arXiv subjects

Zakhar Kabluchko

Publications and source records attributed to Zakhar Kabluchko.

At least 91 records · Page 5Linked to original sources

arXiv2020

Large Deviation Principles for Lacunary Sums

Let $(a_k)_{k\in\mathbb N}$ be a sequence of integers satisfying the Hadamard gap condition $a_{k+1}/a_k>q>1$ for all $k\in\mathbb N$, and let $$ S_n(ω) = \sum_{k=1}^n\cos(2πa_k ω),\qquad n\in\mathbb N,\;ω\in [0,1]. $$ The lacunary trigonometric sum $S_n$ is known to exhibit several properties typical for sums of independent random variables. In this paper we initiate the investigation of large deviation principles (LDPs) for $S_n$. Under the large gap condition $a_{k+1}/a_k\to\infty$, we prove that $(S_n/n)_{n\in\mathbb N}$ satisfies an LDP with speed $n$ and the same rate function $\tilde{I}$ as for sums of independent random variables with the arcsine distribution, but show that the LDP may fail to hold when we only assume the Hadamard gap condition. However, we prove that in the special case $a_k=q^k$ for some $q\in \{2,3,\ldots\}$, $(S_n/n)_{n\in\mathbb N}$ satisfies an LDP with speed $n$ and a rate function $I_q$ different from $\tilde{I}$. We also show that $I_q$ converges pointwise to $\tilde I$ as $q\to\infty$ and construct a random perturbation $(a_k)_{k\in\mathbb N}$ of the sequence $(2^k)_{k\in\mathbb N}$ for which $a_{k+1}/a_k\to 2$ as $k\to\infty$, but for which $(S_n/n)_{n\in\mathbb N}$ satisfies an LDP with the rate function $\tilde{I}$ as in the independent case and not, as one might na{ï}vely expect, with rate function $I_2$. We relate this fact to the number of solutions of certain Diophantine equations. Our results show that LDPs for lacunary trigonometric sums are sensitive to the arithmetic properties of $(a_k)_{k\in\mathbb N}$. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem by Salem and Zygmund or in the law of the iterated logarithm by Erdös and Gál. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems.

math.PR↗

arXiv2020

An identity for the coefficients of characteristic polynomials of hyperplane arrangements

Consider a finite collection of affine hyperplanes in $\mathbb R^d$. The hyperplanes dissect $\mathbb R^d$ into finitely many polyhedral chambers. For a point $x\in \mathbb R^d$ and a chamber $P$ the metric projection of $x$ onto $P$ is the unique point $y\in P$ minimizing the Euclidean distance to $x$. The metric projection is contained in the relative interior of a uniquely defined face of $P$ whose dimension is denoted by $\text{dim}(x,P)$. We prove that for every given $k\in \{0,\ldots, d\}$, the number of chambers $P$ for which $\text{dim}(x,P) = k$ does not depend on the choice of $x$, with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the $k$-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements, Proc. Amer. Math. Soc., 138(8): 2873-2887, 2010].

math.MG↗

arXiv2020

Expected $f$-vector of the Poisson Zero Polytope and Random Convex Hulls in the Half-Sphere

We prove an explicit combinatorial formula for the expected number of faces of the zero polytope of the homogeneous and isotropic Poisson hyperplane tessellation in $\mathbb R^d$. The expected $f$-vector is expressed through the coefficients of the polynomial $$ (1+ (d-1)^2x^2) (1+(d-3)^2 x^2) (1+(d-5)^2 x^2) \ldots. $$ Also, we compute explicitly the expected $f$-vector and the expected volume of the spherical convex hull of $n$ random points sampled uniformly and independently from the $d$-dimensional half-sphere. In the case when $n=d+2$, we compute the probability that this spherical convex hull is a spherical simplex, thus solving an analogue of the Sylvester four-point problem on the half-sphere.

math.PR↗

arXiv2020

How long is the convex minorant of a one-dimensional random walk?

