arXiv subjects

Zakhar Kabluchko

Publications and source records attributed to Zakhar Kabluchko.

At least 109 records · Page 6Linked to original sources

arXiv2019

Asymptotic normality for random simplices and convex bodies in high dimensions

Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $n\to\infty$. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is established also for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the $n$-dimensional $\ell_p$-ball. In particular, this includes the cone and the uniform probability measure.

math.MG↗

arXiv2019

Fluctuations of the Magnetization for Ising Models on Dense Erdős-Rényi Random Graphs

We analyze Ising/Curie-Weiss models on the (directed) Erdős-Rényi random graph on $N$ vertices in which every edge is present with probability $p$. These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993]. We prove a quenched Central Limit Theorem for the magnetization in the high-temperature regime $β<1$ when $p=p(N)$ satisfies $p^3N^2\to +\infty$.

math.PR↗

arXiv2019

Angle sums of random simplices in dimensions $3$ and $4$

Consider a random $d$-dimensional simplex whose vertices are $d+1$ random points sampled independently and uniformly from the unit sphere in $\mathbb R^d$. We show that the expected sum of solid angles at the vertices of this random simplex equals $\frac 18$ if $d=3$ and $\frac{539}{288π^2}-\frac 16$ if $d=4$. The angles are measured as proportions of the full solid angle which is normalized to be $1$. Similar formulae are obtained if the vertices of the simplex are uniformly distributed in the unit ball. These results are special cases of general formulae for the expected angle-sums of random beta simplices in dimensions $3$ and $4$.

math.PR↗

arXiv2019

Angles of the Gaussian simplex

Consider a $d$-dimensional simplex whose vertices are random points chosen independently according to the standard Gaussian distribution on $\mathbb R^d$. We prove that the expected angle sum of this random simplex equals the angle sum of the regular simplex of the same dimension $d$.

math.PR↗

arXiv2019

Asymptotics for the expected number of nodal components for random lemniscates

We determine the true asymptotic behaviour for the expected number of connected components for a model of random lemniscates proposed recently by Lerario and Lundberg. These are defined as the subsets of the Riemann sphere, where the absolute value of certain random, $\text{SO}(3)$-invariant rational function of degree $n$ equals to $1$. We show that the expected number of the connected components of these lemniscates, divided by $n$, converges to a positive constant defined in terms of the quotient of two independent plane Gaussian analytic functions. A major obstacle in applying the novel non-local techniques due to Nazarov and Sodin on this problem is the underlying non-Gaussianity, intristic to the studied model.

math.PR↗

arXiv2019

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $n\to\infty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Bárány, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $\|x\|^{-(d+γ)}$, where $γ=1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$-functional of this random convex hull for arbitrary $γ>0$.

math.PR↗

arXiv2018

An infinite-dimensional helix invariant under spherical projections

We classify all subsets $S$ of the projective Hilbert space with the following property: for every point $\pm s_0\in S$, the spherical projection of $S\backslash\{\pm s_0\}$ to the hyperplane orthogonal to $\pm s_0$ is isometric to $S\backslash\{\pm s_0\}$. In probabilistic terms, this means that we characterize all zero-mean Gaussian processes $Z=(Z(t))_{t\in T}$ with the property that for every $s_0\in T$ the conditional distribution of $(Z(t))_{t\in T}$ given that $Z(s_0)=0$ coincides with the distribution of $(φ(t; s_0) Z(t))_{t\in T}$ for some function $φ(t;s_0)$. A basic example of such process is the stationary zero-mean Gaussian process $(X(t))_{t\in\mathbb R}$ with covariance function $\mathbb E [X(s) X(t)] = 1/\cosh (t-s)$. We show that, in general, the process $Z$ can be decomposed into a union of mutually independent processes of two types: (i) processes of the form $(a(t) X(ψ(t)))_{t\in T}$, with $a: T\to \mathbb R$, $ψ(t): T\to \mathbb R$, and (ii) certain exceptional Gaussian processes defined on four-point index sets. The above problem is reduced to the classification of metric spaces in which in every triangle the largest side equals the sum of the remaining two sides.

