Zakhar Kabluchko
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Zakhar Kabluchko has a total of 108 co-authors. Here are the top co-authors by number of shared publications:
- Alexander Marynych
Department of Mathematics, Kyiv School of Economics, Kyiv, Ukraine
27 shared publications - Christoph Thäle
Ruhr University Bochum, Bochum, Germany
25 shared publications - Dmitry Zaporozhets
St. Petersburg Department of the Steklov Institute of Mathematics, St. Petersburg, Russia
22 shared publications - Alexander Iksanov
Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
17 shared publications - Joscha Prochno
Faculty of Computer Science and Mathematics University of Passau Passau Germany
13 shared publications - Matthias Löwe
Universität Münster
10 shared publications
Latest Publications
- Open Access
- 12 February 2026
- Published by Springer Nature in Discrete & Computational Geometry
- Vol. 76 (2) , 1177-1211
A Refinement of the Sylvester Problem: Probabilities of Combinatorial Types
Let $$X_1,\ldots , X_{d+2}$$ be random points in $$\mathbb {R}^d$$ . The classical Sylvester problem asks to determine the probability that the convex hull of these points, denoted by $$P:= [X_1,\ldots , X_{d+2}]$$ , is a simplex. In the present paper, we study a refined version of this problem which asks to determine the probability that P has a given combinatorial type. It is known that there are $$\lfloor d/2\rfloor +1$$ possible combinatorial types of simplicial d -dimensional polytopes with at most $$d+2$$ vertices. These types are denoted by $$T_0^d, T_1^d, \ldots , T_{\lfloor d/2 \rfloor }^d$$ , where $$T_0^d$$ is a simplex with $$d+1$$ vertices, while the remaining types have exactly $$d+2$$ vertices. Our aim is thus to compute the probability $$ p_{d,m} := \mathbb {P}[P \text { is of type } T_{m}^d], \qquad m\in \{0,1,\ldots , \lfloor d/2 \rfloor \}. $$ The classical Sylvester problem corresponds to the case $$m=0$$ . We shall compute $$p_{d,m}$$ for all m in the following cases: (a) $$X_1,\ldots , X_{d+2}$$ are i.i.d. normal; (b) $$X_1,\ldots , X_{d+2}$$ follow a d -dimensional beta or beta prime distribution, which includes the uniform distribution on the ball or on the sphere as special cases; (c) $$X_1,\ldots , X_{d+2}$$ form a random walk with exchangeable increments. As a by-product of case (a) we recover a recent solution to Youden’s demon problem which asks to determine the probability that, in a one-dimensional i.i.d. normal sample $$\xi _1,\ldots , \xi _n$$ , the empirical mean $$\frac{1}{n} (\xi _1 + \ldots + \xi _n)$$ lies between the k -th and the $$(k+1)$$ -st order statistics. We also consider the conic (or spherical) version of the refined Sylvester problem and solve it in several special cases.- 12 February 2026
- Published by Springer Nature in Discrete & Computational Geometry
- Vol. 76 (2) , 1177-1211
Open AccessA Refinement of the Sylvester Problem: Probabilities of Combinatorial Types
Let $$X_1,\ldots , X_{d+2}$$ be random points in $$\mathbb {R}^d$$ . The classical Sylvester problem asks to determine the probability that the convex hull of these points, denoted by $$P:= [X_1,\ldots , X_{d+2}]$$ , is a simplex. In the present paper, we study a refined version of this problem which asks to determine the probability that P has a given combinatorial type. It is known that there are $$\lfloor d/2\rfloor +1$$ possible combinatorial types of simplicial d -dimensional polytopes with at most $$d+2$$ vertices. These types are denoted by $$T_0^d, T_1^d, \ldots , T_{\lfloor d/2 \rfloor }^d$$ , where $$T_0^d$$ is a simplex with $$d+1$$ vertices, while the remaining types have exactly $$d+2$$ vertices. Our aim is thus to compute the probability $$ p_{d,m} := \mathbb {P}[P \text { is of type } T_{m}^d], \qquad m\in \{0,1,\ldots , \lfloor d/2 \rfloor \}. $$ The classical Sylvester problem corresponds to the case $$m=0$$ . We shall compute $$p_{d,m}$$ for all m in the following cases: (a) $$X_1,\ldots , X_{d+2}$$ are i.i.d. normal; (b) $$X_1,\ldots , X_{d+2}$$ follow a d -dimensional beta or beta prime distribution, which includes the uniform distribution on the ball or on the sphere as special cases; (c) $$X_1,\ldots , X_{d+2}$$ form a random walk with exchangeable increments. As a by-product of case (a) we recover a recent solution to Youden’s demon problem which asks to determine the probability that, in a one-dimensional i.i.d. normal sample $$\xi _1,\ldots , \xi _n$$ , the empirical mean $$\frac{1}{n} (\xi _1 + \ldots + \xi _n)$$ lies between the k -th and the $$(k+1)$$ -st order statistics. We also consider the conic (or spherical) version of the refined Sylvester problem and solve it in several special cases. - Open Access
First passage times for decoupled random walks
