arXiv subjects
Publications and source records attributed to Dmitry Zaporozhets.
For $B\subset\mathbb{R}^k$ denote by $Φ_k(Q;B)$ the number of ordered $k$-tuples in $B$ of real conjugate algebraic numbers of degree $\leq n$ and naive height $\leq Q$. We show that $$ Φ_k(Q;B) = \frac{(2Q)^{n+1}}{2ζ(n+1)} \int_{B} ρ_k(\mathbf{x})\,d\mathbf{x} + O\left(Q^n\right),\quad Q\to \infty, $$ where the function $ρ_k$ will be given explicitly. If $n=2$, then an additional factor $\log Q$ appears in the reminder term.
Consider a random walk $S_i= ξ_1+\ldots+ξ_i$, $i\in\mathbb N$, whose increments $ξ_1,ξ_2,\ldots$ are independent identically distributed random vectors in $\mathbb R^d$ such that $ξ_1$ has the same law as $-ξ_1$ and $\mathbb P[ξ_1\in H] = 0$ for every affine hyperplane $H\subset \mathbb R^d$. Our main result is the distribution-free formula $$ \mathbb E \left[\sum_{1\leq i_1 < \ldots < i_k\leq n} 1_{\{0\notin \text{conv}(S_{i_1},\ldots, S_{i_k})\}}\right] = 2 \binom n k \frac {B(k, d-1) + B(k, d-3) +\ldots} {2^k k!}, $$ where the $B(k,j)$'s are defined by their generating function $$ (t+1) (t+3) \ldots (t+2k-1) = \sum_{j=0}^{k} B(k,j) t^j. $$ The expected number of $k$-tuples above admits the following geometric interpretation: it is the expected number of $k$-dimensional faces of a randomly and uniformly sampled open Weyl chamber of type $B_n$ that are not intersected by a generic linear subspace $L\subset \mathbb R^n$ of codimension $d$. The case $d=1$ turns out to be equivalent to the classical discrete arcsine law for the number of positive terms in a one-dimensional random walk with continuous symmetric distribution of increments. We also prove similar results for random bridges with no central symmetry assumption required.
Consider a sequence of partial sums $S_i= ξ_1+\dots+ξ_i$, $1\leq i\leq n$, starting at $S_0=0$, whose increments $ξ_1,\dots,ξ_n$ are random vectors in $\mathbb R^d$, $d\leq n$. We are interested in the properties of the convex hull $C_n:=\mathrm{Conv}(S_0,S_1,\dots,S_n)$. Assuming that the tuple $(ξ_1,\dots,ξ_n)$ is exchangeable and a certain general position condition holds, we prove that the expected number of $k$-dimensional faces of $C_n$ is given by the formula $$ \mathbb E [f_k(C_n)] = \frac{2\cdot k!}{n!} \sum_{l=0}^{\infty}\genfrac{[}{]}{0pt}{}{n+1}{d-2l} \genfrac{\{}{\}}{0pt}{}{d-2l}{k+1}, $$ for all $0\leq k \leq d-1$, where $\genfrac{[}{]}{0pt}{}{n}{m}$ and $\genfrac{\{}{\}}{0pt}{}{n}{m}$ are Stirling numbers of the first and second kind, respectively. Further, we compute explicitly the probability that for given indices $0\leq i_1<\dots <i_{k+1}\leq n$, the points $S_{i_1},\dots,S_{i_{k+1}}$ form a $k$-dimensional face of $\mathrm{Conv}(S_0,S_1,\dots,S_n)$. This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments. These results generalize the classical one-dimensional discrete arcsine law for the position of the maximum due to E. Sparre Andersen. All our formulae are distribution-free, that is do not depend on the distribution of the increments $ξ_k$'s. The main ingredient in the proof is the computation of the probability that the origin is absorbed by a joint convex hull of several random walks and bridges whose increments are invariant with respect to the action of direct product of finitely many reflection groups of types $A_{n-1}$ and $B_n$. This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position.
