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Friedrich Götze

Publications and source records attributed to Friedrich Götze.

At least 37 records · Page 2Linked to original sources
arXiv2021

Small Values of Indefinite Diagonal Quadratic Forms at Integer Points in at least five Variables

For any $\varepsilon > 0$ we derive effective estimates for the size of a non-zero integral point $m \in \mathbb{Z}^d \setminus \{0\}$ solving the Diophantine inequality $\lvert Q[m] \rvert < \varepsilon$, where $Q[m] = q_1 m_1^2 + \ldots + q_d m_d^2$ denotes a non-singular indefinite diagonal quadratic form in $d \geq 5$ variables. In order to prove our quantitative variant of the Oppenheim conjecture, we extend an approach developed by Birch and Davenport [BD58b] to higher dimensions combined with a theorem of Schlickewei [Sch85]. The result obtained is an optimal extension of Schlickewei's result, giving bounds on small zeros of integral quadratic forms depending on the signature $(r,s)$, to diagonal forms up to a negligible growth factor.

math.NT↗
arXiv2021

Rate of Convergence to the Circular Law via Smoothing Inequalities for Log-Potentials

The aim of this note is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by $n^{-1/2}$. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.

math.PR↗
arXiv2021

An Expansion for a Discrete Non-Lattice Distribution

Much is known about asymptotic expansions for asymptotically normal distributions if these distributions are either absolutely continuous or pure lattice distributions. In this paper we begin an investigation of the discrete but non-lattice case. We tackle one of the simplest examples imaginable and find that curious phenomena occur. Clearly more work is needed.

math.PR↗
arXiv2021

Edgeworth approximations for distributions of symmetric statistics

We study the distribution of a general class of asymptoticallylinear statistics which are symmetric functions of $N$ independent observations. The distribution functions of these statistics are approximated by an Edgeworth expansion with a remainder of order $o(N^{-1})$. The Edgeworth expansion is based on Hoeffding's decomposition which provides a stochastic expansion into a linear part, a quadratic part as well as smaller higher order parts. The validity of this Edgeworth expansion is proved under Cramér's condition on the linear part, moment assumptions for all parts of the statistic and an optimal dimensionality requirement for the non linear part.

math.ST↗
arXiv2020

Characteristic polynomials of products of Wigner matrices: finite-N results and Lyapunov universality

We compute the average characteristic polynomial of the hermitised product of $M$ real or complex Wigner matrices of size $N\times N$ and the average of the characteristic polynomial of a product of $M$ such Wigner matrices times the characteristic polynomial of the conjugate matrix. Surprisingly, the results agree with that of the product of $M$ real or complex Ginibre matrices at finite-$N$, which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two characteristic polynomials involves the kernel of complex eigenvalues. This extends the result of Forrester and Gamburd for one characteristic polynomial of a single random matrix and only depends on the first two moments. In the limit $M\to\infty$ at fixed $N$ we determine the locations of the zeros of a single characteristic polynomial, rescaled as Lyapunov exponents by taking the logarithm of the $M$th root. The position of the $j$th zero agrees asymptotically for large-$j$ with the position of the $j$th Lyapunov exponent for products of Gaussian random matrices, hinting at the universality of the latter.

math.PR↗
arXiv2020

Concentration inequalities for bounded functionals via generalized log-Sobolev inequalities

In this paper we prove multilevel concentration inequalities for bounded functionals $f = f(X_1, \ldots, X_n)$ of random variables $X_1, \ldots, X_n$ that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of $k$-tensors of higher order differences of $f$. We provide applications in both dependent and independent random variables. This includes deviation inequalities for empirical processes $f(X) = \sup_{g \in \mathcal{F}} \lvert g(X) \rvert$ and suprema of homogeneous chaos in bounded random variables in the Banach space case given by $f(X) = \sup_{t} \lVert \sum_{i_1 \neq \ldots \neq i_d} t_{i_1 \ldots i_d} X_{i_1} \cdots X_{i_d}\rVert_{\mathcal{B}}$. The latter application is comparable to earlier results of Boucheron-Bousquet-Lugosi-Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for $U$-statistics with bounded kernels $h$ and for the number of triangles in an exponential random graph model.

math.PR↗
arXiv2020

Grassmann angles and absorption probabilities of Gaussian convex hulls

Let $M$ be an arbitrary subset in $\mathbb R^n$ with a conic (or positive) hull $C$. Consider its Gaussian image $AM$, where $A$ is a $k\times n$-matrix whose entries are independent standard Gaussian random variables. We show that the probability that the convex hull of $AM$ contains the origin in its interior coincides with the $k$-th Grassmann angle of $C$. Also, we prove that the expected Grassmann angles of $AC$ coincide with the corresponding Grassmann angles of $C$. Using the latter result, we show that the expected sum of $j$-th Grassmann angles at $\ell$-dimensional faces of a Gaussian simplex equals the analogous angle-sum for the regular simplex of the same dimension.

