Related papers: Partial geodesics on symmetric groups endowed with…

Developing an idea of M. Gromov, we study the intersection formula for random subsets with density. The \textit{density} of a subset $A$ in a finite set $E$ is defined by $dens A := \log_{|E|}(|A|)$. The aim of this article is to give a…

I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume…

This is supplementary material for the main Geodesics article by the authors. In Appendix A, we present some general results on the construction of Gaussian random fields. In Appendix B, we restate our Shape Theorem, specialized to the…

Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry.…

Metrics in Grassmannians, or distances between subspaces of same dimension, have many uses, and extending them to the Total Grassmannian of subspaces of different dimensions is an important problem, as usual extensions lack good properties…

A sequence $\pi_1,\pi_2,\dots$ of permutations is said to be "quasirandom" if the induced density of every permutation $\sigma$ in $\pi_n$ converges to $1/|\sigma|!$ as $n\to\infty$. We prove that $\pi_1,\pi_2,\dots$ is quasirandom if and…

Intersection distribution and non-hitting index are concepts introduced recently by Li and Pott as a new way to view the behaviour of a collection of finite field polynomials. With both an algebraic interpretation via the intersection of a…

In first-passage percolation, we place i.i.d. continuous weights at the edges of Z^2 and consider the weighted graph metric. A distance-minimizing path between points x and y is called a geodesic, and a bigeodesic is a doubly-infinite path…

We study geodesics in the Brownian map $(\mathcal{S},d,\nu)$, the random metric measure space which arises as the Gromov-Hausdorff scaling limit of uniformly random planar maps. Our results apply to all geodesics including those between…

In the first paper in this series we estimated the probability that a random permutation $\pi\in\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $\pi$ has $m$…

It was shown by G. Pisier that any finite-dimensional normed space admits an $\alpha$-regular $M$-position, guaranteeing not only regular entropy estimates but moreover regular estimates on the diameters of minimal sections of its unit-ball…

Let pi = pi_1 pi_2 ... pi_n be a permutation in the symmetric group S_n written in one-line notation. The pinnacle set of pi, denoted Pin pi, is the set of all pi_i such that pi_{i-1} < pi_i > pi_{i+1}. This is an analogue of the…

Consider an i.i.d. sample from an unknown density function supported on an unknown manifold embedded in a high dimensional Euclidean space. We tackle the problem of learning a distance between points, able to capture both the geometry of…

The Gini index is a popular inequality measure with many applications in social and economic studies. This paper studies semiparametric inference on the Gini indices of two semicontinuous populations. We characterize the distribution of…

We study geodesics of the form $\gamma(t)=\pi(\exp(tX)\exp(tY))$, $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $\pi:G\rightarrow G/K$ is the natural projection. These curves naturally generalise homogeneous…

Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded, concave, connected open set $U$ with $C^0$ boundary and $M\setminus U$ is connected. We assume that either the relative homotopy set $\pi_1(M,M\setminus U)=0$ or the…

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive…

A set of points $S$ in Euclidean space $\mathbb{R}^d$ is called \textit{Ramsey} if any finite partition of $\mathbb{R}^{\infty}$ yields a monochromatic copy of $S$. While characterization of Ramsey set remains a major open problem in the…

In first-passage percolation, one places nonnegative i.i.d. random variables (T(e)) on the edges of Z^d. A geodesic is an optimal path for the passage times T(e). Consider a local property of the time environment. We call it a pattern. We…

H. A. Hayden [1] introduced the idea of semi-symmetric non-metric connection on a Riemannian manifold in (1932). Agashe and Chafle \cite{1} defined and studied semi-symmetric non-metric connection on a Riemannian manifold. In the present…