Let $S_k$ be a random walk in $R^d$ such that its distribution of increments does not assign mass to hyperplanes. We study the probability $p_n$ that the convex hull $conv (S_1, \ldots , S_n)$ of the first $n$ steps of the walk does not include the origin. By providing an explicit formula, we show that for planar symmetrically distributed random walks, $p_n$ does not depend on the distribution of increments. This extends the well known result by Sparre Andersen (1949) that a one-dimensional random walk satisfying the above continuity and symmetry assumptions stays positive with a distribution-free probability. We also find the asymptotics of $p_n$ as $n \to \infty$ for any planar random walk with zero mean square-integrable increments. We further developed our approach from the planar case to study a wide class of geometric characteristics of convex hulls of random walks in any dimension $d \ge 2$. In particular, we give formulas for the expected value of the number of faces, the volume, the surface area, and other intrinsic volumes, including the following multidimensional generalization of the Spitzer--Widom formula (1961) on the perimeter of planar walks: $$ E V_1 (conv(0, S_1, \dots, S_n)) = \sum_{k=1}^n \frac{E \|S_k\|}{k}, $$ where $V_1$ denotes the first intrinsic volume, which is proportional to the mean width. These results have applications to geometry, and in particular, imply the formula by Gao and Vitale (2001) for the intrinsic volumes of special path-simplexes, called canonical orthoschemes, which are finite-dimensional approximations of the closed convex hull of a Wiener spiral. Moreover, there is a direct connection between spherical intrinsic volumes of these simplexes and the probabilities $p_n$. We also prove similar results for convex hulls of random walk bridges, and more generally, for partial sums of exchangeable random vectors.
Let $X_1,\ldots,X_n$ be a standard normal sample in $\mathbb R^d$. We compute exactly the expected volume of the Gaussian polytope $\mathrm{conv}[X_1,\ldots,X_n]$, the symmetric Gaussian polytope $\mathrm{conv}[\pm X_1,\ldots,\pm X_n]$, and the Gaussian zonotope $[0,X_1]+\ldots+[0,X_n]$ by exploiting their connection to the regular simplex, the regular crosspolytope, and the cube with the aid of Tsirelson's formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including $ \mathrm{conv}[l_1X_1,\ldots,l_nX_n] $ and $ \mathrm{conv}[\pm l_1 X_1,\ldots, \pm l_n X_n], $ where $l_1,\ldots,l_n\geq 0$ are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the $k$-th intrinsic volume of the regular simplex $S^{n-1}$ to the expected maximum of independent standard Gaussian random variables $ξ_1,\ldots,ξ_n$ given that the maximum has multiplicity $k$. Namely, we show that $$ V_k(S^{n-1}) = \frac {(2π)^{\frac k2}} {k!} \cdot \lim_{\varepsilon\downarrow 0} \varepsilon^{1-k} \mathbb E [\max\{ξ_1,\ldots,ξ_n\} 1_{\{ξ_{(n)} - ξ_{(n-k+1)}\leq \varepsilon\}}], $$ where $ξ_{(1)} \leq \ldots \leq ξ_{(n)}$ denote the order statistics. A similar result holds for the crosspolytope if we replace $ξ_1,\ldots,ξ_n$ by their absolute values.
We give an explicit formula for the probability that the convex hull of an $n$-step random walk in $R^d$ does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments. This probabilistic problem is shown to be equivalent to either of the two geometric ones: 1) Find the number of Weyl chambers of type $B_n$ intersected by a generic linear subspace of $R^n$ of codimension $d$; 2) Find the conic intrinsic volumes of a Weyl chamber of type $B_n$. We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement. We obtain analogous distribution-free results for Weyl chambers of type $A_{n-1}$ (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type $D_n$, and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form $B_1\times \dots \times B_1$ recovers the Wendel formula (1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin. We also give an asymptotic analysis of the obtained absorption probabilities as $n \to \infty$, in both cases of fixed and increasing dimension $d$.