Motivated by a connection to the infinite Ginibre point process, decoupled random walks were introduced in a recent article Alsmeyer, Iksanov and Kabluchko (2025). The decoupled random walk is a sequence of independent random variables, in which the $n$th variable has the same distribution as the position at time $n$ of a standard random walk with nonnegative increments. We prove distributional convergence in the Skorokhod space equipped with the $J_1$-topology of the running maxima and the first passage times of decoupled random walks. We show that there exist five different regimes, in which distinct limit theorems arise. Rather different functional limit theorems for the number of visits of decoupled standard random walk to the interval $[0,t]$ as $t\to\infty$ were earlier obtained in the aforementioned paper Alsmeyer, Iksanov and Kabluchko (2025). While the limit processes for the first passage times are inverse extremal-like processes, the limit processes for the number of visits are stationary Gaussian.Open AccessFirst passage times for decoupled random walks
Motivated by a connection to the infinite Ginibre point process, decoupled random walks were introduced in a recent article Alsmeyer, Iksanov and Kabluchko (2025). The decoupled random walk is a sequence of independent random variables, in which the $n$th variable has the same distribution as the position at time $n$ of a standard random walk with nonnegative increments. We prove distributional convergence in the Skorokhod space equipped with the $J_1$-topology of the running maxima and the first passage times of decoupled random walks. We show that there exist five different regimes, in which distinct limit theorems arise. Rather different functional limit theorems for the number of visits of decoupled standard random walk to the interval $[0,t]$ as $t\to\infty$ were earlier obtained in the aforementioned paper Alsmeyer, Iksanov and Kabluchko (2025). While the limit processes for the first passage times are inverse extremal-like processes, the limit processes for the number of visits are stationary Gaussian. - Open Access
Zeros of polynomial powers under the heat flow
We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane as $n$ tends to infinity. We describe this limit distribution $μ_t$ as a function of the time parameter $t$ of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution $μ_t$ satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for $μ_t$ is available.Open AccessZeros of polynomial powers under the heat flow
We study the evolution of zeros of high polynomial powers under the heat flow. For any fixed polynomial $P(z)$, we prove that the empirical zero distribution of its heat-evolved $n$-th power converges to a distribution on the complex plane as $n$ tends to infinity. We describe this limit distribution $μ_t$ as a function of the time parameter $t$ of the heat evolution: For small time, zeros start to spread out in approximately semicircular distributions, then intricate curves start to form and merge, until for large time, the zero distribution approaches a widespread semicircle law through the initial center of mass. The Stieltjes transform of the limit distribution $μ_t$ satisfies a self-consistent equation and a Burgers' equation. The present paper deals with general complex-rooted polynomials for which, in contrast to the real-rooted case, no free-probabilistic representation for $μ_t$ is available. - 17 November 2025
- Published by American Mathematical Society (AMS) in Theory of Probability and Mathematical Statistics
- Vol. 113, 65-90
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 f is a Besicovitch almost periodic function and V V is a random variable uniformly distributed on [ − 1 , 1 ] [-1,1] , then the random variables f ( L ⋅ V ) f(L\cdot V) converge in distribution, as L → ∞ L\to \infty , to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes ( f ( L ⋅ V + t ) ) t ∈ R (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. Further we investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of Akbary, Ng and Shahabi (2014) and earlier works.
- 17 November 2025
- Published by American Mathematical Society (AMS) in Theory of Probability and Mathematical Statistics
- Vol. 113, 65-90
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 f is a Besicovitch almost periodic function and V V is a random variable uniformly distributed on [ − 1 , 1 ] [-1,1] , then the random variables f ( L ⋅ V ) f(L\cdot V) converge in distribution, as L → ∞ L\to \infty , to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes ( f ( L ⋅ V + t ) ) t ∈ R (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. Further we investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of Akbary, Ng and Shahabi (2014) and earlier works.