math.PR↗

arXiv2018

Distribution of complex algebraic numbers on the unit circle

For $-π\leqβ_1<β_2\leqπ$ denote by $Φ_{β_1,β_2}(Q)$ the number of algebraic numbers on the unit circle with arguments in $[β_1,β_2]$ of degree $2m$ and with elliptic height at most $Q$. We show that \[ Φ_{β_1,β_2}(Q)=Q^{m+1}\int\limits_{β_1}^{β_2}{p(t)}\,{\rm d}t+O\left(Q^m\,\log Q\right),\quad Q\to\infty, \] where $p(t)$ coincides up to a constant factor with the density of the roots of some random trigonometric polynomial. This density is calculated explicitly using the Edelman--Kostlan formula.

math.NT↗

arXiv2018

Yet another note on the arithmetic-geometric mean inequality

It was shown by E. Gluskin and V.D. Milman in [GAFA Lecture Notes in Math. 1807, 2003] that the classical arithmetic-geometric mean inequality can be reversed (up to a multiplicative constant) with high probability, when applied to coordinates of a point chosen with respect to the surface unit measure on a high-dimensional Euclidean sphere. We present here two asymptotic refinements of this phenomenon in the more general setting of the surface probability measure on a high-dimensional $\ell_p$-sphere, and also show that sampling the point according to either the cone probability measure on $\ell_p$ or the uniform distribution on the ball enclosed by $\ell_p$ yields the same results. First, we prove a central limit theorem, which allows us to identify the precise constants in the reverse inequality. Second, we prove the large deviations counterpart to the central limit theorem, thereby describing the asymptotic behavior beyond the Gaussian scale, and identify the rate function.

math.CA↗

arXiv2018

Sanov-type large deviations in Schatten classes

Denote by $λ_1(A), \ldots, λ_n(A)$ the eigenvalues of an $(n\times n)$-matrix $A$. Let $Z_n$ be an $(n\times n)$-matrix chosen uniformly at random from the matrix analogue to the classical $\ell_ p^n$-ball, defined as the set of all self-adjoint $(n\times n)$-matrices satisfying $\sum_{k=1}^n |λ_k(A)|^p\leq 1$. We prove a large deviations principle for the (random) spectral measure of the matrix $n^{1/p} Z_n$. As a consequence, we obtain that the spectral measure of $n^{1/p} Z_n$ converges weakly almost surely to a non-random limiting measure given by the Ullman distribution, as $n\to\infty$. The corresponding results for random matrices in Schatten trace classes, where eigenvalues are replaced by the singular values, are also presented.

math.PR↗

arXiv2018

Distances between zeroes and critical points for random polynomials with i.i.d. zeroes

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $ξ_0,ξ_1,\ldots,ξ_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\to\infty$, with high probability there is a critical point of $Q_n$ which is very close to $ξ_0$. We localize the position of this critical point by proving that the difference between $ξ_0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(ξ_0))$ and variance of order $\log n \cdot n^{-3}$. Here, $f(z)= \mathbb E[1/(z-ξ_k)]$ is the Cauchy-Stieltjes transform of the $ξ_k$'s. We also state some conjectures on critical points of polynomials with dependent zeroes, for example the Weyl polynomials and characteristic polynomials of random matrices.

math.PR↗

arXiv2018

Real zeros of random analytic functions associated with geometries of constant curvature

Let $ξ_0, ξ_1, \dots$ be i.i.d. random variables with zero mean and unit variance. We study the following four families of random analytic functions: $\sum_{k=0}^n \sqrt{\binom nk} ξ_k z^k$ (spherical polynomials), $\sum_{k=0}^\infty \sqrt{\frac{n^k}{k!}} ξ_k z^k$ (flat random analytic function), $\sum_{k=0}^\infty \sqrt{\binom {n+k-1} k} ξ_k z^k$ (hyperbolic random analytic functions), $\sum_{k=0}^n \sqrt{\frac{n^k}{k!}} ξ_k z^k$ (Weyl polynomials). We compute explicitly the limiting mean density of real zeroes of these random functions. More precisely, we provide a formula for $\lim_{n\to\infty} n^{-1/2} \mathbb{E}N_n[a,b]$, where $N_n[a, b]$ is the number of zeroes in the interval $[a,b]$.

math.PR↗

arXiv2018

Weak convergence of the number of vertices at intermediate levels of random recursive trees

Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We are interested in the asymptotic behavior of $X_n(k)$ for intermediate levels $k=k_n$ satisfying $k_n\to\infty$ and $k_n=o(\log n)$ as $n\to\infty$. In particular, we prove weak convergence of finite-dimensional distributions for the process $(X_n ([k_n u]))_{u>0}$, properly normalized and centered, as $n\to\infty$. The limit is a centered Gaussian process with covariance $(u,v)\mapsto (u+v)^{-1}$. One-dimensional distributional convergence of $X_n(k_n)$, properly normalized and centered, was obtained with the help of analytic tools by Fuchs, Hwang and Neininger in 2006. In contrast, our proofs which are probabilistic in nature exploit a connection of our model with certain Crump-Mode-Jagers branching processes.

math.PR↗

arXiv2018

Intersection of unit balls in classical matrix ensembles

We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschläger for classical $\ell_p$-balls [Schechtman and Schmuckenschläger, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the asymptotic volume of such matrix unit balls, which in turn is based on the theory of logarithmic potentials with external fields.

math.FA↗

arXiv2018

Exact asymptotic volume and volume ratio of Schatten unit balls

The unit ball $B_p^n(\mathbb{R})$ of the finite-dimensional Schatten trace class $\mathcal S_p^n$ consists of all real $n\times n$ matrices $A$ whose singular values $s_1(A),\ldots,s_n(A)$ satisfy $s_1^p(A)+\ldots+s_n^p(A)\leq 1$, where $p>0$. Saint Raymond [Studia Math.\ 80, 63--75, 1984] showed that the limit $$ \lim_{n\to\infty} n^{1/2 + 1/p} \big(\text{Vol}\, B_p^n(\mathbb{R})\big)^{1/n^2} $$ exists in $(0,\infty)$ and provided both lower and upper bounds. In this paper we determine the precise limiting constant based on ideas from the theory of logarithmic potentials with external fields. A similar result is obtained for complex Schatten balls. As an application we compute the precise asymptotic volume ratio of the Schatten $p$-balls, as $n\to\infty$, thereby extending Saint Raymond's estimate in the case of the nuclear norm ($p=1$) to the full regime $1\leq p \leq \infty$ with exact limiting behavior.

math.FA↗

arXiv2018

Limit theorems for the least common multiple of a random set of integers

Let $L_{n}$ be the least common multiple of a random set of integers obtained from $\{1,\ldots,n\}$ by retaining each element with probability $θ\in (0,1)$ independently of the others. We prove that the process $(\log L_{\lfloor nt\rfloor})_{t\in [0,1]}$, after centering and normalization, converges weakly to a certain Gaussian process that is not Brownian motion. Further results include a strong law of large numbers for $\log L_{n}$ as well as Poisson limit theorems in regimes when $θ$ depends on $n$ in an appropriate way.

math.PR↗

arXiv2018

A functional limit theorem for the profile of random recursive trees

Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem for the vector-valued process $(X_{[n^t]}(1),\ldots, X_{[n^t]}(k))_{t\geq 0}$, for each $k\in\mathbb N$. We show that after proper centering and normalization, this process converges weakly to a vector-valued Gaussian process whose components are integrated Brownian motions. This result is deduced from a functional limit theorem for Crump-Mode-Jagers branching processes generated by increasing random walks with increments that have finite second moment.

math.PR↗

arXiv2017

Expected intrinsic volumes and facet numbers of random beta-polytopes

Let $X_1,\ldots,X_n$ be i.i.d.\ random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,β} (x) = \text{const} \cdot (1-\|x\|^2)^β, \quad \|x\|\leq 1, \quad \text{(the beta case)} $$ and $$ \tilde f_{d,β} (x) = \text{const} \cdot (1+\|x\|^2)^{-β}, \quad x\in\mathbb{R}^d, \quad \text{(the beta' case).} $$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,\ldots,X_n$. Asymptotic formulae where obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when $β\downarrow -1$, respectively $β\uparrow +\infty$, we can also cover the models in which $X_1,\ldots,X_n$ are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of $\pm X_1,\ldots,\pm X_n$ and $0,X_1,\ldots,X_n$. One of the main tools used in the proofs is the Blaschke-Petkantschin formula.

math.MG↗