Zeros of random tropical polynomials, random polygons and stick-breaking (Q2796091)

Consider the tropical algebra, where the sum \(a\bigoplus b\) of two numbers is \(\min(a, b)\) and tropical product \(a\bigodot b\) of two numbers is \(a+ b\). A tropical polynomial \({\mathcal T} f: \mathbb R \to \mathbb R\) of degree \(n\) has the form \({\mathcal T} f(x) =\bigoplus_{i=0}^n (C_i\bigodot x^i) = \min_{i=0,\dots, n} (C_i + i x)\) with coefficients \(C_i\in{\mathbb R}\). The zeros of \({\mathcal T} f\) are points in \({\mathbb R}\) where the minimum is achieved at least twice. When the coefficients \(C_i\) are random, the zeros of \({\mathcal T} f\) are random points in \({\mathbb R}\). A natural model of randomness is one where the \(C_i\) are independent and identically distributed (i.i.d.) according to a distribution \(F\). The main result of the paper is the following statement: Theorem 1. Let \(F\) be a continuous distribution, supported on \((0, +\infty)\). Assume \(F(y) \sim C y^{\alpha} +o(y^{\alpha})\) as \(y\to 0\) for some constants \(C\), \(\alpha>0\). Let \(Z_n\) be the number of zeros of tropical polynomial \({\mathcal T} f\). Then as \(n\to \infty\), normalized random variable \({{Z_n - {{2a+2}\over{2a+1}} \log(n)}\over {\sqrt {{{{2a(a+1)(2a^2+ 2a+1)}\over{(2a+1)^3}}} \log(n)}}}\) tends (in distribution) to standard normal distribution. The key idea of the proof of Theorem 1 appeared in [\textit{P. Groeneboom}, Adv. Appl. Probab. 44, No. 2, 330--342 (2012; Zbl 1251.60011)]. The paper is organized as follows: Section 2 gives the connection between zeros of \({\mathcal T} f\) and the lower convex hull of the points \((i, C_i)\). Sections 3-5 treat the case of the exponentially distribution with parameter 1 via two different proofs: discrete stick-breaking and Poisson coupling. Section 6 proves continuous version of Theorem 1 and Section 7 proves Theorem 1 via a coupling argument. The authors summarize the paper and discuss open problems in Section 8.