Recent zbMATH articles in MSC 60Bhttps://zbmath.org/atom/cc/60B2022-09-13T20:28:31.338867ZWerkzeugAsymptotics for averages over classical orthogonal ensembleshttps://zbmath.org/1491.150392022-09-13T20:28:31.338867Z"Claeys, Tom"https://zbmath.org/authors/?q=ai:claeys.tom"Glesner, Gabriel"https://zbmath.org/authors/?q=ai:glesner.gabriel"Minakov, Alexander"https://zbmath.org/authors/?q=ai:minakov.alexander"Yang, Meng"https://zbmath.org/authors/?q=ai:yang.mengSummary: We study the averages of multiplicative eigenvalue statistics in ensembles of orthogonal Haar-distributed matrices, which can alternatively be written as Toeplitz+Hankel determinants. We obtain new asymptotics for symbols with Fisher-Hartwig singularities in cases where some of the singularities merge together and for symbols with a gap or an emerging gap. We obtain these asymptotics by relying on known analogous results in the unitary group and on asymptotics for associated orthogonal polynomials on the unit circle. As consequences of our results, we derive asymptotics for gap probabilities in the circular orthogonal and symplectic ensembles and an upper bound for the global eigenvalue rigidity in the orthogonal ensembles.Singularity of discrete random matriceshttps://zbmath.org/1491.150402022-09-13T20:28:31.338867Z"Jain, Vishesh"https://zbmath.org/authors/?q=ai:jain.vishesh"Sah, Ashwin"https://zbmath.org/authors/?q=ai:sah.ashwin"Sawhney, Mehtaab"https://zbmath.org/authors/?q=ai:sawhney.mehtaab-sAuthors' abstract: Let \(\xi\) be a non-constant real-valued random variable with finite support and let \(M_n(\xi )\) denote an \(n\times n\) random matrix with entries that are independent copies of \(\xi \). For \(\xi\) which is not uniform on its support, we show that
\begin{align*}
{\mathbb{P}}[M_n(\xi )\text{ is singular}] &= {\mathbb{P}}[\text{zero row or column}] \\
&\quad +(1+o_n(1)){\mathbb{P}}[\text{two equal (up to sign) rows or columns}],
\end{align*}
thereby confirming a folklore conjecture. As special cases, we obtain:
\begin{itemize}
\item For \(\xi = {\text{Bernoulli}}(p)\) with fixed \(p \in (0,1/2)\),
\[
{\mathbb{P}}[M_n(\xi )\text{ is singular}] = 2n(1-p)^n + (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^n,
\] which determines the singularity probability to \textit{two} asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. The first asymptotic term confirms a conjecture of Litvak and Tikhomirov.
\item For \(\xi = {\text{Bernoulli}}(p)\) with fixed \(p \in (1/2,1)\),
\[
{\mathbb{P}}[M_n(\xi )\text{ is singular}] = (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^n.
\] Previously, only the much weaker upper bound of \((\sqrt{p} + o_n(1))^n\) was known due to the work of Bourgain-Vu-Wood.
\end{itemize}
For \(\xi\) which is uniform on its support:
\begin{itemize}
\item We show that
\[
{\mathbb{P}}[M_n(\xi )\text{ is singular}]\&= (1+o_n(1))^n{\mathbb{P}}[\text{two rows or columns are equal}].
\]
\item Perhaps more importantly, we provide a sharp analysis of the contribution of the `compressible' part of the unit sphere to the lower tail of the smallest singular value of \(M_n(\xi )\).
