Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials

Oxford Mathematician Dmitry Belyaev is interested in the interface between analysis and probability. Here he discusses his latest work.

"There are two areas of mathematics that clearly have nothing to do with each other: projective geometry and conformally invariant critical models of statistical physics. It turns out that the situation is not as simple as it looks and these two areas might be connected.

We start with projective geometry. Let $g(x):\mathbb{R}^{m+1} \to \mathbb{R}$ be a homogeneous polynomial of degree $n$ in $m + 1$ variables. Although the values of the polynomial are not well defined in homogeneous coordinates $[x_0 : x_1 : \dotsm : x_m]$, but the zero locus, the set where $g([x_0 : x_1 : \dotsm : x_m]) = 0$, is well defined. The set $S = \{x ∈ \mathbb{PR}^m : g(x) = 0\}$ is a projective variety.

We can ask what a typical projective variety looks like. The answer to this question very much depends on the meaning of the word ‘typical’. One possibility is to define some ‘natural’ probability measure on the space of all homogeneous polynomials $g$ and treat ‘typical’ behaviour as almost sure behaviour with respect to this measure. Since the space of polynomials is too large, there is no canonical way to define the most natural uniform measure. Second best choice is a Gaussian measure. This still does not completely determine the measure, but there is one Gaussian measure which stands out: this is the only Gaussian measure which is the real trace of a complex Gaussian measure on space of homogeneous polynomials on $\mathbb{CP}^m$ which is invariant with respect to the unitary group. A random polynomial of degree n with respect to this measure could be written as

$$f_n(x) = f_{n;m}(x) = \sum_{|J|=n}\sqrt{\binom{n}{J}} a_J x^J,$$

where $J = (j_0, . . . , j_m)$ is the multi-index, $|J| = j_0 + \dotsb + j_m$, $\binom{n}{J} = \frac{n!}{j_0! \dotsb j_m!}$, and $\{a_J\}$ are i.i.d. standard Gaussian random variables. This random function is called the Kostlan ensemble or complex Fubini-Study ensemble. We can think that a ‘typical’ variety of degree $n$ is the nodal set of the Kostlan ensemble of degree $n$. We are mostly interested in the two-dimensional case $m = 2$.

It has been shown by V. Beffara and D. Gayet that there is Russo-Seymour-Welsh type estimate for Bargmann-Fock random function which is the scaling limit of the Kostlan ensemble. This means that if one fixes a nice domain with two marked boundary arcs, then the probability that there is a nodal line connecting two arcs inside the domain is bounded from below by a constant which depends on the shape of the domain, but not on its scale. These types of estimates first appeared in the study of critical percolation models and are a strong indication that the corresponding curves have conformally invariant scaling limits.

In the recent work with S. Muirhead and I. Wigman we have extended this result to the Kostlan ensemble on the sphere. Namely, we have obtained a lower bound on the probability to cross a domain which is uniform in the degree of the polynomial and in the scale of the domain. This suggests that large components of a ‘typical’ projective curve have a scaling limit which is conformally invariant and should be described by the Schramm-Loewner Evolution."

For a fuller explanation of Dmitry and colleagues' work please clck here.