We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized.

math.PR↗

arXiv2020

Angle sums of random polytopes

For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family are the Gaussian polytopes defined as convex hulls of i.i.d. samples from a non-degenerate Gaussian distribution in $\mathbb R^d$. The second family are convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are non-trivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a special case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known but it seems that no detailed proofs can be found in the existing literature.

math.PR↗

arXiv2020

The maximum entropy principle and volumetric properties of Orlicz balls

We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a result of Schechtman and Schmuckenschläger [GAFA, Lecture notes in Math. 1469 (1991), 174--178] for $\ell_p^d$-balls. As another application, we determine the precise asymptotic volume ratio for $2$-concave Orlicz spaces $\ell_M^d$. Our method rests on ideas from statistical mechanics and large deviations theory, more precisely the maximum entropy or Gibbs principle for non-interacting particles, and presents a natural approach and fresh perspective to such geometric and volumetric questions. In particular, our approach explains how the $p$-generalized Gaussian distribution occurs in problems related to the geometry of $\ell_p^d$-balls, which are Orlicz balls when the Orlicz function is $M(t) = |t|^p$.

math.FA↗

arXiv2020

Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes

Consider a random simplex $[X_1,\ldots,X_n]$ defined as the convex hull of independent identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^{n-1}$ with the following beta density: $$ f_{n-1,β} (x) \propto (1-\|x\|^2)^β 1_{\{\|x\| < 1\}}, \qquad x\in\mathbb{R}^{n-1}, \quad β>-1. $$ Let $J_{n,k}(β)$ be the expected internal angle of the simplex $[X_1,\ldots,X_n]$ at its face $[X_1,\ldots,X_k]$. Define $\tilde J_{n,k}(β)$ analogously for i.i.d. random points distributed according to the beta' density $$ \tilde f_{n-1,β} (x) \propto (1+\|x\|^2)^{-β}, \qquad x\in\mathbb{R}^{n-1}, \quad β> \frac{n-1}{2}. $$ We derive formulae for $J_{n,k}(β)$ and $\tilde J_{n,k}(β)$ which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of $β$. For $J_{n,1}(\pm 1/2)$ we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry. (i) We compute the expected $f$-vectors of the typical Poisson-Voronoi cells in dimensions up to $10$. (ii) Consider the random polytope $K_{n,d} := [U_1,\ldots,U_n]$ where $U_1,\ldots,U_n$ are i.i.d. random points sampled uniformly inside some $d$-dimensional convex body $K$ with smooth boundary and unit volume. M. Reitzner proved the existence of the limit of the normalized expected $f$-vector of $K_{n,d}$: $$ \lim_{n\to\infty} n^{-{\frac{d-1}{d+1}}}\mathbb E \mathbf f(K_{n,d}) = \mathbf c_d \cdot Ω(K), $$ where $Ω(K)$ is the affine surface area of $K$, and $\mathbf c_d$ is an unknown vector not depending on $K$. We compute $\mathbf c_d$ explicitly in dimensions up to $d=10$ and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.

math.PR↗

arXiv2020

Limit theorems for random points in a simplex

In this work the $\ell_q$-norms of points chosen uniformly at random in a centered regular simplex in high dimensions are studied. Berry-Esseen bounds in the regime $1\leq q < \infty$ are derived and complemented by a non-central limit theorem together with moderate and large deviations in the case where $q=\infty$. A comparison with corresponding results for $\ell_p^n$-balls is carried out as well.

math.PR↗

arXiv2020

Faces in random great hypersphere tessellations

The concept of typical and weighted typical spherical faces for tessellations of the $d$-dimensional unit sphere, generated by $n$ independent random great hyperspheres distributed according to a non-degenerate directional distribution, is introduced and studied. Probabilistic interpretations for such spherical faces are given and their directional distributions are determined. Explicit formulas for the expected $f$-vector, the expected spherical Quermaßintegrals and the expected spherical intrinsic volumes are found in the isotropic case. Their limiting behaviour as $n\to\infty$ is discussed and compared to the corresponding notions and results in the Euclidean case. The expected statistical dimension and a problem related to intersection probabilities of spherical random polytopes is investigated.