\end{itemize}
Reviewer: Sho Matsumoto (Kagoshima)Joint global fluctuations of complex Wigner and deterministic matriceshttps://zbmath.org/1491.150412022-09-13T20:28:31.338867Z"Male, Camile"https://zbmath.org/authors/?q=ai:male.camile"Mingo, James A."https://zbmath.org/authors/?q=ai:mingo.james-a"Péché, Sandrine"https://zbmath.org/authors/?q=ai:peche.sandrine"Speicher, Roland"https://zbmath.org/authors/?q=ai:speicher.rolandRandom perturbations of matrix polynomialshttps://zbmath.org/1491.150422022-09-13T20:28:31.338867Z"Pagacz, Patryk"https://zbmath.org/authors/?q=ai:pagacz.patryk"Wojtylak, Michał"https://zbmath.org/authors/?q=ai:wojtylak.michalAuthors' abstract: A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of the resolvent of the sum is derived, and the eigenvalues are localised. Four instances are considered: a low-rank matrix perturbed by the Wigner matrix, a product \(HX\) of a fixed diagonal matrix \(H\) and the Wigner matrix \(X\) and two special matrix polynomials of higher degree. The results are illustrated with various examples and numerical simulations.
Reviewer: Sen Zhu (Changchun)Nonlinear parabolic equations for measureshttps://zbmath.org/1491.354212022-09-13T20:28:31.338867Z"Manita, O. A."https://zbmath.org/authors/?q=ai:manita.oxana-a"Shaposhnikov, S. V."https://zbmath.org/authors/?q=ai:shaposhnikov.stanislav-v(no abstract)Optimal local law and central limit theorem for \(\beta\)-ensembleshttps://zbmath.org/1491.370512022-09-13T20:28:31.338867Z"Bourgade, Paul"https://zbmath.org/authors/?q=ai:bourgade.paul"Mody, Krishnan"https://zbmath.org/authors/?q=ai:mody.krishnan"Pain, Michel"https://zbmath.org/authors/?q=ai:pain.michelThe \(\beta\)-ensemble is a collection of real random variables (eigenvalues) \(\lambda_1< \dots <\lambda_N\) whose joint density is proportional to \[ e^{-\frac {\beta N}{2}\sum_{k=1}^N V(\lambda_k)} \prod_{1\leq j < k \leq N} |\lambda_k-\lambda_j|^\beta, \] where \(V(\lambda)\) is a function satisfying some hypotheses. The authors study the fluctuations of \(\lambda_k\) around their expected values as \(N,k\to\infty\). They prove a rigidity result on the eigenvalues which is an estimate on the maximum of \(|\lambda_k - \gamma_k|\) taken over all \(k\in [a_N, N-a_N]\) with \(a_N/\log N\to\infty\), where \(\gamma_k\) is the \(k/N\)-th quantile of the equilibrium measure associated with \(V\). Another result is a multivariate central limit theorem for the fluctuations of the \(\lambda_k\)'s in the bulk of the spectrum wih covariance function corresponding to a logarithmically correlated field. In particular, the authors show that each eigenvalue in the bulk fluctuates on scale \(\sqrt {\log N}/N\). They also establish a similar CLT (again, with logarithmic correlations), for the logarithmic potential generated by the eigenvalues. An important ingredient in the proofs of these theorems is an upper bound on \(\mathbb E |s_N(z) - m_V(z)|^q\), where \(s_N(z) = \frac 1N \sum_{k=1}^N \frac{1}{\lambda_k-z}\) is the Stieltjes transform of the empirical eigenvalue distribution and \(m_V(z)\) is the Stieltjes transform of the equilibrium measure associated with the external potential \(V\). Contrary to much progress on random matrix universality, the proofs do not proceed by comparison with the Gaussian case. In fact, the results are new even for the Gaussian \(\beta\)-ensembles (corresponding to quadratic external potential \(V(x) = x^2/2\)).