Consider a random polynomial $$ G(z):=ξ_0+ξ_1z+\dots+ξ_nz^n,\quad z\in\mathbb{C}, $$ where $ξ_0,ξ_1,\dots,ξ_{n}$ are independent real-valued random variables with probability density functions $f_0,\dots,f_n$. We give an explicit formula for the mixed $(k,l)$-correlation function $ρ_{k,l}:\mathbb{R}^k\times\mathbb{C}_+^l \to\mathbb{R}_+$ between $k$ real and $l$ complex zeros of $G_n$.
An old conjecture states that among all simplices inscribed in the unit sphere the regular one has the maximal mean width. An equivalent formulation is that for any centered Gaussian vector $(ξ_1,\dots,ξ_n)$ satisfying $\mathbb Eξ_1^2= \dots =\mathbb Eξ_n^2=1$ one has $$ \mathbb E\,\max\{ξ_1,\dots,ξ_n\}\leq\sqrt{\frac{n}{n-1}}\, \mathbb E\,\max\{η_1,\dots,η_n\}, $$ where $η_1,η_2,\dots,$ are independent standard Gaussian variables. Using this probabilistic interpretation we derive an asymptotic version of the conjecture. We also show that the mean width of the regular simplex with $2n$ vertices is remarkably close to the mean width of the regular crosspolytope with the same number of vertices. Interpreted probabilistically, our result states that $$ 1\leq\frac{\mathbb E\,\max\{|η_1|,\dots,|η_n|\}}{\mathbb E\,\max\{η_1,\dots,η_{2n}\}} \leq\min\left\{\sqrt{\frac{2n}{2n-1}}, \, 1+\frac{C}{n\, \log n} \right\}, $$ where $C>0$ is an absolute constant. We also compute the higher moments of the projection length $W$ of the regular cube, simplex and crosspolytope onto a line with random direction, thus proving several formulas conjectured by S. Finch. Finally, we prove distributional limit theorems for the length of random projection as the dimension goes to $\infty$. In the case of the $n$-dimensional unit cube $Q_n$, we prove that $$ W_{Q_n} - \sqrt{\frac{2n}π} \overset{d}{\underset{n\to\infty}\longrightarrow} {\mathcal{N}} \left(0, \frac{π-3}π\right), $$ whereas for the simplex and the crosspolytope the limiting distributions are related to the Gumbel double exponential law.
For a region $Ω\subset\mathbb{C}$ denote by $Ψ(Q;Ω)$ the number of complex algebraic numbers in $Ω$ of degree $\leq n$ and naive height $\leq Q$. We show that $$ Ψ(Q;Ω)=\frac{Q^{n+1}}{2ζ(n+1)}\int_Ωψ(z)\,ν(dz)+O\left(Q^n \right),\quad Q\to\infty, $$ where $ν$ is the Lebesgue measure on the complex plane and the function $ψ$ will be given explicitly.
Consider $n$ points $X_1,\ldots,X_n$ in $\mathbb R^d$ and denote their convex hull by $Π$. We prove a number of inclusion-exclusion identities for the system of convex hulls $Π_I:=conv(X_i\colon i\in I)$, where $I$ ranges over all subsets of $\{1,\ldots,n\}$. For instance, denoting by $c_k(X)$ the number of $k$-element subcollections of $(X_1,\ldots,X_n)$ whose convex hull contains a point $X\in\mathbb R^d$, we prove that $$ c_1(X)-c_2(X)+c_3(X)-\ldots + (-1)^{n-1} c_n(X) = (-1)^{\dim Π} $$ for all $X$ in the relative interior of $Π$. This confirms a conjecture of R. Cowan [Adv. Appl. Probab., 39(3):630--644, 2007] who proved the above formula for almost all $X$. We establish similar results for the number of polytopes $Π_J$ containing a given polytope $Π_I$ as an $r$-dimensional face, thus proving another conjecture of R. Cowan [Discrete Comput. Geom., 43(2):209--220, 2010]. As a consequence, we derive inclusion-exclusion identities for the intrinsic volumes and the face numbers of the polytopes $Π_I$. The main tool in our proofs is a formula for the alternating sum of the face numbers of a convex polytope intersected by an affine subspace. This formula generalizes the classical Euler--Schläfli--Poincaré relation and is of independent interest.
