arXiv subjects

Zakhar Kabluchko

Publications and source records attributed to Zakhar Kabluchko.

At least 37 records · Page 2Linked to original sources

arXiv2025

On the shape of the typical Poisson-Voronoi cell in high dimensions

We study the typical cell of the Poisson-Voronoi tessellation. We show that when divided by the $d$-th root of the intensity parameter $λ$ of the Poisson process times the volume of the unit ball, the inradius, outradius, diameter and mean width of the typical cell converge in probability to the constants $1/2, 1, 2, 2$ respectively, as the dimension $d\to\infty$. We also show that the width of the typical cell, when rescaled in the same way, is bounded between $2\sqrt{5}/(2+\sqrt{5})-o_d(1)$ and $3/2+o_d(1)$, with probability $1-o_d(1)$. These results in particular imply that, with probability $1-o_d(1)$, the Hausdorff distance between the typical cell and any ball is at least of the order of the diameter of the typical cell. In addition, we show that for all $k$ with $d-k\to\infty$, with probability $1-o_d(1)$, all faces of dimension $k$ have a diameter that is of a much smaller order than the diameter, inradius, etc., of the full typical cell. The same is true for ''almost all'' faces of dimension $d-k$ with $k$ fixed. And, we show that the number of such faces is $\left( (k+1)^{(k+1)/2} / k^{k/2} \pm o_d(1) \right)^d$ with probability $1-o_d(1)$.

math.PR↗

arXiv2025

Sylvester's problem for beta-type distributions

Consider $d+2$ i.i.d. random points $X_1,\ldots, X_{d+2}$ in $\mathbb R^d$. In this note, we compute the probability that their convex hull is a simplex focusing on three specific distributional settings: (i) the distribution of $X_1$ is multivariate standard normal; (ii) the density of $X_1$ is proportional to $(1-\|x\|^2)^β$ on the unit ball (the beta distribution); (iii) the density of $X_1$ is proportional to $(1+\|x\|^2)^{-β}$ (the beta prime distribution). In the Gaussian case, we show that this probability equals twice the sum of the solid angles of a regular $(d+1)$-dimensional simplex.

math.PR↗

arXiv2025

Angles of orthocentric simplices

A $d$-dimensional simplex in Euclidean space is called orthocentric if all of its altitudes intersect at a single point, referred to as the orthocenter. We explicitly compute the internal and external angles at all faces of an orthocentric simplex. To this end, we introduce a parametric family of polyhedral cones, called orthocentric cones, and derive formulas for their angles and, more generally, for their conic intrinsic volumes. We characterize the tangent and normal cones of orthocentric simplices in terms of orthocentric cones with explicit parameters. Depending on whether the orthocenter lies inside the simplex, on its boundary, or outside, the simplex is classified as acute, rectangular, or obtuse, respectively. The solid angle formulas differ in these three cases. As a probabilistic application of the angle formulas, we explicitly compute the expected number of $k$-dimensional faces and the expected volume of the random polytope $[g_1/τ_1, \ldots, g_n/τ_n]$, where $g_1, \ldots, g_n$ are independent standard Gaussian vectors in $\mathbb{R}^d$, and $τ_1, \ldots, τ_n > 0$ are constants.

math.MG↗

arXiv2025

Zeros and exponential profiles of polynomials I: Limit distributions, finite free convolutions and repeated differentiation

Given a sequence of polynomials $(P_n)_{n \in \mathbb{N}}$ with only nonpositive zeros, the aim of this article is to present a user-friendly approach for determining the limiting zero distribution of $P_n$ as $\mathrm{deg}\, P_n \to \infty$. The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution $μ$ and the existence of an exponential profile $g$ associated with the coefficients of the polynomials $(P_n)_{n \in \mathbb{N}}$. The exponential profile $g$, which can be roughly described by $[z^k]P_n(z) \approx \exp(n g(k/n))$, offers a direct route to computing the Cauchy transform $G$ of $μ$: the functions $t \mapsto tG(t)$ and $α\mapsto \exp(-g'(α))$ are mutual inverses. This relationship, in various forms, has previously appeared in the literature, most notably in the paper [Van Assche, Fano and Ortolani, SIAM J. Math. Anal., 1987]. As a first contribution, we present a self-contained probabilistic proof of this equivalence by representing the polynomials as generating functions of sums of independent Bernoulli random variables. This probabilistic framework naturally lends itself to tools from large deviation theory, such as the exponential change of measure. The resulting theorems generalize and unify a range of previously known results, which were traditionally established through analytic or combinatorial methods. Secondly, using the profile-based approach, we investigate how the exponential profile and the limiting zero distribution behave under certain operations on polynomials, including finite free convolutions, Hadamard products, and repeated differentiation. In particular, our approach yields new proofs of the convergence results `$\boxplus_n \to \boxplus$' and `$\boxtimes_n \to \boxtimes$', extending them to cases where the distributions are not necessarily compactly supported.