Reviewer: Zakhar Kabluchko (Münster)Weighted Gaussian entropy and determinant inequalitieshttps://zbmath.org/1491.600052022-09-13T20:28:31.338867Z"Stuhl, I."https://zbmath.org/authors/?q=ai:stuhl.izabella"Kelbert, M."https://zbmath.org/authors/?q=ai:kelbert.mark-ya"Suhov, Y."https://zbmath.org/authors/?q=ai:suhov.yurii-m"Yasaei Sekeh, S."https://zbmath.org/authors/?q=ai:yasaei-sekeh.salimehSummary: We produce a series of results extending information-theoretical inequalities (discussed by Dembo-Cover-Thomas in [\textit{A. Dembo} et al., IEEE Trans. Inf. Theory 37, No. 6, 1501--1518 (1991; Zbl 0741.94001)]) to a weighted version of entropy. Most of the resulting inequalities involve the Gaussian weighted entropy; they imply a number of new relations for determinants of positive-definite matrices. Unlike the Shannon entropy where the contribution of an outcome depends only upon its probability, the weighted (or context-dependent) entropy takes into account a `value' of an outcome determined by a given weight function \(\varphi \). An example of a new result is a weighted version of the strong Hadamard inequality (SHI) between the determinants of a positive-definite \(d\times d\) matrix and its square blocks (sub-matrices) of different sizes. When \(\varphi \equiv 1\), the weighted inequality becomes a `standard' SHI; in general, the weighted version requires some assumptions upon \(\varphi \). The SHI and its weighted version generalize a widely known `usual' Hadamard inequality \(\det\mathbf{C}\le \prod \nolimits_{j=1}^dC_{jj} \).Infinite \(p\)-adic random matrices and ergodic decomposition of \(p\)-adic Hua measureshttps://zbmath.org/1491.600072022-09-13T20:28:31.338867Z"Assiotis, Theodoros"https://zbmath.org/authors/?q=ai:assiotis.theodoros\textit{Y. A. Neretin} [Izv. Math. 77, No. 5, 941--953 (2013; Zbl 1284.22013); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 5, 95--10 (2013)] has constructed Hua measures on the set of infinite \(p\)-adic matrices. \textit{A. I. Bufetov} and \textit{Y. Qiu} [Compos. Math. 153, No. 12, 2482--2533 (2017; Zbl 1386.37004)] described the measures on the same space that are ergodic with respect to a certain natural action. In this paper, the author gives explicit ergodic decomposition of Neretin's Hua measures.
Reviewer: Tomas Persson (Lund)Extreme eigenvalue statistics of \(m\)-dependent heavy-tailed matriceshttps://zbmath.org/1491.600082022-09-13T20:28:31.338867Z"Basrak, Bojan"https://zbmath.org/authors/?q=ai:basrak.bojan"Cho, Yeonok"https://zbmath.org/authors/?q=ai:cho.yeonok"Heiny, Johannes"https://zbmath.org/authors/?q=ai:heiny.johannes"Jung, Paul"https://zbmath.org/authors/?q=ai:jung.paul-m|jung.paul-hConsider a stationary random field \((X_{i,j})_{(i,j)\in\mathbb Z_+^2}\) which is \(m\)-dependent, that is, for any \(A, B\subset \mathbb Z_+^2\) such that
\[
\max(|i-k|,|j-l|)>m\quad \text{for all}\quad (i,j)\in A, (k,l)\in B,
\]
the families \((X_{i,j})_{(i,j)\in A}\) and \((X_{k,l})_{(k,l)\in B}\) are independent. Also consider its diagonally symmetrized version \((\hat X_{i,j})_{(i,j)\in\mathbb Z_+^2}\), where \(\hat X_{i,j}= X_{i,j}\) for \(i\leq j\) and \(\hat X_{i,j}= X_{j,i}\) for \(i> j\). Assume that the common distribution of the random variables satisfies
\[
\mathbb P[|X_{i,j}|>t]=t^{-\alpha} L(t),\quad \alpha\in(0,4),
\]
for some slowly varying at infinity function \(L\) and in addition assume that \(\mathbb E X_{i,j}=0\) if \(\alpha\in[2,4)\). Choose a normalizing sequence \(a_n\) such that
\[
n\mathbb P[|X_{i,j}|>a_n]\to 1,\quad n\to\infty,
\]
and construct 2 sequences of random matrices
\[
\mathbf A_n= (X_{i,j}/a_{np_n})_{1\leq i\leq p_n,1\leq j\leq n }\quad\text{and}\quad\hat{\mathbf A}_n=(\hat X_{i,j}/a_{n^2})_{1\leq i,j\leq n} ,
\]
where the integer sequence \(p_n\) satisfies \(p_n/n\to\gamma\in(0,\infty)\).
