The arc length and topology of a random lemniscate

Abstract

A polynomial lemniscate is a curve in the complex plane defined by {z ∈ C : |p(z)| = t}. Erdös, Herzog and Piranian posed the extremal problem of determining the maximum length of a lemniscate Λ = {z ∈ C : |p(z)| = 1} when p is a monic polynomial of degree n. In this paper, we study the length and topology of a random lemniscate whose defining polynomial has independent Gaussian coefficients. When the polynomial is sampled from the Kac ensemble we show that the length approaches a nonzero constant as n → ∞. Concerning the connected components of a random lemniscate, we prove that the average number of them is asymptotically n and that there is a positive probability (independent of n) of a giant component, that is, a connected component whose length is some prescribed portion (say half) of the expected total length.

Expected number and distribution of critical points of real Lefschetz pencils

“…In the real case, Kac [7], Kostlan [8] and Shub and Smale [16] computed the expected number of real roots of a random real polynomial. In higher dimensions, Podkorytov [14] and Bürgisser [1] computed the expected Euler characteristic of random real algebraic submanifolds and Letendre [11] the expected volume (see [12] for the expected length of a random lemniscate). In [3,4,5] Gayet and Welschinger estimated from above and below the Betti numbers of the real locus of real algebraic submanifolds (see also [9]).…”

Some Recent Results on the Geometry of Complex Polynomials: The Gauss--Lucas Theorem, Polynomial Lemniscates, Shape Analysis, and Conformal Equivalence

“…In 2017, E. Lundberg and K. Ramachandran [15] studied the lengths of the lemniscates Λ 1 (p n ), for a random sequence polynomials {p n }, proving the following.…”

The lemniscate tree of a random polynomial

“…Motivated by recent studies on the topology of random real algebraic varieties [14,15,17,16,21,22,24,25,26,29], we investigate a probabilistic version of Arnold's problem by studying the topological properties of the landscape generated by the modulus of a random polynomial. We focus specifically on the statistical properties of a certain binary tree, the lemniscate tree defined below, of a random polynomial with independent identically distributed (i.i.d.)…”