arXiv subjects

Léo Mathis

Publications and source records attributed to Léo Mathis.

8 recordsLinked to original sources

arXiv2025

Real Gaussian exponential sums via a real moment map

We study the expected number of solutions of a system of identically distributed exponential sums with centered Gaussian coefficient and arbitrary variance. We use the Adler and Taylor theory of Gaussian random fields to identify a moment map which allows to express the expected number of solution as an integral over the Newton polytope, in analogy with the Bernstein Khovanskii Kushnirenko Theorem. We apply this result to study the monotonicity of the expected number of solution with respect to the support of the exponential sum in an open set. We find that, when a point is added in the support in the interior of the Newton polytope there exists an open sets where the expected number of solutions decreases, answering negatively to a local version of a conjecture by Bürgisser. When the point added in the support is far enough away from the Newton polytope we show that there is an unbounded open set where the number of solution decreases. We also prove some new lower bounds for the Aronszajn multiplication of exponential sums.

math.PR↗

arXiv2025

Probabilistic intersection theory in Riemannian homogeneous spaces

Let $M=G/H$ be a Riemannian homogeneous space, where $G$ is a compact Lie group with closed subgroup $H$. Classical intersection theory states that the de Rham cohomology ring of $M$ describes the signed count of intersection points of submanifolds $Y_1, \ldots, Y_s$ of $M$ in general position, when the codimensions add up to $\dim M$. We introduce the probabilistic intersection ring $\mathrm{H}_{\mathbb E}(M)$, whose multiplication describes the unsigned count of intersection points, when the $Y_i$ are randomly moved by independent uniformly random elements of $G$. The probabilistic intersection ring $\mathrm{H}_{\mathbb E}(M)$ has the structure of a graded commutative and associative real Banach algebra. It is defined as a quotient of the ring of Grassmann zonoids of a fixed cotangent space $V$ of $M$. The latter was introduced by the authors in [Adv. Math. 402, 2022]. There is a close connection to valuations of convex bodies: $\mathrm{H}_{\mathbb E}(M)$ can be interpreted as a subspace of the space of translation invariant, even, continuous valuations on $V$, whose multiplication coincides with Alesker's multiplication for smooth valuations. We describe the ring structure of the probabilistic intersection ring for spheres, real projective space and complex projective space, relying on Fu [J. Diff. Geo. 72(3), 2006] for the latter case. From this, we derive an interesting probabilistic intersection formula in complex projective space. Finally, we initiate the investigation of the probabilistic intersection ring for real Grassmannians, outlining the construction of a probabilistic version of Schubert Calculus.

math.DG↗

arXiv2024

Gaussian Zonoids, Gaussian determinants and Gaussian random fields

We study the Vitale zonoid (a convex body associated to a probability distribution) associated to a non--centered Gaussian vector. This defines a family of convex bodies, that contains and generalizes ellipsoids, which we call Gaussian zonoids. We show that each Gaussian zonoid can be approximated by an ellipsoid that we compute explicitely. We use this result to give new estimates for the expectation of the absolute value of the determinant of a non--centered Gaussian matrix in terms of mixed volume of ellipsoids. Finally, exploiting a recent link between random fields and zonoids uncovered by Stecconi and the author, we apply our results to the study of the zero set of non--centered Gaussian random fields. We show how these can be approximated by a suitable centered Gaussian random field and give a quantitative asymptotic in the limit where the variance goes to zero.

math.PR↗

arXiv2023

Expectation of a random submanifold: the zonoid section

We develop a calculus based on zonoids - a special class of convex bodies - for the expectation of functionals related to a random submanifold $Z$ defined as the zero set of a smooth vector valued random field on a Riemannian manifold. We identify a convenient set of hypotheses on the random field under which we define its zonoid section, an assignment of a zonoid $ζ(p)$ in the exterior algebra of the cotangent space at each point $p$ of the manifold. We prove that the first intrinsic volume of $ζ(p)$ is the Kac-Rice density of the expected volume of $Z$, while its center computes the expected current of integration over $Z$. We show that the intersection of random submanifolds corresponds to the wedge product of the zonoid sections and that the preimage corresponds to the pull-back. Combining this with the recently developed zonoid algebra, it allows to give a multiplication structure to the Kac-Rice formulas, resembling that of the cohomology ring of a manifold. Moreover, it establishes a connection with the theory of convex bodies and valuations, which includes very deep and difficult results such as the Alexandrov-Fenchel inequality and the Brunn-Minkowsky inequality. We export them to this context to prove two analogous new inequalities for random submanifolds. Applying our results in the context of Finsler geometry, we prove some new Crofton formulas for the length of curves and the Holmes-Thompson volumes of submanifolds in a Finsler manifold.

math.PR↗

arXiv2022

Fiber Convex Bodies

In this paper we study the fiber body, that is the extension of the notion of fiber polytopes for more general convex bodies. After giving an overview of the properties of the fiber body, we focus on three particular classes of convex bodies. First we describe the strict convexity of the fiber body of the so called puffed polytopes. Then we provide an explicit equation for the support function of the fiber body of some smooth convex bodies. Finally we give a formula that allows to compute the fiber body of a zonoid with a particular focus on the so called discotopes. Throughout the paper we illustrate our results with detailed examples.

math.MG↗

arXiv2022

The zonoid algebra, generalized mixed volumes, and random determinants

We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordered ring, which we call the zonoid algebra. This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. We also analyze a similar construction based on the complex wedge product, which leads to the new notion of mixed $J$-volume. These ideas connect to the theory of random determinants.

math.MG↗

arXiv2021

On tameness of zonoids

We prove that in a globally subanalytic family of convex bodies the set of zonoids is log-analytic, and in particular it is definable in the o-minimal structure generated by globally subanalytic sets and the graph of the exponential function.

math.MG↗

arXiv2019

Probabilistic Schubert Calculus: asymptotics

In the recent paper [arXiv:1612.06893] P. Bürgisser and A. Lerario introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by $δ_{k,n}$ the average number of projective $k$-planes in $\mathbb{R}\textrm{P}^n$ that intersect $(k+1)(n-k)$ many random, independent and uniformly distributed linear projective subspaces of dimension $n-k-1$. They called $δ_{k,n}$ the expected degree of the real Grassmannian $\mathbb{G}(k,n)$ and, in the case $k=1$, they proved that: $$ δ_{1,n}= \frac{8}{3π^{5/2}} \cdot \left(\frac{π^2}{4}\right)^n \cdot n^{-1/2} \left( 1+\mathcal{O}\left(n^{-1}\right)\right) .$$ Here we generalize this result and prove that for every fixed integer $k>0$ and as $n\to \infty$, we have \begin{equation*} δ_{k,n}=a_k \cdot \left(b_k\right)^n\cdot n^{-\frac{k(k+1)}{4}}\left(1+\mathcal{O}(n^{-1})\right) \end{equation*} where $a_k$ and $b_k$ are some (explicit) constants, and $a_k$ involves an interesting integral over the space of polynomials that have all real roots. For instance: $$δ_{2,n}= \frac{9\sqrt{3}}{2048\sqrt{2π}} \cdot 8^n \cdot n^{-3/2} \left( 1+\mathcal{O}\left(n^{-1}\right)\right).$$ Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and we give an explicit formula for $δ_{1,n}$ involving a one dimensional integral of certain combination of Elliptic functions.

math.AG↗