The authors show that for the empirical measures of the singular values \(\sigma_i(\mathbf A)\) of \(\mathbf A\) and the eigenvalues \(\lambda_i(\hat {\mathbf A}_n)\) of \(\hat {\mathbf A}_n\) we have
\[
\sum_{i=1}^n\delta_{\sigma_i({\mathbf A}_n)} \stackrel{d}{\to} \mathcal P,\quad \left(\sum_{i=1}^n\delta_{\lambda_i(\hat {\mathbf A}_n)}\mathrm1_{\lambda_i(\hat {\mathbf A}_n)>0}, \sum_{i=1}^n\delta_{\lambda_i(\hat {\mathbf A}_n)}\mathrm1_{\lambda_i(\hat {\mathbf A}_n)<0}\right) \stackrel{d}{\to} (\mathcal P, -\mathcal P),
\]
where \(\mathcal P\) is some certain Poisson cluster process in \(\mathbb R_+^1\) with a given precise and detailed albeit complicated structure.
This result generalizes the one from [\textit{A. Auffinger} et al., Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 3, 589--610 (2009; Zbl 1177.15037)], where the case of i.i.d. \(X_{i,j}\) has been considered.
Reviewer: Dmitry Zaporozhets (Sankt-Peterburg)Erratum to: ``Some patterned matrices with independent entries''https://zbmath.org/1491.600092022-09-13T20:28:31.338867Z"Bose, Arup"https://zbmath.org/authors/?q=ai:bose.arup"Saha, Koushik"https://zbmath.org/authors/?q=ai:saha.koushik"Sen, Priyanka"https://zbmath.org/authors/?q=ai:sen.priyankaErratum to the authors' paper [ibid. 10, No. 3, Article ID 2150030, 46 p. (2021; Zbl 1479.60013)].Singularity of random symmetric matrices revisitedhttps://zbmath.org/1491.600102022-09-13T20:28:31.338867Z"Campos, Marcelo"https://zbmath.org/authors/?q=ai:campos.marcelo"Jenssen, Matthew"https://zbmath.org/authors/?q=ai:jenssen.matthew-o"Michelen, Marcus"https://zbmath.org/authors/?q=ai:michelen.marcus"Sahasrabudhe, Julian"https://zbmath.org/authors/?q=ai:sahasrabudhe.julianSummary: Let \(M_n\) be drawn uniformly from all \(\pm 1\) symmetric \(n\times n\) matrices. We show that the probability that \(M_n\) is singular is at most \(\exp(-c(n\log n)^{1/2})\), which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of \(\exp (-cn^{1/2})\) on the singularity probability, our method is different and considerably simpler: we prove a ``rough'' inverse Littlewood-Offord theorem by a simple combinatorial iteration.Eigenstate thermalization hypothesis for Wigner matriceshttps://zbmath.org/1491.600112022-09-13T20:28:31.338867Z"Cipolloni, Giorgio"https://zbmath.org/authors/?q=ai:cipolloni.giorgio"Erdős, László"https://zbmath.org/authors/?q=ai:erdos.laszlo"Schröder, Dominik"https://zbmath.org/authors/?q=ai:schroder.dominikThe authors prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with high probability and with an optimal error inversely proportional to the square root of the dimension. More precisely, let $W$ be an $N\times N$ random Wigner matrix whose entries $w_{ij}$ are i.i.d. up to the symmetry constraint $w_{ij} = \overline{w_{ji}}$. Let $u_1,\ldots, u_N$ be an orthonormal eigenbasis of $W$. One of the main results of the paper states that for any deterministic matrix $A$ with $\|A\|\leq 1$ it holds that $\max_{i,j}|\langle u_i, A u_j\rangle - \delta_{ij} \mathrm{Tr}(A)/N|=\mathcal O(N^{\varepsilon-1/2})$ with very high probability. This result thus rigorously verifies the Eigenstate Thermalisation Hypothesis by \textit{J. Deutsch} [``Quantum statistical mechanics in a closed system'', Phys. Rev. A 43, No. 4, 2046--2049 (1991; \url{doi:10.1103/PhysRevA.43.2046})] for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, the authors prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in [\textit{P. Bourgade} and \textit{H. T. Yau}, Commun. Math. Phys. 350, No. 1, 231--278 (2017; Zbl 1379.58014); \textit{P. Bourgade} et al., Commun. Pure Appl. Math. 73, No. 7, 1526--1596 (2020; Zbl 1446.60005)]. On a technical level, a key contribution of this work is to identify improved multi-resolvent local laws that arise when intersplicing resolvents with traceless deterministic matrices by a careful analysis of a cumulant expansion via Feynman diagrams. The improvement also arises on the level of a single resolvent for $\mathrm{Tr}(GA)$ with $A$ traceless and deterministic, as shown in Theorem 3.