math.PR↗

arXiv2020

A new approach to weak convergence of random cones and polytopes

A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation generated by $n$ independent and uniformly distributed random linear hyperplanes in $\mathbb{R}^{d+1}$ weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in $\mathbb{R}^d$, as $n \to \infty$.

math.PR↗

arXiv2020

Angles of Random Simplices and Face Numbers of Random Polytopes

Pick $d+1$ points uniformly at random on the unit sphere in $\mathbb R^d$. What is the expected value of the angle sum of the simplex spanned by these points? Choose $n$ points uniformly at random in the $d$-dimensional ball. What is the expected number of faces of their convex hull? We answer these and some related questions of stochastic geometry. To this end, we compute expected internal angles of random simplices whose vertices are independent random points sampled from one of the following $d$-dimensional distributions: (i) the beta distribution with the density proportional to $(1-\|x\|^2)^β$, where $x$ is belongs to the unit ball in $\mathbb R^d$; (ii) the beta' distribution with the density proportional to $(1+\|x\|^2)^{-β}$, where $x\in\mathbb{R}^{d}$. These results imply explicit formulae for the expected face numbers of the following random polytopes: (a) the typical Poisson-Voronoi cell; (b) the zero cell of the Poisson hyperplane tessellation; (c) beta and beta' polytopes defined as convex hulls of i.i.d. samples from the corresponding distributions.

math.PR↗

arXiv2020

Fluctuations for the partition function of Ising models on Erdös-Rényi random graphs

We analyze Ising/Curie-Weiss models on the Erdős-Rényi graph with $N$ vertices and edge probability $p=p(N)$ that were introduced by Bovier and Gayrard [J.\ Statist.\ Phys., 72(3-4):643--664, 1993] and investigated in two previous articles by the authors. We prove Central Limit Theorems for the partition function of the model and -- at other decay regimes of $p(N)$ -- for the logarithmic partition function. We find critical regimes for $p(N)$ at which the behavior of the fluctuations of the partition function changes.

math.PR↗

arXiv2020

Beta polytopes and Poisson polyhedra: $f$-vectors and angles

We study random polytopes of the form $[X_1,\ldots,X_n]$ defined as convex hulls of independent and identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^d$ with one of the following densities: $$ f_{d,β} (x) = c_{d,β} (1-\|x\|^2)^β, \qquad \|x\| < 1, \quad \text{(beta distribution, $β>-1$)} $$ or $$ \tilde f_{d,β} (x) = \tilde{c}_{d,β} (1+\|x\|^2)^{-β}, \qquad x\in\mathbb{R}^d, \quad \text{(beta' distribution, $β>d/2$)}. $$ This setting also includes the uniform distribution on the unit sphere and the standard normal distribution as limiting cases. We derive exact and asymptotic formulae for the expected number of $k$-faces of $[X_1,\ldots,X_n]$ for arbitrary $k\in\{0,1,\ldots,d-1\}$. We prove that for any such $k$ this expected number is strictly monotonically increasing with $n$. Also, we compute the expected internal and external angles of these polytopes at faces of every dimension and, more generally, the expected conic intrinsic volumes of their tangent cones. By passing to the large $n$ limit in the beta' case, we compute the expected $f$-vector of the convex hull of Poisson point processes with power-law intensity function. Using convex duality, we derive exact formulae for the expected number of $k$-faces of the zero cell for a class of isotropic Poisson hyperplane tessellations in $\mathbb R^d$. This family includes the zero cell of a classical stationary and isotropic Poisson hyperplane tessellation and the typical cell of a stationary Poisson--Voronoi tessellation as special cases. In addition, we prove precise limit theorems for this $f$-vector in the high-dimensional regime, as $d\to\infty$. Finally, we relate the $d$-dimensional beta and beta' distributions to the generalized Pareto distributions known in extreme-value theory.