math.PR↗
arXiv2019

Second Order Concentration via Logarithmic Sobolev Inequalities

We show sharpened forms of the concentration of measure phenomenon centered at first order stochastic expansions. The bound are based on second order difference operators and second order derivatives. Applications to functions on the discrete cube and stochastic Hoeffding type expansions in mathematical statistics are studied as well as linear eigenvalue statistics in random matrix theory.

math.PR↗
arXiv2019

Higher order concentration in presence of Poincaré-type inequalities

We show sharpened forms of the concentration of measure phenomenon typically centered at stochastic expansions of order $d-1$ for any $d \in \mathbb{N}$. Here we focus on differentiable functions on the Euclidean space in presence of a Poincaré-type inequality. The bounds are based on $d$-th order derivatives.

math.PR↗
arXiv2019

Joint distribution of conjugate algebraic numbers: a random polynomial approach

We count the algebraic numbers of fixed degree by their $\mathbf{w}$-weighted $l_p$-norm which generalizes the naïve height, the length, the Euclidean and the Bombieri norms. For non-negative integers $k,l$ such that $k+2l\leq n$ and a Borel subset $B\subset \mathbb{R}\times\mathbb{C}_+^l$ denote by $Φ_{p,\mathbf{w},k,l}(Q,B)$ the number of ordered $(k+l)$-tuples in $B$ of conjugate algebraic numbers of degree $n$ and $\mathbf{w}$-weighted $l_p$-norm at most $ Q$. We show that $$ \lim_{ Q\to\infty}\frac{Φ_{p,\mathbf{w},k,l}( Q,B)}{ Q^{n+1}}=\frac{\mathrm{Vol}_{n+1}(\mathbb{B}_{p,\mathbf{w}}^{n+1})}{2ζ(n+1)}\int_B ρ_{p,\mathbf{w},k,l}(\mathbf{x},\mathbf{z}){\rm d}\mathbf{x}{\rm d}\mathbf{z}, $$ where ${\mathrm{Vol}}_{n+1}(\mathbb{B}_{p,\mathbf{w}}^{n+1})$ is the volume of the unit $\mathbf{w}$-weighted $l_p$-ball and $ρ_{p,\mathbf{w},k,l}$ will denote the correlation function of $k$ real and $l$ complex zeros of the random polynomial $\sum_{j=1}^n \frac{η_j}{w_j} z^j$, where $η_j $ are i.i.d. random variables with density $c_p e^{-|t|^p}$ for $0<p<\infty$ and with constant density on $[-1,1]$ for $p=\infty$. If the boundary of $B$ is of Lipschitz type, we also estimate the rate of convergence. We give an explicit formula for $ρ_{p,\mathbf{w},k,l}$, which in the case $k+2l=n$ has a very simple form. To this end, we obtain a general formula for the correlations between real and complex zeros of a random polynomial with arbitrary independent absolutely continuous coefficients.

math.NT↗
arXiv2019

Local semicircle law under fourth moment condition

We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that $\max_{jk} {\mathbb E} |X_{jk}|^{4+δ} < \infty, δ> 0$, it was proved in [Götze, Naumov and Tikhomirov, Bernoulli, 2018] that with high probability the typical distance between the Stieltjes transforms $m_n(z), z = u + i v$, of the empirical spectral distribution (ESD) and the Stieltjes transforms $m_{sc}(z)$ of the semicircle law is of order $(nv)^{-1} \log n$. The aim of this paper is to remove $δ>0$ and show that this result still holds if we assume that $\max_{jk} {\mathbb E} |X_{jk}|^{4} < \infty$. We also discuss applications to the rate of convergence of the ESD to the semicircle law in the Kolmogorov distance, rates of localization of the eigenvalues around the classical positions and rates of delocalization of eigenvectors.

math.PR↗
arXiv2019

On the distribution of Salem numbers

In this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals $I_1,\ldots, I_{k}\subset \left[0;π\right]$, $1\leq k\leq m$ let $Sal_{m,k}(Q,I_1,\ldots,I_{k})$ denote the set of ordered $(k+1)$-tuples $\left(α_0,\ldots,α_{k}\right)$ of conjugate algebraic integers, such that $α_0$ is a Salem numbers of degree $2m+2$ satisfying $α\leq Q$ for some positive real number $Q$ and $\argα_i\in I_i$. We derive the following asymptotic approximation \[ \# Sal_{m,k}(Q,I_1,\ldots,I_{k})=ω_m\,Q^{m+1}\,\int\limits_{I_1}\ldots\int\limits_{I_{k}}ρ_{m,k}(\boldsymbolθ)\rm d\boldsymbolθ+O\left(Q^{m}\right),\quad Q\rightarrow\infty, \] providing explicit expressions for the constant $ω_m$ and the function $ρ_{m,k}(\boldsymbolθ)$. Moreover we derive a similar asymptotic formula for the set of all Salem numbers of fixed degree and absolute value bounded by $Q$ as $Q\rightarrow\infty$.