Reviewer: Marius Lemm (Cambridge)Universality for random matrices with equi-spaced external source: a case study of a biorthogonal ensemblehttps://zbmath.org/1491.600122022-09-13T20:28:31.338867Z"Claeys, Tom"https://zbmath.org/authors/?q=ai:claeys.tom"Wang, Dong"https://zbmath.org/authors/?q=ai:wang.dongSummary: We prove the edge and bulk universality of random Hermitian matrices with equi-spaced external source. One feature of our method is that we use neither a Christoffel-Darboux type formula, nor a double-contour formula, which are standard methods to prove universality results for exactly solvable models. This matrix model is an example of a biorthogonal ensemble, which is a special kind of determinantal point process whose kernel generally does not have a Christoffel-Darboux type formula or double-contour representation. Our methods may showcase how to handle universality problems for biorthogonal ensembles in general.An extension of the angular synchronization problem to the heterogeneous settinghttps://zbmath.org/1491.600132022-09-13T20:28:31.338867Z"Cucuringu, Mihai"https://zbmath.org/authors/?q=ai:cucuringu.mihai"Tyagi, Hemant"https://zbmath.org/authors/?q=ai:tyagi.hemantSummary: Given an undirected measurement graph \(G = ([n], E)\), the classical angular synchronization problem consists of recovering unknown angles \(theta_1, \dots, \theta_n\) from a collection of noisy pairwise measurements of the form \((\theta_i - \theta_j) \mod 2\pi\), for each \(\{i, j\} \in E \). This problem arises in a variety of applications, including computer vision, time synchronization of distributed networks, and ranking from preference relationships. In this paper, we consider a generalization to the setting where there exist \(k\) unknown groups of angles \(\theta_{l, 1}, \dots, \theta_{l, n}\), for \(l = 1, \dots, k\). For each \({\left\{{{i, j}}\right\}} \in E\), we are given noisy pairwise measurements of the form \(\theta_{\ell, i} - \theta_{\ell, j}\) for an unknown \(\ell \in \{1, 2, \dots, k\}\). This can be thought of as a natural extension of the angular synchronization problem to the heterogeneous setting of multiple groups of angles, where the measurement graph has an unknown edge-disjoint decomposition \(G = G_1 \cup G_2 \dots \cup G_k\), where the \(G_i\)'s denote the subgraphs of edges corresponding to each group. We propose a probabilistic generative model for this problem, along with a spectral algorithm for which we provide a detailed theoretical analysis in terms of robustness against both sampling sparsity and noise. The theoretical findings are complemented by a comprehensive set of numerical experiments, showcasing the efficacy of our algorithm under various parameter regimes. Finally, we consider an application of bi-synchronization to the graph realization problem, and provide along the way an iterative graph disentangling procedure that uncovers the subgraphs \(G_i, i = 1, \dots, k\) which is of independent interest, as it is shown to improve the final recovery accuracy across all the experiments considered.The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular elementhttps://zbmath.org/1491.600142022-09-13T20:28:31.338867Z"Hall, Brian C."https://zbmath.org/authors/?q=ai:hall.brian-c"Ho, Ching-Wei"https://zbmath.org/authors/?q=ai:ho.ching-weiSummary: We compute the Brown measure of \(x_0+i\sigma_t\), where \(\sigma_t\) is a free semicircular Brownian motion and \(x_0\) is a freely independent self-adjoint element that is not a multiple of the identity. The Brown measure is supported in the closure of a certain bounded region \(\Omega_t\) in the plane. In \(\Omega_t,\) the Brown measure is absolutely continuous with respect to Lebesgue measure, with a density that is constant in the vertical direction. Our results refine and rigorize