A New Approach to Strong Convergence of Random Matrices
Speaker: Jorge Garza VargasDate of Talk: November 22, 2024
Upstream link: UCLA Functional Analysis Seminar
1. The Problem
For each \(N\), let \(X_N\) be a random matrix, and let \(\lambda _{1, N},\ldots, \lambda _{N, N}\) be its eigenvalues. Let \(\hat \mu _N = \frac{1}{N} \sum _{i=1}^{N} \delta _{\lambda _{i, N}}\), and let \[\mu _N = \mathbb{E} \left[ \hat \mu _N \right]. \]
The trace of the measure is defined as \(\operatorname{tr} (\mu ) = \int x\ d \mu \), so \(\operatorname{tr} \left( \mu _N \right) = \mathbb{E}\left[ \operatorname{tr}_N X_N \right]\), where \(\operatorname{tr}_N\) is the normalised trace of the matrix.
In 1984, McKay showed that if \(X_N\) was the adjacency matrix of a random \(d\)-regular graph of \(N\) vertices, then the measures \(\mu _N\) converge to some \(\mu _\infty\) weakly, a measure supported in the interval \(\left[ -2\sqrt{d-1}, 2\sqrt{d-1} \right]\) (I think).
While this suggests that the normalised trace lies within the same interval above, this does not mean that all the eigenvalues do too; in fact, one frequently runs into the eigenvalue \(d\) with the eigenvector of all ones. Friedman’s theorem in 2004 shows that this is the only such eigenvalue and eigenvector — the restriction of these matrices to the orthogonal complement of this vector will have a spectrum contained in the above interval.
2. The Polynomial Method and…Approximation Theory?
The speaker’s main contribution was giving a simpler proof of Friedman’s theorem. The original proof was encumbered by delicate combinatorics when handling small cycles. The new proof gives a (in my eyes) surprising application of the polynomial method.
They showed that for a polynomial \(h\), one has \(\mathbb{E} \left[ \operatorname{tr} _N h\left( X_N \right) \right] = \psi _h\left( \frac{1}{N} \right)\) for some rational function \(\psi _h\) with degree not exceeding \(\deg h\) (a result first showed by Nica in 1994), then used approximation theory to get control on how it behaves near \(0\).
I had only heard of the polynomial method being used (1) on polynomials instead of rational functions and (2) for proving combinatorial things rather than these sorts of asymptotics. I suppose the problem is combinatorial in nature, and I know less than \(\epsilon \) about the polynomial method; perhaps it’s not so surprising to more well-read mathematicians.
3. Keeping the Talk Down-to-Earth
The talk mostly focused on random matrices constructed as adjacency matrices of random \(d\)-regular graphs, which I think is an intuitive or at least accessible topic to plebians such as myself. Apparently, the paper has consequences for other less tangible fields (i.e. functional analysis), a more or less completely foreign field to someone such as myself.
The point is, I appreciate how the talk was able to convey the significance of the speaker’s work without wading waist-deep into complicated swampy territory. I’m put off by a lot of seminar talks because of this — they wander off way into the weeds, and I’m separated from the pack within a few sips of coffee. This one was very well communicated and had a really friendly guise!