- Open Access
On the distribution patterns of zeros for random polynomials with regularly varying coefficients
This paper investigates asymptotic distribution of complex zeros of random polynomials $P_n(z):=\sum_{k=0}^{n}b(k)ξ_k z^k$, as $n\to\infty$, where $b$ is a regularly varying function at infinity with index $α\in \mathbb{R}$ and $(ξ_k)_{k\geq 0}$ is a sequence of independent copies of a complex-valued random variable $ξ$. The limiting distribution of zeros both inside and outside the unit disk is determined assuming $\mathbb{E}[\log^{+}|ξ|]<\infty$. Under the additional assumptions $\mathbb{E}[ξ]=0$ and $\mathbb{E}[|ξ|^2]<\infty$, local universality results for zeros near the boundary of the unit disk are established. Notably, it is shown that the point process of zeros undergoes a transition from liquid-like to crystalline phases as $α$ crosses the critical value $α_c = -1/2$ from right to left. In the liquid phase ($α> α_c$), the limiting point process of zeros is universal. In the crystalline phase, it is universal if and only if $α= α_c$ and $\sum_k b^2(k) = +\infty$ (the weak crystalline phase), and non-universal when $\sum_k b^2(k) < +\infty$ (the strong crystalline phase). The zeros of the so-called random self-inversive polynomials on the unit circle exhibit a similar phase transition.Open AccessOn the distribution patterns of zeros for random polynomials with regularly varying coefficients
This paper investigates asymptotic distribution of complex zeros of random polynomials $P_n(z):=\sum_{k=0}^{n}b(k)ξ_k z^k$, as $n\to\infty$, where $b$ is a regularly varying function at infinity with index $α\in \mathbb{R}$ and $(ξ_k)_{k\geq 0}$ is a sequence of independent copies of a complex-valued random variable $ξ$. The limiting distribution of zeros both inside and outside the unit disk is determined assuming $\mathbb{E}[\log^{+}|ξ|]<\infty$. Under the additional assumptions $\mathbb{E}[ξ]=0$ and $\mathbb{E}[|ξ|^2]<\infty$, local universality results for zeros near the boundary of the unit disk are established. Notably, it is shown that the point process of zeros undergoes a transition from liquid-like to crystalline phases as $α$ crosses the critical value $α_c = -1/2$ from right to left. In the liquid phase ($α> α_c$), the limiting point process of zeros is universal. In the crystalline phase, it is universal if and only if $α= α_c$ and $\sum_k b^2(k) = +\infty$ (the weak crystalline phase), and non-universal when $\sum_k b^2(k) < +\infty$ (the strong crystalline phase). The zeros of the so-called random self-inversive polynomials on the unit circle exhibit a similar phase transition. - Open Access
- 3 November 2025
- Published by Elsevier in Discrete Mathematics
- Vol. 349 (3)
Multinomial random combinatorial structures and r-versions of Stirling, Eulerian and Lah numbers
- 3 November 2025
- Published by Elsevier in Discrete Mathematics
- Vol. 349 (3)
Open AccessMultinomial random combinatorial structures and r-versions of Stirling, Eulerian and Lah numbers
- Top Cited Papers
- 25 September 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 379 (3) , 2217-2253
Repeated differentiation and free unitary Poisson process
We investigate the hydrodynamic behavior of zeroes of trigonometric polynomials under repeated differentiation. We show that if the zeroes of a real-rooted, degreetrigonometric polynomial are distributed according to some probability measure in the large limit, then the zeroes of its -th derivative, where is fixed, are distributed according to the free multiplicative convolution of and the free unitary Poisson distribution with parameter . In the simplest special case, our result states that the zeroes of the -th derivative of the trigonometric polynomial (which can be thought of as the trigonometric analogue of the Laguerre polynomials) are distributed according to the free unitary Poisson distribution with parameter , in the large limit. The latter distribution is defined in terms of the function which solves the implicit equation and satisfies - 25 September 2025
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 379 (3) , 2217-2253
Top Cited PapersRepeated differentiation and free unitary Poisson process
We investigate the hydrodynamic behavior of zeroes of trigonometric polynomials under repeated differentiation. We show that if the zeroes of a real-rooted, degreetrigonometric polynomial are distributed according to some probability measure in the large limit, then the zeroes of its -th derivative, where is fixed, are distributed according to the free multiplicative convolution of and the free unitary Poisson distribution with parameter . In the simplest special case, our result states that the zeroes of the -th derivative of the trigonometric polynomial (which can be thought of as the trigonometric analogue of the Laguerre polynomials) are distributed according to the free unitary Poisson distribution with parameter , in the large limit. The latter distribution is defined in terms of the function which solves the implicit equation and satisfies - Open Access
Zeros and exponential profiles of polynomials II: Examples
In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little $q$-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices.Open AccessZeros and exponential profiles of polynomials II: Examples
In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little $q$-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices. - Top Cited PapersOpen Access
- 29 June 2025
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 269 (2) , 545-586
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 ≤ λ ≤ 1. The case λ = 0 corresponds to genuine geodesic hyperplanes, the case λ = 1 to horospheres and λ ∈ (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 → ∞. 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 ≥ 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 ≥ 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.
- 29 June 2025
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 269 (2) , 545-586
Top Cited PapersOpen AccessFluctuations 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 ≤ λ ≤ 1. The case λ = 0 corresponds to genuine geodesic hyperplanes, the case λ = 1 to horospheres and λ ∈ (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 → ∞. 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 ≥ 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 ≥ 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.