Stochastic tropical geometry I: Zeros of random tropical polynomials
Speaker: Tran Mai Ngoc (University of Texas, Austin)

Time: 9 am, 08/01/2014
Venue: Room 6, Buiding A14, Institute of Mathematics

Abstract: Tropical geometry is the study of the tropicalization of classical algebraic varieties. Stochastic geometry is the study of random closed set. At their intersection, stochastic tropical geometry, the objects of study are random tropical varieties. The most basic of such varieties are zeros of random tropical polynomials $mathcal{T}f_n(x) = min_{i=1,ldots,n}(c_i + ix)$ whose coefficients $c_i$ are independent and identically distributed. Recently, Tao and Vu studied the non-tropical version of this problem and proved a local universality theorem on the number of zeros. We conjecture that a similar universality theorem also holds in the tropical case.

In this talk, we consider the specific case where the $c_i$'s are standard exponentials. Using techniques from the areas of random polytopes and random permutations, we derive the exact and asymptotic joint distribution of the zeros of $mathcal{T}f_n$. Furthermore, we show that the expected number of zeros of $mathcal{T}f_n$ is asymptotically normal, with expectation $4/3log(n)$ and variance $8/27 log(n)$. This is remarkably similar to known results on expected number of faces of random polytopes in the plane, which lies at the basis of our conjecture. Our work leads to many open problems lying at the interface of stochastic and tropical geometry.  .


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