has already been studied and found to be in bijection to so-called nested Catalan tables, this thesis unravels the full combinatorial structure of n1 + n2 -point functions with n1 , n2 odd for the first time.