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Zeros of random tropical polynomials, random polygons and stick-breaking


Authors: Francois Baccelli and Ngoc Mai Tran
Journal: Trans. Amer. Math. Soc. 368 (2016), 7281-7303
MSC (2010): Primary 11S05, 60C05, 60D05, 14T05
DOI: https://doi.org/10.1090/tran/6565
Published electronically: February 2, 2016
MathSciNet review: 3471091
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Abstract: For $ i = 0, 1, \ldots , n$, let $ C_i$ be independent and identically distributed random variables with distribution $ F$ with support $ (0,\infty )$. The number of zeros of the random tropical polynomials $ \mathcal {T}f_n(x) = \min _{i=1,\ldots ,n}(C_i + ix)$ is also the number of faces of the lower convex hull of the $ n+1$ random points $ (i,C_i)$ in $ \mathbb{R}^2$. We show that this number, $ Z_n$, satisfies a central limit theorem when $ F$ has polynomial decay near 0. Specifically, if $ F$ near 0 behaves like a $ gamma(a,1)$ distribution for some $ a > 0$, then $ Z_n$ has the same asymptotics as the number of renewals on the interval $ [0,\log (n)/a]$ of a renewal process with inter-arrival distribution $ -\log (Beta(a,2))$. Our proof draws on connections between random partitions, renewal theory and random polytopes. In particular, we obtain generalizations and simple proofs of the central limit theorem for the number of vertices of the convex hull of $ n$ uniform random points in a square. Our work leads to many open problems in stochastic tropical geometry, the study of functionals and intersections of random tropical varieties.


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Additional Information

Francois Baccelli
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712

Ngoc Mai Tran
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712

DOI: https://doi.org/10.1090/tran/6565
Received by editor(s): April 1, 2014
Received by editor(s) in revised form: August 11, 2014, and September 20, 2014
Published electronically: February 2, 2016
Additional Notes: This work was supported by an award from the Simons Foundation ($#197982$ to The University of Texas at Austin). The authors would like to thank an anonymous referee for the careful reading and helpful suggestions.
Article copyright: © Copyright 2016 American Mathematical Society