Recent zbMATH articles in MSC 60Bhttps://zbmath.org/atom/cc/60B2021-06-15T18:09:00+00:00WerkzeugBayesian mean-parameterized nonnegative binary matrix factorization.https://zbmath.org/1460.620372021-06-15T18:09:00+00:00"Lumbreras, Alberto"https://zbmath.org/authors/?q=ai:lumbreras.alberto"Filstroff, Louis"https://zbmath.org/authors/?q=ai:filstroff.louis"Févotte, Cédric"https://zbmath.org/authors/?q=ai:fevotte.cedricSummary: Binary data matrices can represent many types of data such as social networks, votes, or gene expression. In some cases, the analysis of binary matrices can be tackled with nonnegative matrix factorization (NMF), where the observed data matrix is approximated by the product of two smaller nonnegative matrices. In this context, probabilistic NMF assumes a generative model where the data is usually Bernoulli-distributed. Often, a link function is used to map the factorization to the \([0, 1]\) range, ensuring a valid Bernoulli mean parameter. However, link functions have the potential disadvantage to lead to uninterpretable models. Mean-parameterized NMF, on the contrary, overcomes this problem. We propose a unified framework for Bayesian mean-parameterized nonnegative binary matrix factorization models (NBMF). We analyze three models which correspond to three possible constraints that respect the mean-parameterization without the need for link functions. Furthermore, we derive a novel collapsed Gibbs sampler and a collapsed variational algorithm to infer the posterior distribution of the factors. Next, we extend the proposed models to a nonparametric setting where the number of used latent dimensions is automatically driven by the observed data. We analyze the performance of our NBMF methods in multiple datasets for different tasks such as dictionary learning and prediction of missing data. Experiments show that our methods provide similar or superior results than the state of the art, while automatically detecting the number of relevant components.Parameter symmetry in perturbed GUE corners process and reflected drifted Brownian motions.https://zbmath.org/1460.600062021-06-15T18:09:00+00:00"Petrov, Leonid"https://zbmath.org/authors/?q=ai:petrov.leonid"Tikhonov, Mikhail"https://zbmath.org/authors/?q=ai:tikhonov.mikhailSummary: The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix \(G+\operatorname{diag}(\mathbf{a})\), where \(G\) is the random matrix from the Gaussian Unitary Ensemble (GUE), and \(\operatorname{diag}(\mathbf{a})\) is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arithmetic progression. The construction we present may be viewed as a random matrix analogue of the recent results of the authors and \textit{A. Saenz} [``Mapping TASEP back in time'', Preprint, \url{arXiv:1907.09155}].Random walk on the symplectic forms over a finite field.https://zbmath.org/1460.600032021-06-15T18:09:00+00:00"He, Jimmy"https://zbmath.org/authors/?q=ai:he.jimmySummary: Random transvections generate a walk on the space of symplectic forms on \(\mathbb{F}_q^{2n} \). The main result is to establish cutoff for this Markov chain. After \(n + c\) steps, the walk is close to uniform while before \(n - c\) steps, it is far from uniform. The upper bound is proved by explicitly finding and bounding the eigenvalues of the random walk. The lower bound is found by showing that the support of the walk is exponentially small if only \(n - c\) steps are taken. The result can be viewed as a \(q\)-deformation of a result of Diaconis and Holmes on a random walk on matchings.On the correlation functions of the characteristic polynomials of real random matrices with independent entries.https://zbmath.org/1460.150382021-06-15T18:09:00+00:00"Afanasiev, Ievgenii"https://zbmath.org/authors/?q=ai:afanasiev.ievgeniiSummary: The paper is concerned with the correlation functions of the characteristic polynomials of real random matrices with independent entries. The asymptotic behavior of the correlation functions is established in the form of a certain integral over unitary self-dual matrices with respect to the invariant measure. The integral is computed in the case of the second order correlation function. From the obtained asymptotics it is clear that the correlation functions behave like that for the Real Ginibre Ensemble up to a factor depending only on the fourth absolute moment of the common probability law of the matrix entries.Reduction principle for functionals of vector random fields.https://zbmath.org/1460.600462021-06-15T18:09:00+00:00"Olenko, Andriy"https://zbmath.org/authors/?q=ai:olenko.andriy-ya"Omari, Dareen"https://zbmath.org/authors/?q=ai:omari.dareenSummary: We prove a version of the reduction principle for functionals of vector long-range dependent random fields. The components of the fields may have different long-range dependent behaviours. The results are illustrated by an application to the first Minkowski functional of the Fisher-Snedecor random fields. Simulation studies confirm the obtained theoretical results and suggest some new problems.On the law of large numbers for the empirical measure process of generalized Dyson Brownian motion.https://zbmath.org/1460.601082021-06-15T18:09:00+00:00"Li, Songzi"https://zbmath.org/authors/?q=ai:li.songzi"Li, Xiang-Dong"https://zbmath.org/authors/?q=ai:li.xiang-dong"Xie, Yong-Xiao"https://zbmath.org/authors/?q=ai:xie.yong-xiaoSummary: We study the generalized Dyson Brownian motion (GDBM) of an interacting \(N\)-particle system with logarithmic Coulomb interaction and general potential \(V\). Under reasonable condition on \(V\), we prove the existence and uniqueness of strong solution to SDE for GDBM. We then prove that the family of the empirical measures of GDBM is tight on \(\mathcal{C}([0,T],\mathscr{P}(\mathbb{R}))\) and all the large \(N\) limits satisfy a nonlinear McKean-Vlasov equation. Inspired by previous works due to Biane and Speicher, Carrillo, McCann and Villani, and Blower, we prove that the McKean-Vlasov equation is indeed the gradient flow of the Voiculescu free entropy on the Wasserstein space of probability measures over \(\mathbb{R} \). Using the optimal transportation theory, we prove that if \(V''\ge K\) for some constant \(K\in \mathbb{R} \), the McKean-Vlasov equation has a unique weak solution in the space of probability measures \(\mathscr{P}(\mathbb{R})\). This establishes the Law of Large Numbers and the propagation of chaos for the empirical measures of GDBM with non-quadratic external potentials which are not necessarily convex. Finally, we prove the longtime convergence of the McKean-Vlasov equation for \(C^2\)-convex potentials \(V\).Fluctuations of the product of random matrices and generalized Lyapunov exponent.https://zbmath.org/1460.600082021-06-15T18:09:00+00:00"Texier, Christophe"https://zbmath.org/authors/?q=ai:texier.christopheSummary: I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products \(\varPi_n=M_nM_{n-1}\ldots M_1\), where \(M_i\)'s are i.i.d. Following [\textit{V. N. Tutubalin}, ``On limit theorems for the product of random matrices'', Theory Probab. Appl. 10, No. 1, 15--27 (1965; \url{doi:10.1137/1110002})], the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group. The formalism is illustrated by considering products of random matrices from the group \(\mathrm{SL}(2,\mathbb{R})\) where explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the one-dimensional Schrödinger equation where the random potential is a Lévy noise (derivative of a Lévy process). In this case, I obtain a general formula for the variance of \(\ln \|\varPi_n\|\) and for the variance of \(\ln |\psi (x)|\), where \(\psi (x)\) is the wavefunction, in terms of a single integral involving the Fourier transform of the invariant density of the matrix product. Finally I discuss the continuum limit of random matrix products (matrices close to the identity). In particular, I investigate a simple case where the spectral problem providing the generalized Lyapunov exponent can be solved exactly.Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information.https://zbmath.org/1460.460572021-06-15T18:09:00+00:00"Banna, Marwa"https://zbmath.org/authors/?q=ai:banna.marwa"Mai, Tobias"https://zbmath.org/authors/?q=ai:mai.tobiasSummary: This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information, highlighting the special case of Lipschitz conjugate variables. For the first time in this generality, it is shown that the analytic distributions of those evaluations have Hölder continuous cumulative distribution functions with an explicit Hölder exponent that depends only on the degree of the considered polynomial. For linear polynomials, we reach in the case of finite non-microstates free Fisher information the optimal Hölder exponent \(\frac{ 2}{ 3} \), and get Lipschitz continuity in the case of Lipschitz conjugate variables. In particular, our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy, which partially settles a conjecture of \textit{I.~Charlesworth} and \textit{D.~Shlyakhtenko} [J. Funct. Anal. 271, No.~8, 2274--2292 (2016; Zbl 1376.46051)].
We further provide a very general criterion that gives for weak approximations of measures having Hölder continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance.
Finally, we combine these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general Gibbs laws. For Gibbs laws, this extends the corresponding result obtained in
[\textit{A.~Guionnet} and \textit{D.~Shlyakhtenko}, Geom. Funct. Anal. 18(2008), No.~6, 1875--1916 (2009; Zbl 1187.46056)]
from convergence in distribution to convergence in Kolmogorov distance; in the GUE case, we even provide explicit rates, which quantify results of
[\textit{U.~Haagerup} and \textit{S.~Thorbjørnsen}, Ann. Math. (2) 162, No.~2, 711--775 (2005; Zbl 1103.46032)] and
[\textit{U.~Haagerup} et al., Adv. Math. 204, No.~1, 1--83 (2006; Zbl 1109.15020)] in terms of the Kolmogorov distance.The best constants in the multiple Khintchine inequality.https://zbmath.org/1460.460072021-06-15T18:09:00+00:00"Núñez-Alarcón, Daniel"https://zbmath.org/authors/?q=ai:nunez-alarcon.daniel"Serrano-Rodríguez, Diana Marcela"https://zbmath.org/authors/?q=ai:serrano-rodriguez.diana-marcelaFirst, the authors show that the Khintchine inequality and the mixed \((\ell _{p/(p-1)}\),\(\ell _{2})\)-Littlewood inequality are equivalent, that is, one can be obtained from the other one. In this way, the cotype constants \(C_{2,p}\) of \(\ell _{p}\)-spaces for \(1\leq p\leq 2\) are obtained. Among other results, the authors provide estimates of the optimal constants in the multiple Khintchine inequality and the optimal constants of the multilinear mixed \((\ell _{p/(p-1)}\),\(\ell _{2})\)-Littlewood inequality.
