Let q^n be a continuous density function in n-dimensional Euclidean space.
We think of q^n as the density function of some random sequence X^n with values in \BbbR^n. For I\subset[1,n], let X_I denote the collection of coordinates X_i, i\in I, and let \bar X_I denote the collection of coordinates
X_i, i\notin I. We denote by Q_I(x_I|\bar x_I) the joint conditional density function of X_I, given \bar X_I. We prove measure concentration for q^n in the case when, for an appropriate class of sets I, (i) the conditional densities Q_I(x_I|\bar x_I), as functions of x_I, uniformly satisfy a logarithmic
Sobolev inequality and (ii) these conditional densities also satisfy a contractivity condition related to Dobrushin and Shlosman's strong mixing condition.
Concentration for random Euclidean combinatorial optimization
Institut Élie Cartan de Lorraine, CNRS, Université de Lorraine（洛林埃利嘉当研究所，法国国家科学研究中心，洛林大学）；Department of Mathematics, CIT School, Technical University of Munich & Munich Center for Machine Learning (MCML)（慕尼黑工业大学计算与信息技术学院数学系，慕尼黑机器学习中心）；Università di Pisa（比萨大学）
Given two vectors in Euclidean space, how unlikely is it that a random vector has a larger inner product with the shorter vector than with the longer one? When the random vector has independent, identically distributed components, we conjecture that this probability is no more than a constant multiple of the ratio of the Euclidean norms of the two given vectors, up to an additive term to allow for the possibility that the longer vector has more arithmetic structure. We give some partial results to support the basic conjecture.
We show sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order $d-1$ for any $d \in \mathbb{N}$. Here we focus on differentiable functions on the Euclidean space in presence of a Poincaré-type inequality. The bounds are based on $d$-th order derivatives.
The ellipses model is a continuum percolation process in which ellipses with random orientation and eccentricity are placed in the plane according to a Poisson point process. A parameter $α$ controls the tail distribution of the major axis' distribution and we focus on the regime $α\in (1,2)$ for which there exists a unique infinite cluster of ellipses and this cluster fulfills the so called highway property. We prove that the distance within this infinite cluster behaves asymptotically like the (unrestricted) Euclidean distance in the plane. We also show that the chemical distance between poi…
We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this set: a union of disjoint disks, a union of disjoint balls, a union of disjoint line segments, and the boundary of a polyhedron. In the first three cases, we provide closed-form expressions of the cumulative distribution function and the density. In the last case, we propose an algorithm with complexity O(n ln n), n being the number of edges of the polyhedron…
Nonhomogeneous Euclidean first-passage percolation and distance learning
IMAS-CONICET, Departamento de Matemática, Fac. Cs. Exactas y Naturales, Universidad de Buenos Aires（IMAS-CONICET，数学系，理学院，布宜诺斯艾利斯大学）；NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学-华东师范大学上海纽约大学数学科学研究所）；Intituto de Cálculo, Fac. Cs. Exactas y Naturales, Universidad de Buenos Aires（计算研究所，理学院，布宜诺斯艾利斯大学）；Aristas S.R.L.（Aristas有限责任公司）；Departamento de Matemática, Fac. Cs. Exactas y Naturales, Universidad de Buenos Aires（数学系，理学院，布宜诺斯艾利斯大学）
Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on i.i.d.\ vertex weights and the Euclidean distance $|x-y|$. Depending on the various parameters involved, we get a rich phase diagram. We study graph distances and compare it to the Euclidean distance of the vertices. Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correc…
We study nodal component count of the following Gaussian Laplace eigenfunctions: monochromatic random waves (MRW) on $\mathbb{R}^2$, arithmetic random waves (ARW) on $\mathbb{T}^2$ and random spherical harmonics (RSH) on $\mathbb{S}^2$. Exponential concentration for nodal component count of RSH on $\mathbb{S}^2$ and ARW on $\mathbb{T}^2$ were established by Nazarov-Sodin and Rozenshein respectively. We prove exponential concentration for nodal component count in the following three cases: MRW on growing Euclidean balls in $\mathbb{R}^2$; RSH and ARW on geodesic balls, in $\mathbb{S}^2$ and $\m…
We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the {\it Swiss cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.