results of Janik, Nowak, Papp, Wambach, and Zahed and of Jarosz and Nowak in the physics literature. We also show that pushing forward the Brown measure of \(x_0+i\sigma_t\) by a certain map \(Q_t:\Omega_t \rightarrow{\mathbb{R}}\) gives the distribution of \(x_0+\sigma_t\). We also establish a similar result relating the Brown measure of \(x_0+i\sigma_t\) to the Brown measure of \(x_0+c_t\), where \(c_t\) is the free circular Brownian motion.Non-Gaussian hyperplane tessellations and robust one-bit compressed sensinghttps://zbmath.org/1491.600182022-09-13T20:28:31.338867Z"Dirksen, Sjoerd"https://zbmath.org/authors/?q=ai:dirksen.sjoerd"Mendelson, Shahar"https://zbmath.org/authors/?q=ai:mendelson.shaharAuthors' abstract: We show that a tessellation generated by a small number of random affine hyperplanes can be used to approximate Euclidean distances between any two points in an arbitrary bounded set \(T\), where the random hyperplanes are generated by subgaussian or heavy-tailed normal vectors and uniformly distributed shifts. The number of hyperplanes needed for constructing such tessellations is determined by natural metric complexity measures of the set \(T\) and the wanted approximation error. In comparison, previous results in this direction were restricted to Gaussian hyperplane tessellations of subsets of the Euclidean unit sphere.
As an application, we obtain new reconstruction results in memoryless one-bit compressed sensing with non-Gaussian measurement matrices: by quantizing at uniformly distributed thresholds, it is possible to accurately reconstruct low-complexity signals from a small number of one-bit quantized measurements, even if the measurement vectors are drawn from a heavy-tailed distribution. These reconstruction results are uniform in nature and robust in the presence of pre-quantization noise on the analog measurements as well as adversarial bit corruptions in the quantization process. Moreover, if the measurement matrix is subgaussian then accurate recovery can be achieved via a convex program.
Reviewer: Ilya S. Molchanov (Bern)A CLT in Stein's distance for generalized Wishart matrices and higher-order tensorshttps://zbmath.org/1491.600412022-09-13T20:28:31.338867Z"Mikulincer, Dan"https://zbmath.org/authors/?q=ai:mikulincer.danSummary: We study a central limit theorem for sums of independent tensor powers, \(\frac{1}{\sqrt{d}}\sum\limits_{i=1}^dX_i^{\otimes p}\). We focus on the high-dimensional regime where \(X_i\in\mathbb{R}^n\) and \(n\) may scale with \(d\). Our main result is a proposed threshold for convergence. Specifically, we show that, under some regularity assumption, if \(n^{2p-1}\ll d\), then the normalized sum converges to a Gaussian. The results apply, among others, to symmetric uniform log-concave measures and to product measures. This generalizes several results found in the literature. Our main technique is a novel application of optimal transport to Stein's method, which accounts for the low-dimensional structure, which is inherent in \(X_i^{\otimes p}\).On approximation theorems for the Euler characteristic with applications to the bootstraphttps://zbmath.org/1491.600492022-09-13T20:28:31.338867Z"Krebs, Johannes"https://zbmath.org/authors/?q=ai:krebs.johannes-t-n"Roycraft, Benjamin"https://zbmath.org/authors/?q=ai:roycraft.benjamin"Polonik, Wolfgang"https://zbmath.org/authors/?q=ai:polonik.wolfgangThe authors study approximation theorems for the Euler characteristic of the Vietrois-Rips and Čech filtrations obtained from a Poisson or binomial sampling scheme in the critical regime. The main results concern to functional central limits theorems for the Euler characteristic curve associated to such filtrations, the results are applied to the smooth bootstrap of the Euler characteristic where the rate of convergence is determined relative to the Kantorovich-Wasserstein and Kolmogorov metrics. Computer simulations are also provided.