We give an explicit formula for the correlation functions of real zeros of a random polynomial with arbitrary independent continuously distributed coefficients.
Consider a random polynomial $$ G_Q(x)=ξ_{Q,n}x^n+ξ_{Q,n-1}x^{n-1}+...+ξ_{Q,0} $$ with independent coefficients uniformly distributed on $2Q+1$ integer points $\{-Q, ..., Q\}$. Denote by $D(G_Q)$ the discriminant of $G_Q$. We show that there exists a constant $C_n$, depending on $n$ only such that for all $Q\ge 2$ the distribution of $D(G_Q)$ can be approximated as follows $$ \sup_{-\infty\leq a\leq b\leq\infty}|\mathbb{P}(a\leq \frac{D(G_Q)}{Q^{2n-2}}\leq b)-\int_a^bφ_n(x)\, dx|\leq\frac{C_n}{\log Q}, $$ where $φ_n$ denotes the distribution function of the discriminant of a random polynomial of degree $n$ with independent coefficients which are uniformly distributed on $[-1,1]$. Let $Δ(G_Q)$ denote the minimal distance between the complex roots of $G_Q$. As an application we show that for any $\varepsilon>0$ there exists a constant $δ_n>0$ such that $Δ(G_Q)$ is stochastically bounded from below/above for all sufficiently large $Q$ in the following sense $$ \mathbb{P}(δ_n<Δ(G_Q)<\frac1{δ_n})>1-\varepsilon . $$
Let $ξ_0,ξ_1,\ldots$ be independent identically distributed complex- valued random variables such that $\mathbb{E}\log(1+|ξ_0|)<\infty$. We consider random analytic functions of the form \[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}ξ_kf_{k,n}z^k,\] where $f_{k,n}$ are deterministic complex coefficients. Let $μ_n$ be the random measure counting the complex zeros of $\mathbf{G}_n$ according to their multiplicities. Assuming essentially that $-\frac{1}{n}\log f_{[tn],n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $\frac{1}{n}μ_n$ converges in probability to some deterministic measure $μ$ which is characterized in terms of the Legendre-Fenchel transform of $u$. The limiting measure $μ$ does not depend on the distribution of the $ξ_k$'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.
A formula due to Sudakov relates the first intrinsic volume of a convex set in a Hilbert space to the maximum of the isonormal Gaussian process over this set. Using this formula we compute the first intrinsic volumes of infinite-dimensional convex compact sets including unit balls with respect to Sobolev-type seminorms and ellipsoids in the Hilbert space. We relate the distribution of the random one-dimensional projections of these sets to the distributions $S_1,S_2,C_1,C_2$ studied by Biane, Pitman, Yor [Bull.\ AMS 38 (2001)]. We show that the $k$-th intrinsic volume of the set of all functions on $[0,1]$ which have Lipschitz constant bounded by $1$ and which vanish at $0$ (respectively, which have vanishing integral) is given by $$ V_k = \frac{π^{k/2}}{Γ\left(\frac 32 k +1 \right)}, \text{ respectively } V_k = \frac{π^{(k+1)/2}}{2Γ\left(\frac 32 k +\frac 32\right)}. $$ This is related to the results of Gao and Vitale [Discrete Comput. Geom. 26 (2001), Elect. Comm. Probab. 8 (2003)] who considered a similar question for functions with a restriction on the total variation instead of the Lipschitz constant. Using the results of Gao and Vitale we give a new proof of the formula for the expected volume of the convex hull of the $d$-dimensional Brownian motion which is due to Eldan [Elect. J. Probab., to appear]. Additionally, we prove an analogue of Eldan's result for the Brownian bridge. Similarly, we show that the results on the intrinsic volumes of the Lipschitz balls can be translated into formulae for the expected volumes of zonoids (Aumann integrals) generated by the Brownian motion and the Brownian bridge. Also, these results have discrete versions for Gaussian random walks and bridges. Our proofs exploit Sudakov's and Tsirelson's theorems which establish a connection between the intrinsic volumes and the isonormal Gaussian process.
