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Zakhar Kabluchko

Publications and source records attributed to Zakhar Kabluchko.

At least 19 recordsLinked to original sources
arXiv2026

Zeros of polynomial powers under the heat flow

We study the zero evolution of high powers of polynomials 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 holomorphic heat flow operator: For small time, zeros of the heat-evolved polynomial start to spread out from the initial zeros $λ_j$ of $P$, and $μ_t$ approximates the superposition of semicircle laws around $λ_j$. Then for arbitrary time, the support of $μ_t$ forms intricate curves, which merge as $t$ grows, until for large time, the limit distribution $μ_t$ approaches a widespread semicircle law through the initial center of mass of the $λ_j$. We further show that the Stieltjes transform of $μ_t$ 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.

math.PR↗
arXiv2026

Absorption Probabilities for Random Convex Hulls: Distribution-Freeness via the Wall-Crossing Method

We consider the probability that the convex hull of the first $n$ partial sums of a $d$-dimensional random walk contains the origin. Under symmetric exchangeability of the increments and a general-position assumption, this absorption probability is distribution-free and admits an explicit formula, previously obtained by Kabluchko, Vysotsky and Zaporozhets [Geom. Funct. Anal. 27 (2017)] using characteristic polynomials of hyperplane arrangements. We give a different proof, based on a wall-crossing method which we develop here. Starting from a deterministic configuration of increments, we count the signed permutations for which the convex hull of the corresponding partial sums contains the origin and show that this count remains unchanged under generic deformations of the increments, and hence is the same for all configurations outside a natural exceptional set of measure zero. Evaluating the invariant at a single well-chosen configuration reduces the remaining calculation to the enumeration of permutation records combined with Wendel's theorem. Our method also reproves Wendel's theorem on convex hulls of random points with a sign-flip-invariant joint distribution and, in dimension one, Sparre Andersen's theorem. Finally, we derive new probabilistic representations and recurrence relations for the absorption probabilities of random-walk convex hulls and their random-bridge analogues.

math.PR↗
arXiv2026

Intrinsic volumes of $\ell_p$-balls and a continuum of Maxwell--Poincaré--Borel laws for their curvature measures

For $p>1$, we derive explicit formulas for the intrinsic volumes $V_0(\mathbb B_p^n),\dots,V_{n-1}(\mathbb B_p^n)$ of the $n$-dimensional $\ell_p$-balls $$ \mathbb B_p^n = \{x\in\mathbb R^n:\ |x_1|^p+\ldots+|x_n|^p\le 1\} $$ and, more generally, of their coordinate-weighted analogues. The formula is given in terms of a one-dimensional integral involving the special function $$ \mathcal F_p(t;ν) = \int_{\mathbb R}|u|^νe^{-|u|^p-t|u|^{2p-2}}\,du. $$ Previously known formulas for the intrinsic volumes of ellipsoids, weighted crosspolytopes, and rectangular boxes arise as special or limiting cases. We also obtain asymptotic formulas for $V_{j(n)}(\mathbb B_p^n)$ in the high-dimensional regime $n\to\infty$, where the index $j(n)$ is allowed to depend on $n$. We further investigate the curvature measures of $\mathbb B_p^n$. These are finite measures $$ Φ_0(\mathbb B_p^n,\cdot),\dots,Φ_{n-1}(\mathbb B_p^n,\cdot) $$ on $\partial\mathbb B_p^n$ that localize the intrinsic volumes. We prove a Maxwell--Poincaré--Borel type limit theorem: if $X_n$ is a random boundary point of $\mathbb B_p^n$ distributed according to the normalized curvature measure $Φ_{j(n)}(\mathbb B_p^n,\cdot)/V_{j(n)}(\mathbb B_p^n)$, where $j(n)/n\toα\in[0,1]$ as $n\to\infty$, then for every fixed $r\in\mathbb N$, the joint distribution of the first $r$ coordinates of $n^{1/p}X_n$ converges weakly to the product measure $ν_{p,α}^{\otimes r}$. Here $ν_{p,α}$ is an explicit probability measure on $\mathbb R$ depending on $p>1$ and $α\in[0,1]$. The main tool underlying these results is an explicit characterization of the curvature measures of coordinate-weighted $\ell_p$-balls, and in particular an explicit formula for their mixed moments.