math.PR↗

arXiv2020

Grassmann angles and absorption probabilities of Gaussian convex hulls

Let $M$ be an arbitrary subset in $\mathbb R^n$ with a conic (or positive) hull $C$. Consider its Gaussian image $AM$, where $A$ is a $k\times n$-matrix whose entries are independent standard Gaussian random variables. We show that the probability that the convex hull of $AM$ contains the origin in its interior coincides with the $k$-th Grassmann angle of $C$. Also, we prove that the expected Grassmann angles of $AC$ coincide with the corresponding Grassmann angles of $C$. Using the latter result, we show that the expected sum of $j$-th Grassmann angles at $\ell$-dimensional faces of a Gaussian simplex equals the analogous angle-sum for the regular simplex of the same dimension.

math.PR↗

arXiv2019

Absorption probabilities for Gaussian polytopes and regular spherical simplices

The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of $n$ independent standard normally distributed points in $\mathbb R^d$. We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb R^d$ as a function of the Euclidean norm of $x$, and the probability that $\mathcal P_{n,d}$ contains the point $σX$, where $σ\geq 0$ is constant and $X$ is a standard normal vector independent of $\mathcal P_{n,d}$. As a by-product, we also compute the expected number of $k$-faces and the expected volume of $\mathcal P_{n,d}$, thus recovering the results of Affentranger and Schneider [Discr. and Comput. Geometry, 1992] and Efron [Biometrika, 1965], respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $Φ(z)$ and its complex version $Φ(iz)$. The main tool used in the proofs is the conic version of the Crofton formula.

math.PR↗

arXiv2019

Fluctuations of the Magnetization for Ising Models on Erdős-Rényi Random Graphs -- the Regimes of Small p and the Critical Temperature

We continue our analysis of Ising models on the (directed) Erdős-Rényi random graph. This graph is constructed on $N$ vertices and every edge has probability $p$ to be present. These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993] and analyzed by the authors in a previous note, in which we consider the case of $p=p(N)$ satisfying $p^3N^2\to +\infty$ and $β<1$. In the current note we prove a quenched Central Limit Theorem for the magnetization for $p$ satisfying $pN \to \infty$ in the high-temperature regime $β<1$. We also show a non-standard Central Limit Theorem for $p^4N^3 \to \infty$ at the critical temperature $β=1$. For $p^4N^3 \to 0$ we obtain a Gaussian limiting distribution for the magnetization. Finally, on the critical line $p^4N^3 \to c$ the limiting distribution for the magnetization contains a quadratic component as well as a $x^4$-term. Hence, at $β=1$ we observe a phase transition in $p$ for the fluctuations of the magnetization.

math.PR↗

arXiv2019

A new look at random projections of the cube and general product measures

A strong law of large numbers for $d$-dimensional random projections of the $n$-dimensional cube is derived. It shows that with respect to the Hausdorff distance a properly normalized random projection of $[-1,1]^n$ onto $\mathbb{R}^d$ almost surely converges to a centered $d$-dimensional Euclidean ball of radius $\sqrt{2/π}$, as $n\to\infty$. For every point inside this ball we determine the asymptotic number of vertices and the volume of the part of the cube projected `close' to this point. Moreover, large deviations for random projections of general product measures are studied. Let $ν^{\otimes n}$ be the $n$-fold product measure of a Borel probability measure $ν$ on $\mathbb{R}$, and let $I$ be uniformly distributed on the Stiefel manifold of orthogonal $d$-frames in $\mathbb{R}^n$. It is shown that the sequence of random measures $ν^{\otimes n}\circ(n^{-1/2}I^*)^{-1}$, $n\in\mathbb{N}$, satisfies a large deviations principle with probability $1$. The rate function is explicitly identified in terms of the moment generating function of $ν$. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube $[-1,1]^n$ and the discrete cube $\{-1,1\}^n$ as a special cases.

math.PR↗

arXiv2019

High-dimensional limit theorems for random vectors in $\ell_p^n$-balls. II

In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $\ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guédon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of $\ell_p^n$-balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Kabluchko, Prochno, Thäle: High-dimensional limit theorems for random vectors in $\ell_p^n$-balls, Commun. Contemp. Math. (2019)].

math.PR↗