As Usual, Gil gave a fantastic talk.
The pivot, as far as I'm concerned, is the fact that the spectral gap of the zig-zag product of a family of $d$-regular graphs $G$ with a fixed $d$-vertex graph $H$ may be larger than the spectral gap of $H$ itself.
Recall that the said gap is larger than the product of the gap of $G$ and the gap of $H^2$ (see an exposition following a presentation by Salil Vadhan.
The analysis refers to all the eigenvalues of the graph $H$, rather than only to its second largest eigenvalue (aka rate of rapid mixing of random walk). Specifically, it is obtained by maximizing an algebriac expression that depends on the chacacteristic polynomial of the graph (i.e., of its adjacency matrix).
The well-known Zig-Zag product and related graph operators, like derandomized squaring, are fundamentally combinatorial in nature. Classical bounds on their behavior often rely on a mix of combinatorics and linear algebra. However, these traditional bounds are not tight and frequently fail to align with experimental results. In this talk, we will present a more refined analysis that utilizes the full spectrum of the graph, rather than relying solely on its spectral expansion. This approach produces results that both match experimental observations and, in a sense, are proved to be optimal. Our technique is analytic, diverging from classical methods: for the upper bound, we apply finite free probability, while for the lower bound, we draw on results from analytic combinatorics.