math.PR↗

arXiv2025

Beta Polytopes and Beta Cones: An Exactly Solvable Model in Geometric Probability

Let $X_1,\ldots, X_n$ be independent random points in the unit ball of $\mathbb R^d$ such that $X_i$ follows a beta distribution with the density proportional to $(1-\|x\|^2)^{β_i}1_{\{\|x\| <1\}}$. Here, $β_1,\ldots, β_n> -1$ are parameters. We study random polytopes of the form $[X_1,\ldots,X_n]$, called beta polytopes. We determine explicitly expected values of several functionals of these polytopes including the number of $k$-dimensional faces, the volume, the intrinsic volumes, the total $k$-volume of the $k$-skeleton, various angle sums, and the $S$-functional which generalizes and unifies many of the above examples. We identify and study the central object needed to analyze beta polytopes: beta cones. For these, we determine explicitly expected values of several functionals including the solid angle, conic intrinsic volumes and the number of $k$-dimensional faces. We identify expected conic intrinsic volumes of beta cones as a crucial quantity needed to express all the functionals mentioned above. We obtain a formula for these expected conic intrinsic volumes in terms of a function $Θ$ for which we provide an explicit integral representation. The proofs combine methods from integral and stochastic geometry with the study of the analytic properties of the function $Θ$.

math.PR↗

arXiv2025

Almost periodic stochastic processes with applications to analytic number theory

A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if $f$ is a Besicovitch almost periodic function and $V$ is a random variable uniformly distributed on $[-1,1]$, then the random variables $f(L\cdot V)$ converge in distribution, as $L\to\infty$, to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes $(f(L\cdot V+t))_{t\in\mathbb{R}}$ in the space of Besicovitch almost periodic functions, and also in the sense of weak convergence of finite-dimensional distributions. We further investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of [Limiting distributions of the classical error terms of prime number theory, Quart. J. Math. 65 (2014), 743--780] and earlier works.

math.PR↗

arXiv2024

Lee--Yang zeroes of the Curie--Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow

The backward heat flow on the real line started from the initial condition $z^n$ results in the classical $n$-th Hermite polynomial whose zeroes are distributed according to the Wigner semicircle law in the large $n$ limit. Similarly, the backward heat flow with the periodic initial condition $(\sin \frac θ2)^n$ leads to trigonometric or unitary analogues of the Hermite polynomials. These polynomials are closely related to the partition function of the Curie--Weiss model and appeared in the work of Mirabelli on finite free probability. We relate the $n$-th unitary Hermite polynomial to the expected characteristic polynomial of a unitary random matrix obtained by running a Brownian motion on the unitary group $U(n)$. We identify the global distribution of zeroes of the unitary Hermite polynomials as the free unitary normal distribution. We also compute the asymptotics of these polynomials or, equivalently, the free energy of the Curie--Weiss model in a complex external field. We identify the global distribution of the Lee--Yang zeroes of this model. Finally, we show that the backward heat flow applied to a high-degree real-rooted polynomial (respectively, trigonometric polynomial) induces, on the level of the asymptotic distribution of its roots, a free Brownian motion (respectively, free unitary Brownian motion).

math.PR↗

arXiv2024

Strange shadows of $\ell_p$-balls

We prove a large deviations principle for orthogonal projections of the unit ball $\mathbb{B}_p^n$ of $\ell_p^n$ onto a random $k$-dimensional linear subspace of $\mathbb{R}^n$ as $n\to\infty$ in the case $2<p\le \infty$ and for the intersection of $\mathbb{B}_p^n$ with a random $k$-dimensional subspace in the case $1\le p <2$. The corresponding rate function is finite only on $L_q$-zonoids and their duals, respectively, and given in terms of the maximum entropy over suitable measures generating the $L_q$-zonoid, where $\frac{1}{p}+\frac{1}{q}=1$. In particular, we obtain that the renormalized projections/sections almost surely tend to a $k$-dimensional Euclidean ball of certain radius. Moreover, we identify the asymptotic probability that the random orthogonal projection remains within a ball of smaller radius. As a byproduct we obtain an interesting inequality for the Gamma function.

math.PR↗

arXiv2024

Propagation of chaos and residual dependence in Gibbs measures on finite sets

We compare a mean-field Gibbs distribution on a finite state space on $N$ spins to that of an explicit simple mixture of product measures. This illustrates the situation beyond the so-called increasing propagation of chaos introduced by Ben Arous and Zeitouni 1999, where marginal distributions of size $k=o(N)$ are compared to product measures.