math.PR↗
arXiv2026

Expected hyperbolic volumes of random beta polytopes

Let $X_1,\ldots,X_n$ be independent random points in the closed unit ball of $\mathbb{R}^d$. Assume that each $X_i$ has a beta distribution with parameter $β_i \ge -1$: if $β_i>-1$, then $X_i$ has Lebesgue density proportional to $(1-\|x\|^2)^{β_i}$ on $\{\|x\|<1\}$, whereas the case $β_i=-1$ corresponds to the uniform distribution on the unit sphere $\{\|x\|=1\}$. Let $[X_1,\ldots,X_n]$ denote the convex hull of these points. Interpreting the unit ball as the Klein model of hyperbolic geometry, we derive closed-form formulas for the expected hyperbolic volume of the random hyperbolic polytope $[X_1,\ldots,X_n]$. As a special case, if $X_1,\ldots,X_n$ are independent and uniformly distributed on the unit sphere in $\mathbb{R}^3$, then for every $n\ge 4$, \[ \mathbb{E}\,\operatorname{Vol}_{3}^{\mathrm{hyp}}\!\bigl([X_1,\ldots,X_n]\bigr) = π\left(\frac{n}{2}-\sum_{j=1}^{n-1}\frac{1}{j}\right). \]

math.PR↗
arXiv2026

Random Polyhedral Cones I: Distributional Results via Gale Duality

Let $U_1,\ldots,U_n$ be independent random vectors uniformly distributed on the unit sphere $\mathbb S^{d-1}\subseteq\mathbb R^d$, where $n\ge d$, and consider the random polyhedral cone \[ \mathcal W_{n,d}:=\mathop{\mathrm{pos}} (U_1,\ldots,U_n) = \{λ_1 U_1+ \ldots + λ_n U_n: λ_1\geq 0, \ldots, λ_n \geq 0\}. \] We establish several distributional results for $\mathcal W_{n,d}$ and the associated spherical polytope $\mathcal W_{n,d}\cap\mathbb S^{d-1}$. Our main contributions include: (i) Let $α_d$ denote the solid angle of $\mathcal W_{d,d}$ and write $m(d,k):=\mathbb E[α_d^k]$ for its $k$-th moment. We prove the symmetry $m(d,k)=m(k,d)$. As an application, we compute $\mathop{\mathrm{Var}}[α_d]=2^{-d}(d+1)^{-1}-4^{-d}$ and derive a closed formula for the third moment. (ii) For $n=d+1,d+2,d+3$ we determine the probability that $\mathcal W_{n,d}\cap\mathbb S^{d-1}$ is a spherical simplex, a spherical analogue of the classical Sylvester problem. In the case $n=d+2$ we also determine the distribution of the number of vertices of $\mathcal W_{d+2,d}\cap\mathbb S^{d-1}$. (iii) Let $f_\ell(\mathcal W_{n,d})$ denote the number of $\ell$-dimensional faces of $\mathcal W_{n,d}$. We prove a distributional limit theorem for $f_\ell(\mathcal W_{n,d})$ in the regime $n=d+k$ and $\ell=d-q$, where $k,q\in\mathbb N$ are fixed and $d\to\infty$. The limit law is a weighted sum of independent chi squared variables, with weights given by explicit eigenvalues of a convolution operator on the sphere. A unifying ingredient is an explicit coupling producing i.i.d. uniform vectors $U_1,\ldots,U_n\in\mathbb S^{d-1}$ together with i.i.d. uniform vectors $V_1,\ldots,V_n\in\mathbb S^{n-d-1}$ whose associated oriented matroids are Gale dual.