It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_n(z)=\sum_{k=0}^nξ_kz^k$ with i.i.d. coefficients $ξ_0,\ldots,ξ_n$ concentrate a.s. near the unit circle as $n\to\infty$ if and only if ${\mathbb{E}\log_+}|ξ_0|<\infty$. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like $L({\log}|t|)({\log}|t|)^{-α}$ as $t\to\infty$, where $α\geq0$, and $L$ is a slowly varying function. Under this assumption, the structure of complex and real roots of $G_n$ is described in terms of the least concave majorant of the Poisson point process on $[0,1]\times (0,\infty)$ with intensity $αv^{-(α+1)}\,du\,dv$.
Consider a $d\times d$ matrix $M$ whose rows are independent centered non-degenerate Gaussian vectors $ξ_1,...,ξ_d$ with covariance matrices $Σ_1,...,Σ_d$. Denote by $\mathcal{E}_i$ the location-dispersion ellipsoid of $ξ_i:\mathcal{E}_i={\mathbf{x}\in\mathbb{R}^d : \mathbf{x}^\topΣ_i^{-1} \mathbf{x}\leqslant1}$. We show that $$ \mathbb{E}\,|\det M|=\frac{d!}{(2π)^{d/2}}V_d(\mathcal{E}_1,...,\mathcal{E}_d), $$ where $V_d(\cdot,...,\cdot)$ denotes the {\it mixed volume}. We also generalize this result to the case of rectangular matrices. As a direct corollary we get an analytic expression for the mixed volume of $d$ arbitrary ellipsoids in $\mathbb{R}^d$. As another application, we consider a smooth centered non-degenerate Gaussian random field $X=(X_1,...,X_k)^\top:\mathbb{R}^d\to\mathbb{R}^k$. Using Kac-Rice formula, we obtain the geometric interpretation of the intensity of zeros of $X$ in terms of the mixed volume of location-dispersion ellipsoids of the gradients of $X_i/\sqrt{\mathbf{Var} X_i}$. This relates zero sets of equations to mixed volumes in a way which is reminiscent of the well-known Bernstein theorem about the number of solutions of the typical system of algebraic equations.
Let $ξ_0,ξ_1,...$ be independent identically distributed (i.i.d.) random variables such that $\E \log (1+|ξ_0|)<\infty$. We consider random analytic functions of the form $$ G_n(z)=\sum_{k=0}^{\infty} ξ_k f_{k,n} z^k, $$ where $f_{k,n}$ are deterministic complex coefficients. Let $ν_n$ be the random measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming essentially that $-\frac 1n \log f_{[tn], n}\to u(t)$ as $n\to\infty$, where $u(t)$ is some function, we show that the measure $ν_n$ converges weakly to some deterministic measure which is characterized in terms of the Legendre--Fenchel transform of $u$. The limiting measure is universal, that is it does not depend on the distribution of the $ξ_k$'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.
For a Borel set $A$ and a homogeneous Poisson point process $η$ in $\R^d$ of intensity $λ>0$, define the Poisson--Voronoi approximation $ A_η$ of $A$ as a union of all Voronoi cells with nuclei from $η$ lying in $A$. If $A$ has a finite volume and perimeter we find an exact asymptotic of $\E\Vol(AΔA_η)$ as $λ\to\infty$ where $\Vol$ is the Lebesgue measure. Estimates for all moments of $\Vol(A_η)$ and $\Vol(AΔA_η)$ together with their asymptotics for large $λ$ are obtained as well.