math.PR↗

arXiv2024

Generalised convexity with respect to families of affine maps

The standard convex closed hull of a set is defined as the intersection of all images, under the action of a group of rigid motions, of a half-space containing the given set. In this paper we propose a generalisation of this classical notion, that we call a $(K,\mathbb{H})$-hull, and which is obtained from the above construction by replacing a half-space with some other convex closed subset $K$ of the Euclidean space, and a group of rigid motions by a subset $\mathbb{H}$ of the group of invertible affine transformations. The main focus is put on the analysis of $(K,\mathbb{H})$-convex hulls of random samples from $K$.

math.MG↗

arXiv2024

Zeros of random polynomials undergoing the heat flow

We investigate the evolution of the empirical distribution of the complex roots of high-degree random polynomials, when the polynomial undergoes the heat flow. In one prominent example of Weyl polynomials, the limiting zero distribution evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law, as was recently conjectured for characteristic polynomials of random matrices by Hall and Ho, 2022. Moreover, for a general family of random polynomials with independent coefficients and isotropic limiting distribution of zeros, we determine the zero distribution of the heat-evolved polynomials in terms of its logarithmic potential. Furthermore, we explicitly identify two critical time thresholds, at which singularities develop and at which the limiting distribution collapses to the semicircle law. We completely characterize the limiting root distribution of the heat-evolved polynomials before singularities develop as the push-forward of the initial distribution under a transport map. Finally, we discuss the results from the perspectives of partial differential equations (in particular Hamilton-Jacobi equation and Burgers' equation), optimal transport, and free probability. The theory is accompanied by explicit examples, simulations, and conjectures.

math.PR↗

arXiv2024

Asymptotic theory of Schatten classes

The study of Schatten classes has a long tradition in geometric functional analysis and related fields. In this paper we study a variety of geometric and probabilistic aspects of finite-dimensional Schatten classes of not necessarily square matrices. Among the main results are the exact and asymptotic volume of the Schatten-$\infty$ unit ball, the boundedness of its isotropy constant, a Poincaré-Maxwell-Borel lemma for the uniform distribution on the Schatten-$\infty$ ball, and Sanov-type large deviations principles for the singular values of matrices sampled uniformly from the Schatten-$p$ unit ball for any $0 < p \leq \infty$.

math.FA↗

arXiv2024

Multinomial random combinatorial structures and $r$-versions of Stirling, Eulerian and Lah numbers

We introduce multinomial and $r$-variants of several classic objects of combinatorial probability, such as the random recursive and Hoppe trees, random set partitions and compositions, the Chinese restaurant process, Feller's coupling, and some others. Just as various classic combinatorial numbers - like Stirling, Eulerian and Lah numbers - emerge as essential ingredients defining the distributions of the mentioned processes, the so-called $r$-versions of these numbers appear in exact distributional formulas for the multinomial and $r$-counterparts. This approach allows us to offer a concise probabilistic interpretation for various identities involving $r$-versions of these combinatorial numbers, which were either unavailable or meaningful only for specific values of the parameter $r$. We analyze the derived distributions for fixed-size structures and establish distributional limit theorems as the size tends to infinity. Utilizing the aforementioned generalized Stirling numbers of both kinds, we define and analyze $(r,s)$-Lah distributions, which have arisen in the existing literature on combinatorial probability in various contexts.

math.PR↗

arXiv2024

A quantitative central limit theorem for Poisson horospheres in high dimensions

Consider a stationary Poisson process of horospheres in a $d$-dimensional hyperbolic space. In the focus of this note is the total surface area these random horospheres induce in a sequence of balls of growing radius $R$. The main result is a quantitative, non-standard central limit theorem for these random variables as the radius $R$ of the balls and the space dimension $d$ tend to infinity simultaneously.

math.PR↗

arXiv2024

Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space

Poisson processes of so-called $λ$-geodesic hyperplanes in $d$-dimensional hyperbolic space are studied for $0\leqλ\leq 1$. The case $λ=0$ corresponds to genuine geodesic hyperplanes, the case $λ=1$ to horospheres and $λ\in(0,1)$ to $λ$-equidistants. In the focus are the fluctuations of the centred and normalized total surface area of the union of all $λ$-geodesic hyperplanes in the Poisson process within a hyperbolic ball of radius $R$ centred at some fixed point, as $R\to\infty$. It is shown that for $λ<1$ these random variables satisfy a quantitative central limit theorem precisely for $d=2$ and $d=3$. The exact form of the non-Gaussian, infinitely divisible limiting distribution is determined for all higher space dimensions $d\geq 4$. The special case $λ=1$ is in sharp contrast to this behaviour. In fact, for the total surface area of Poisson processes of horospheres, a non-standard central limit theorem with limiting variance $1/2$ is established for all space dimensions $d\geq 2$. We discuss the analogy between the problem studied here and the Random Energy Model whose partition function exhibits a similar structure of possible limit laws.

