# A new proof of Friedman's second eigenvalue theorem with strong implications

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In 2004 J. Friedman wrote a ~100 page paper proving a conjecture of Alon which stated that random d-regular graphs are nearly optimal expanders. Since then, Friedman's result has been refined and generalized in several directions, perhaps most notably by Bordenave and Collins who in 2019 established strong convergence of independent permutation matrices (a massive generalization of Friedman's theorem), a result that led to groundbreaking results in spectral theory and geometry.

In this talk I will present joint work with C. Chen, J. Tropp and R. van Handel, where we introduce a new proof technique that allows one to convert qualitative results in random matrix theory into quantitative ones. This technique yields a fundamentally new approach to the study of strong convergence which is more flexible and significantly simpler than the existing techniques. Concretely, we're able to obtain (1) a remarkably short of Friedman's theorem (2) a quantitative version of the result of Bordenave and Collins (3) a proof of strong convergence for arbitrary stable representations of the symmetric group, which constitutes a substantial generalization of the result of Bordenave and Collins.