A new approach to strong convergence
Chi-Fang Chen, Jorge Garza-Vargas, Joel A. Tropp, Ramon van Handel
TL;DR
This work introduces a soft, broadly applicable approach to strong convergence for sequences of random matrices by exploiting two inputs: that the expected traces of polynomials are rational in $1/N$ with accessible leading terms, and a first-order $1/N$ expansion that defines limiting functionals $\nu_0$ and $\nu_1$. The method leverages Markov-type inequalities and Fourier analysis to translate moment information into operator-norm control, avoiding problem-specific resolvent techniques. The authors obtain quantitative strong-convergence results for random permutation matrices and random regular graphs, including an effective Friedman bound with sharp large-deviation structure (the staircase theorem), and extend the approach to stable representations of the symmetric group, yielding a large family of new strong-convergence examples. These results sharpen convergence rates and broaden the scope of strong convergence, with potential impact on random lifts, graph limits, and operator-algebraic contexts where noncommutative polynomials of random matrices arise.
Abstract
A family of random matrices $\boldsymbol{X}^N=(X_1^N,\ldots,X_d^N)$ is said to converge strongly to a family of bounded operators $\boldsymbol{x}=(x_1,\ldots,x_d)$ when $\|P(\boldsymbol{X}^N,\boldsymbol{X}^{N*})\|\to\|P(\boldsymbol{x}, \boldsymbol{x}^*)\|$ for every noncommutative polynomial $P$. This phenomenon plays a key role in several recent breakthroughs on random graphs, geometry, and operator algebras. However, proofs of strong convergence are notoriously delicate and have relied largely on problem-specific methods. In this paper, we develop a new approach to strong convergence that uses only soft arguments. Our method exploits the fact that for many natural models, the expected trace of $P(\boldsymbol{X}^N,\boldsymbol{X}^{N*})$ is a rational function of $\frac{1}{N}$ whose lowest order asymptotics are easily understood. We develop a general technique to deduce strong convergence directly from these inputs using the inequality of A. and V. Markov for univariate polynomials and elementary Fourier analysis. To illustrate the method, we develop the following applications. 1. We give a short proof of the result of Friedman that random regular graphs have a near-optimal spectral gap, and obtain a sharp understanding of the large deviations probabilities of the second eigenvalue. 2. We prove a strong quantitative form of the strong convergence property of random permutation matrices due to Bordenave and Collins. 3. We extend the above to any stable representation of the symmetric group, providing many new examples of the strong convergence phenomenon.