math.PR↗

arXiv2024

On decoupled standard random walks

Let $S_{n}=\sum_{k=1}^{n}ξ_{k}$, $n\in\mathbb{N}$, be a standard random walk with i.i.d. nonnegative increments $ξ_{1},ξ_{2},\ldots$ and associated renewal counting process $N(t)=\sum_{n\ge 1}1_{\{S_{n}\le t\}}$, $t\ge 0$. A decoupling of $(S_{n})_{n\ge 1}$ is any sequence $\hat{S}_{1}$, $\hat{S}_{2},\ldots$ of independent random variables such that, for each $n\in\mathbb{N}$, $\hat{S}_{n}$ and $S_{n}$ have the same law. Under the assumption that the law of $\hat{S}_{1}$ belongs to the domain of attraction of a stable law with finite mean, we prove a functional limit theorem for the \emph{decoupled renewal counting process} $\hat{N}(t)=\sum_{n\ge 1}1_{\{\hat{S}_{n}\le t\}}$, $t\ge 0$, after proper scaling, centering and normalization. We also study the asymptotics of $\log \mathbb{P}\{\min_{n\ge 1}\hat{S}_{n}>t\}$ as $t\to\infty$ under varying assumptions on the law of $\hat{S}_{1}$. In particular, we recover the assertions which were previously known in the case when $\hat{S}_{1}$ has an exponential law. These results, which were formulated in terms of an infinite Ginibre point process, served as an initial motivation for the present work. Finally, we prove strong law of large numbers type results for the sequence of decoupled maxima $M_{n}=\max_{1\le k\le n}\hat{S}_{k}$, $n\in\mathbb{N}$, and the related first passage time process $\hatτ(t)=\inf\{n\in\mathbb{N}: M_{n}>t\}$, $t\ge 0$. In particular, we provide a tail condition on the law of $\hat{S}_{1}$ in the case when the latter has finite mean but infinite variance that implies $\lim_{t\to\infty}t^{-1}\hatτ(t)=\lim_{t\to\infty}t^{-1}\mathbb{E}\hatτ(t)=0$. In other words, $t^{-1}\hatτ(t)$ may exhibit a different limit behavior than $t^{-1}τ(t)$, where $τ(t)$ denotes the level-$t$ first passage time of $(S_{n})_{n\ge 1}$.

math.PR↗

arXiv2024

Limit theorems for mixed-norm sequence spaces with applications to volume distribution

Let $p, q \in (0, \infty]$ and $\ell_p^m(\ell_q^n)$ be the mixed-norm sequence space of real matrices $x = (x_{i, j})_{i \leq m, j \leq n}$ endowed with the (quasi-)norm $\Vert x \Vert_{p, q} := \big\Vert \big( \Vert (x_{i, j})_{j \leq n} \Vert_q \big)_{i \leq m} \Vert_p$. We shall prove a Poincaré-Maxwell-Borel lemma for suitably scaled matrices chosen uniformly at random in the $\ell_p^m(\ell_q^n)$ unit balls $\mathbb{B}_{p, q}^{m, n}$, and obtain both central and non-central limit theorems for their $\ell_p(\ell_q)$-norms. We use those limit theorems to study the asymptotic volume distribution in the intersection of two mixed-norm sequence balls. Our approach is based on a new probabilistic representation of the uniform distribution on $\mathbb{B}_{p, q}^{m, n}$.

math.PR↗

arXiv2023

Propagation of chaos in the Random field Curie-Weiss model

We prove propagation of chaos in the Random field mean-field Ising model, also known ad the Random field Curie-Weiss model. We show that in the paramagnetic phase, i.e.\ in the regime where temperature and distribution of the external field admit a unique minimizer of the expected Helmholtz free energy, propagation of chaos holds. By the latter we mean that the finite-dimensional marginals of the Gibbs measure converge towards a product measure with the correct expectation as the system size goes to infinity. This holds independently of whether the system is in a high-temperature phase or at a phase transition point. If the Helmholtz free energy possesses several minima, there are several possible equilibrium measures. In this case, we show that the system picks one of them at random (depending on the realization of the random external field) and propagation of chaos with respect to a product measure with the same marginals as the one randomly picked holds true. We illustrate our findings in a simple example.

math.PR↗