The paper is well written and recommended to those researchers interested in topological data analysis from its statistical point of view.
Let \(X_n\) be a random process that generates point clouds of \(n\) points in \([0,T]^d\). Over these point clouds we can consider the Vietoris-Rips or the Čech filtrations, denoted by \(\mathcal{K}_t(X_n)\), \(t\in[0,T]\). For each \(t\) we can consider the mean Euler characteristic of these filtrations, that define the so called mean Euler characteristic curve. This curve depends only in the underlying distribution of \(X_n\) and the main problem to solve is to provide good estimations for this curve depending on \(n\).
The functional central limit theorem presented in this work is the first to consider the Čech complex, in this sense it is an extension of [\textit{A. M. Thomas} and \textit{T. Owada}, Adv. Appl. Probab. 53, No. 1, 57--80 (2021; Zbl 07458579)]. Moreover, the application to the binomial case is also new and it is achieved by means of an approximation by Possion schemes. Remark that central limit theorems for persistent Betti numbers obtained from a stationary Poisson process can be found in [\textit{Y. Hiraoka} et al., Ann. Appl. Probab. 28, No. 5, 2740--2780 (2018; Zbl 1402.60059)].
The authors use a bootstrap technique to estimate the Euler characteristic curve: Given an iid generated cloud point of size \(n\) relative to an unknown density, the authors use a density estimate to replicate the cloud point and compute a bootstrapped Euler characteristic curve from these replicates. The authors show precise estimations for the Kantorovich-Wasserstein and Kolmogorov distances between the bootstrapped and the true curves depending on \(n\) and the supremum distance of the densities. The use of bootstrap for estimation of persistent invariants is not new, see for instance [\textit{F. Chazal} et al., J. Mach. Learn. Res. 18, Paper No. 159, 40 p. (2018; Zbl 1435.62452)].
Reviewer: Carlos Meniño (Vigo)Domino tilings of the aztec diamond with doubly periodic weightingshttps://zbmath.org/1491.600662022-09-13T20:28:31.338867Z"Berggren, Tomas"https://zbmath.org/authors/?q=ai:berggren.tomasSummary: In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular, a family of models which, for any \(k\in \mathbb{N} \), includes models with \(k\) smooth regions is analyzed as the size of the Aztec diamond tends to infinity. We use a nonintersecting paths formulation and give a double integral formula for the correlation kernel of the Aztec diamond of finite size. By a classical steepest descent analysis of the correlation kernel, we obtain the local behavior in the smooth and rough regions, as the size of the Aztec diamond tends to infinity. From the mentioned limit the macroscopic picture, such as the arctic curves and, in particular, the number of smooth regions, is deduced. Moreover, we compute the limit of the height function, and, as a consequence, we confirm in the setting of this paper that the limit in the rough region fulfills the complex Burgers' equation, as stated by Kenyon and Okounkov.Stochastic integration with respect to cylindrical semimartingaleshttps://zbmath.org/1491.600752022-09-13T20:28:31.338867Z"Fonseca-Mora, Christian A."https://zbmath.org/authors/?q=ai:fonseca-mora.christian-aSummary: In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on a locally convex space \(\Phi\). Our construction of the stochastic integral is based on the theory of tensor products of topological vector spaces and the property of good integrators of real-valued semimartingales. This theory is further developed in the case where \(\Phi\) is a complete, barrelled, nuclear space, where we obtain a complete description of the class of integrands as \(\Phi\)-valued locally bounded and weakly predictable processes. Several