math.PR↗
arXiv2026

Seeing Through Hyperbolic Space: Visibility for $λ$-Geodesic Hyperplanes

We study visibility from a fixed point in the presence of a Poisson process of $λ$--geodesic hyperplanes in a $d$-dimensional hyperbolic space. The family of $λ$--geodesic hyperplanes interpolates between totally geodesic hyperplanes and horospheres. Our main result establishes a universality principle for this model: we prove that the fundamental visibility properties are invariant with respect to the parameter $λ\in[0,1]$. Namely, there is a critical intensity $γ_{\mathrm{crit}}>0$ such that the visible region is unbounded with positive probability for $γ< γ_{\mathrm{crit}}$ and almost surely bounded for $γ> γ_{\mathrm{crit}}$. For $d=2$ we establish almost sure boundedness also at criticality. The value for $γ_{\mathrm{crit}}$ is explicit and does not depend on $λ$. In the bounded phase, we show that the mean visible volume is identical with the known formula for $λ=0$. The key integral-geometric step is an explicit computation showing that the measure of $λ$-geodesic hyperplanes hitting a geodesic segment is a linear function of the length of the segment, independent of~$λ$.

math.PR↗
arXiv2026

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 $ξ_1,\ldots, ξ_n$, the empirical mean $\frac 1n (ξ_1 + \ldots + ξ_n)$ 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.

math.PR↗
arXiv2026

Zero distribution of multiplicative Hermite and Laguerre polynomials

It is well-known that, as $n\to\infty$, the zero distribution of the $n$-th Hermite polynomial converges to the semicircular law (the free normal distribution), while the zero distribution of the associated Laguerre polynomials converges to the Marchenko--Pastur law (the free Poisson distribution). In this paper, we establish multiplicative analogues of these results. We define the multiplicative Hermite and Laguerre polynomials by \begin{align*} H_n^*(x;s) &:= e^{-\frac 12 s ((x\partial_x)^2 - n x \partial_x) } (x-1)^n = \sum_{j=0}^n (-1)^{n-j} \binom nj e^{-\frac 12 s (j^2 - nj)} x^j, \\ L_n^*(x; b,c) &:= (x\partial_x + b)^c (x-1)^n = \sum_{j=0}^n (-1)^{n-j} \binom nj (j+b)^c x^j, \end{align*} where $n\in \mathbb N_0$, $\partial_x$ denotes the differentiation operator w.r.t. $x$, and $s\in \mathbb R$, $b\in \mathbb C$, $c\in \mathbb N_0$ are parameters. In the Hermite case, we show that, as $n\to\infty$, the zero distribution of $H_n^*(x;s/n)$ converges weakly to the free multiplicative normal distribution on the positive half-line (when $s>0$) or to the free unitary normal distribution on the unit circle $\{|z| = 1\}$ (when $s<0$). In the Laguerre case, we show that the zero distribution of $L_n^*(x; nβ, \lfloor n γ\rfloor)$ converges to the free multiplicative Poisson distribution on the positive half-line (when $γ>0$ and $β\in \mathbb R\backslash[0,1]$) or on the unit circle (when $γ>0$ and $β\in -\frac 12 + \sqrt{-1} \, \mathbb R$). All these results are obtained by essentially the same method, which treats the Hermite/Laguerre cases and the unitary/positive settings in a unified way.

math.PR↗
arXiv2026

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.

math.PR↗
arXiv2025

Arithmetic sensitivity of cumulant growth in lacunary sums: transcendental versus algebraic ratio limits

