Universality of the minimum modulus for random trigonometric polynomials

Universality of the minimum modulus for random trigonometric polynomials, Discrete Analysis 2021:20, 46 pp.

This paper belongs to a long tradition of study of the behaviour of random polynomials that goes back at least to Littlewood. Littlewood was mainly interested in polynomials of the form $\sum_{i=0}^n\epsilon_iz^i$ where the $\epsilon_i$ are chosen independently at random from $\{-1,1\}$. Because such polynomials have real coefficients, we can expect them to have real roots, and a great deal of work has been done on how these roots are typically distributed.

One general aim of research in this area is to prove universality results – that is, results that state that various statistics about the macroscopic behaviour of a random model are insensitive to the precise details of how the random model is defined. Universality is a phenomenon that appears all over probability: perhaps the most basic example is the central limit theorem, which tells us that if we add together a large number of independent random variables with the same mean and variance, then the resulting distribution will be approximately Gaussian regardless of what those variables are. But often it is very hard to prove rigorously that it occurs. For example, it is pretty clear that the limiting behaviour of a self-avoiding walk on the integer lattice in two dimensions as the length gets large is the same as that of a self-avoiding walk on the triangular lattice, or indeed of any “reasonable” model of a two-dimensional self-avoiding walk, but this has not been proved rigorously. One famous universality result that has been proved is that several statistics concerning the distribution of the eigenvalues of random matrices depend only on the first four moments of the matrix entries.

Returning to random polynomials, one would expect that, for example, the random polynomials studied by Littlewood should behave similarly in many ways to random polynomials $\sum_{i=0}^ng_iz^i$, where now the coefficients $g_i$ are independent Gaussians with standard deviation 1. Results of such a kind exist for the behaviour of the roots, but this paper concerns a different question, namely what one can say about the distribution of the minimum modulus of the values taken by the polynomial on the unit circle.

Oren Yakir and Ofer Zeitouni recently proved in the case of Gaussian coefficients that the minimum modulus is exponentially distributed in the limit (once it is appropriately scaled). However, their proof made essential use of the fact that the coefficients were Gaussian, both in certain computations of second-moment type and also in enabling them to use an argument of Liggett characterizing Poisson processes.

This paper dramatically extends the result of Yakir and Zeitouni by proving a universality result, and thereby showing that the Gaussian coefficients, though needed for the proof of Yakir and Zeitouni, are not needed for the conclusion to hold. The authors prove their result under the much weaker assumption that the coefficients are sub-Gaussian: a random variable $X$ is sub-Gaussian if there exist constants $C$ and $\delta>0$ such that $\mathbb P[X\geq t]\leq C\exp(-\delta t^2)$ for every $t>0$. However, as they also comment, all they actually need is that the coefficients have bounded $k$th moments for some sufficiently high $k$. In particular, their result holds for random $\pm 1$ coefficients, which turns out to be in a certain sense the hardest case.

In order to prove their main result, the authors relate the joint distribution of small values of the random polynomial at $m$ points to a random walk in $\mathbb R^{4m}$. In order to analyse the random walk, they develop a high-dimensional local central limit theorem for sums of independent random variables satisfying a condition that ensures that they avoid a certain number-theoretic obstruction. They also obtain small-ball estimates – that is, anticoncentration results that give upper bounds for the probability that the random walk lands inside a certain small ball. These results go well beyond the previous state of the art and are of interest in themselves.

The following video is of a talk given by one of the authors about the results in their paper.