math.NT↗
arXiv2019

Random affine simplexes

For a fixed $k\in\{1,\dots,d\}$ consider random vectors $X_0,\dots, X_{k}\in\mathbb R^d$ with an arbitrary spherically symmetric joint density function. Let $A$ be any non-singular $d\times d$ matrix. We show that the $k$-dimensional volume of the convex hull of affinely transformed $X_{i}$'s satisfies \[ |\mathrm{conv}(AX_0,\dots,AX_{k})|\stackrel{d}{=}\frac{|P_ξ\mathcal{E}|}{κ_k}\cdot|\mathrm{conv}(X_0,\dots,X_{k})|, \] where $\mathcal{E}:=\{\mathbf{x}\in\mathbb R^d:{\mathbf{x}^\top (A^\top A)^{-1}\mathbf{x}}\leq 1\}$ is an ellipsoid, $P_ξ$ denotes the orthogonal projection to a random uniformly chosen $k$-dimensional linear subspace $ξ$ independent of $X_0,\dots, X_{k}$, and $κ_k$ is the volume of the unit $k$-dimensional ball. We express $|P_ξ\mathcal{E}|$ in terms of Gaussian random matrices. The important special case $k=1$ corresponds to the distance between two random points: \[ |AX_0-AX_1|\stackrel{d}{=}\sqrt{\frac{λ_1^2N_1^2+\dots+λ_d^2N_d^2}{N_1^2+\dots+N_d^2}}\cdot|X_0-X_1|, \] where $N_1,\dots,N_d$ are i.i.d. standard Gaussian variables independent of $X_0,X_1$ and $λ_1,\dots,λ_d$ are the singular values of $A$. As an application, we derive the following integral geometry formula for ellipsoids: \[ \frac{κ_{d}^{k+1}}{κ_k^{d+1}}\,\frac{κ_{k(d+p)+k}}{κ_{k(d+p)+d}}\,\int\limits_{A_{d,k}}|\mathcal{E}\cap E|^{p+d+1}\,μ_{d,k}(dE)=|\mathcal{E}|^{k+1}\,\int\limits_{G_{d,k}}|P_L\mathcal{E}|^p\,ν_{d,k}(dL), \] where $p> -d+k-1$ and $A_{d,k}$ and $G_{d,k}$ are the affine and the linear Grassmannians equipped with their respective Haar measures. The case $p=0$ reduces to an affine version of the integral formula of Furstenberg and Tzkoni.

math.PR↗
arXiv2019

Concentration inequalities for polynomials in $α$-sub-exponential random variables

In this work we derive multi-level concentration inequalities for polynomial functions in independent random variables with a $α$-sub-exponential tail decay. A particularly interesting case is given by quadratic forms $f(X_1, \ldots, X_n) = \langle X,A X \rangle$, for which we prove Hanson-Wright-type inequalities with explicit dependence on various norms of the matrix $A$. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in $α$-sub-exponential random variables, such as quadratic Poisson chaos. We provide various applications of these inequalities. Among these are generalizations the results given by Rudelson-Vershynin from sub-Gaussian to $α$-sub-exponential random variables, i. e. concentration of the Euclidean norm of the linear image of a random vector, small ball probability estimates and concentration inequalities for the distance between a random vector and a fixed subspace. Moreover, we obtain concentration inequalities for the excess loss in a fixed design linear regression and the norm of a randomly projected random vector.

math.PR↗
arXiv2019

Non-crossing partitions

Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical non-crossing partitions are associated to Coxeter and Artin groups of type $\mathsf{A}_n$, which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments.

math.GR↗
arXiv2018

Estimates for the closeness of convolutions of probability distributions on convex polyhedra

The aim of the present work is to show that the results obtained earlier on the approximation of distributions of sums of independent summands by the accompanying compound Poisson laws and the estimates of the proximity of sequential convolutions of multidimensional distributions may be transferred to the estimation of the closeness of convolutions of probability distributions on convex polyhedra.

math.PR↗
arXiv2018

On Local laws for non-Hermitian random matrices and their products

The aim of this paper is to prove a local version of the circular law for non-Hermitian random matrices and its generalization to the product of non-Hermitian random matrices under weak moment conditions. More precisely we assume that the entries $X_{jk}^{(q)}$ of non-Hermitian random matrices ${\bf X}^{(q)}, 1 \le j,k \le n, q = 1, \ldots, m, m \geq 1$ are i.i.d. r.v. with $\mathbb E X_{jk} =0, \mathbb E X_{jk}^2 = 1$ and $\mathbb E |X_{jk}|^{4+δ} < \infty$ for some $δ> 0$. It is shown that the local law holds on the optimal scale $n^{-1+2a}, a > 0$, up to some logarithmic factor. We further develop a Stein type method to estimate the perturbation of the equations for the Stieltjes transform of the limiting distribution. We also generalize the recent results [Bourgade--Yau-Yin, 2014], [Tao--Vu, 2015] and [Nemish, 2017]. An extension to the case of non-i.i.d. entries is discussed.

math.PR↗