other properties of the stochastic integral are proven, including a Riemann representation, a stochastic integration by parts formula and a stochastic Fubini theorem. Our theory is then applied to provide sufficient and necessary conditions for existence and uniqueness of solutions to linear stochastic evolution equations driven by semimartingale noise taking values in the strong dual \(\Phi^{\prime}\) of \(\Phi\). In the last part of this article we apply our theory to define stochastic integrals with respect to a sequence of real-valued semimartingales.Long-time influence of small perturbations and motion on the simplex of invariant probability measureshttps://zbmath.org/1491.601402022-09-13T20:28:31.338867Z"Freidlin, Mark I."https://zbmath.org/authors/?q=ai:freidlin.mark-iSummary: We present a general approach to a broad class of asymptotic problems related to the long-time influence of small perturbations, of both the deterministic and stochastic type. The main characteristic of this influence is a limiting motion on the simplex of invariant probability measures of the non-perturbed system in an appropriate time scale. We consider perturbations of dynamical systems in \(\mathbb{R}^n\), linear and nonlinear perturbations of PDEs, wave fronts in the reaction-diffusion equations, homogenization problems and perturbations caused by small time delay. The main tools we use in these problems are limit theorems for large deviations, modified averaging principle and diffusion approximation.Perturbations of multiple Schramm-Loewner evolution with two non-colliding Dyson Brownian motionshttps://zbmath.org/1491.601472022-09-13T20:28:31.338867Z"Chen, Jiaming"https://zbmath.org/authors/?q=ai:chen.jiaming.1"Margarint, Vlad"https://zbmath.org/authors/?q=ai:margarint.vladSummary: In this article we study multiple \(\mathrm{SLE}_\kappa\), for \(\kappa \in (0, 4]\), driven by Dyson Brownian motions. This model was introduced in the unit disk by \textit{J. Cardy} [J. Phys. A, Math. Gen. 36, No. 24, L379--L386 (2003; Zbl 1038.82074)] in connection with the Calogero-Sutherland model. We prove the Carathéodory convergence of perturbed Loewner chains under different initial conditions and under different diffusivity \(\kappa \in (0, 4]\) for the case of \(N = 2\) driving forces. Our proofs use the analysis of Bessel processes and estimates on Loewner differential equation with multiple driving forces. In the last section, we estimate the Hausdorff distance of the hulls under perturbations of the driving forces, with assumptions on the modulus of the derivative of the multiple Loewner maps.Analysis of Langevin Monte Carlo via convex optimizationhttps://zbmath.org/1491.650092022-09-13T20:28:31.338867Z"Durmus, Alain"https://zbmath.org/authors/?q=ai:durmus.alain"Majewski, Szymon"https://zbmath.org/authors/?q=ai:majewski.szymon"Miasojedow, Błażej"https://zbmath.org/authors/?q=ai:miasojedow.blazejSummary: In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as the first order optimization algorithm for an objective functional defined on the Wasserstein space of order 2. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from log-concave smooth target distribution on \(\mathbb{R}^d\). Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution. These new algorithms are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm, which is a popular extension of the Unadjusted Langevin Algorithm for largescale Bayesian inference. Using the optimization perspective, we provide non-asymptotic convergence analysis for the newly proposed methods.Information volume fractal dimensionhttps://zbmath.org/1491.940292022-09-13T20:28:31.338867Z"Gao, Qiuya"https://zbmath.org/authors/?q=ai:gao.qiuya"Wen, Tao"https://zbmath.org/authors/?q=ai:wen.tao"Deng, Yong"https://zbmath.org/authors/?q=ai:deng.yong