We study the asymptotic behavior of cumulants of lacunary trigonometric sums $S_n(ω) := \sum_{k=1}^n \cos (2 πa_k ω)$, $ω\in[0,1]$, and show that cumulant growth is highly sensitive to the arithmetic structure of the sequence $(a_k)_{k \geq 1}$ of positive integers. In particular, if $\lim_{k \to \infty} a_{k+1}/a_k = η> 1$ for some transcendental number $η$, we prove that for every $m\in \mathbb N$ the $m$-th cumulant of $S_n$ is asymptotically equivalent to the $m$-th cumulant of the ``independent model'' $\widetilde{S}_n := \sum_{k=1}^n \cos (2 πa_k U_k)$, where $U_1, U_2, \dots$ are independent random variables having uniform distribution on $[0,1]$. In particular, the order of growth of the cumulants as $n \to \infty$ is linear in this case. We also show that the transcendence condition for $\lim_{k \to \infty} a_{k+1}/a_k$ is in general necessary: when the ratio limit $η$ is algebraic, the cumulants of $S_n$ may have a different asymptotic order from those of $\widetilde{S}_n$. For instance, for $a_k = 2^k+1$ (with $η= 2$), the sixth cumulant of $S_n$ grows quadratically in $n$. In contrast, for $a_k = 2^k$ (again $η= 2$) or when $(a_k)_{k \geq 1}$ is the Fibonacci sequence (with $η= (1+\sqrt 5)/2$), the $m$-th cumulant of $S_n$ grows linearly as $n\to\infty$, but with a growth rate that differs from the one of the independent model $\widetilde{S}_n$. Overall, our results show that the asymptotic behavior of the cumulants of lacunary trigonometric sums depends on arithmetic effects in a very delicate way. This is particularly remarkable since many other probabilistic limit theorems, such as the Central Limit Theorem, hold for lacunary trigonometric sums in a universal way without any such sensitivity towards arithmetic effects.

math.NT↗
arXiv2025

Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators

We start with a random polynomial $P^{N}(z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_{t}^{N}$ obtained by $\lceil Nt\rceil$ applications of a fractional differential operator of the form $z^{a} (d/dz)^{b},$ where $a$ and $b$ are real numbers. When $b>0,$ we compute the limiting root distribution $μ_{t}$ of $P_{t}^{N}$ as $N\rightarrow\infty.$ We show that $μ_{t}$ is the push-forward of the limiting root distribution of $P^{N}$ under a transport map $T_{t}$. The map $T_{t}$ is defined by flowing along the characteristic curves of a PDE satisfied by the log potential of $μ_{t}.$ In the special case of repeated differentiation, our results may be interpreted as saying that the roots evolve radially \textit{with constant speed} until they hit the origin, at which point, they cease to exist. For general $a$ and $b,$ the transport map $T_{t}$ has a free probability interpretation as multiplication of an $R$-diagonal operator by an $R$-diagonal \textquotedblleft transport operator.\textquotedblright As an application, we obtain a push-forward characterization of the free self-convolution semigroup $\oplus$ of radial measures on $\mathbb{C}$. We also consider the case $b<0,$ which includes the case of repeated integration. More complicated behavior of the roots can occur in this case.

math.PR↗
arXiv2025

The heat flow, GAF, and SL(2;R)

We establish basic properties of the heat flow on entire holomorphic functions that have order at most 2. We then look specifically at the action of the heat flow on the Gaussian analytic function (GAF). We show that applying the heat flow to a GAF and then rescaling and multiplying by an exponential of a quadratic function gives another GAF. It follows that the zeros of the GAF are invariant in distribution under the heat flow, up to a simple rescaling. We then show that the zeros of the GAF evolve under the heat flow approximately along straight lines, with an error whose distribution is independent of the starting point. Finally, we connect the heat flow on the GAF to the metaplectic representation of the double cover of the group $SL(2;\mathbb{R}).$

math.PR↗
arXiv2025

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.

math.PR↗
arXiv2025

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.

math.CA↗
arXiv2025

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, degree $d$ trigonometric polynomial are distributed according to some probability measure $ν$ in the large $d$ limit, then the zeroes of its $[2td]$-th derivative, where $t>0$ is fixed, are distributed according to the free multiplicative convolution of $ν$ and the free unitary Poisson distribution with parameter $t$. In the simplest special case, our result states that the zeroes of the $[2td]$-th derivative of the trigonometric polynomial $(\sin \frac θ2)^{2d}$ (which can be thought of as the trigonometric analogue of the Laguerre polynomials) are distributed according to the free unitary Poisson distribution with parameter $t$, in the large $d$ limit. The latter distribution is defined in terms of the function $ζ=ζ_t(θ)$ which solves the implicit equation $ζ- t \tan ζ= θ$ and satisfies $$ ζ_t(θ)= θ+ t \tan (θ+ t \tan (θ+ t \tan (θ+\ldots))), \qquad \mathrm{Im}\, θ>0, \;\; t>0. $$

math.PR↗
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↗