Reviewer: Elhadj Dahia (Bou Saâda)Dimensionality reduction in Euclidean space.https://zbmath.org/1460.622112021-06-15T18:09:00+00:00"Nelson, Jelani"https://zbmath.org/authors/?q=ai:nelson.jelani(no abstract)Spectral rigidity for addition of random matrices at the regular edge.https://zbmath.org/1460.460582021-06-15T18:09:00+00:00"Bao, Zhigang"https://zbmath.org/authors/?q=ai:bao.zhigang"Erdős, László"https://zbmath.org/authors/?q=ai:erdos.laszlo"Schnelli, Kevin"https://zbmath.org/authors/?q=ai:schnelli.kevinSummary: We consider the sum of two large Hermitian matrices \(A\) and \(B\) with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of \(A\) and \(B\) as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [the authors, Commun. Math. Phys. 349, No. 3, 947--990 (2017; 1416.60015); Adv. Math. 319, 251--291 (2017; Zbl 1383.46048)] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.Spherical functions approach to sums of random Hermitian matrices.https://zbmath.org/1460.150392021-06-15T18:09:00+00:00"Kuijlaars, Arno B. J."https://zbmath.org/authors/?q=ai:kuijlaars.arno-b-j"Román, Pablo"https://zbmath.org/authors/?q=ai:roman.pablo-manuelSummary: We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair \((\mathrm{U}(n) \ltimes \mathrm{Herm}(n), \mathrm{U}(n))\). It is inspired by a similar approach of \textit{M. Kieburg} and \textit{H. Kösters} [Random Matrices Theory Appl. 5, No. 4, Article ID 1650015, 57 p. (2016; Zbl 1375.60020)]
for products of random matrices. The spherical functions have determinantal expressions because of the Harish-Chandra/Itzykson-Zuber integral formula. It leads to remarkably simple expressions for the spherical transform and its inverse. The spherical transform is applied to sums of unitarily invariant random matrices from polynomial ensembles and the subclass of polynomial ensembles of derivative type (in the additive sense), which turns out to be closed under addition. We finally present additional detailed calculations for the sum with a random matrix from a Laguerre unitary ensemble.Edge scaling limit of the spectral radius for random normal matrix ensembles at hard edge.https://zbmath.org/1460.600072021-06-15T18:09:00+00:00"Seo, Seong-Mi"https://zbmath.org/authors/?q=ai:seo.seong-miSummary: We investigate local statistics of eigenvalues for random normal matrices, represented as 2D determinantal Coulomb gases, in the case when the eigenvalues are forced to be in the support of the equilibrium measure associated with an external field. For radially symmetric external fields with sufficient growth at infinity, we show that the fluctuations of the spectral radius around a hard edge tend to follow an exponential distribution as the number of eigenvalues tends to infinity. As a corollary, we obtain the order statistics of the moduli of eigenvalues.Scaling limit for determinantal point processes on spheres.https://zbmath.org/1460.600402021-06-15T18:09:00+00:00"Katori, Makoto"https://zbmath.org/authors/?q=ai:katori.makoto"Shirai, Tomoyuki"https://zbmath.org/authors/?q=ai:shirai.tomoyukiSummary: The unitary group with the Haar probability measure is called circular unitary ensemble. All the eigenvalues lie on the unit circle in the complex plane and they can be regarded as a determinantal point process on \(\mathbb{S}^1\). It is also known that the scaled point processes converge weakly to the determinantal point process associated with the so-called sine kernel as the size of matrices tends to \(\infty\). We extend this result to the case of high-dimensional spheres and show that the scaling limit processes are determinantal point processes associated with the kernels expressed by the Bessel functions of the first kind.Tree representations of \(\alpha\)-determinantal point processes.https://zbmath.org/1460.600412021-06-15T18:09:00+00:00"Osada, Shota"https://zbmath.org/authors/?q=ai:osada.shotaSummary: We introduce tree representations for \(\alpha\)-determinantal point processes. The \(\alpha\)-determinantal point processes is introduced in \textit{T. Shirai} and \textit{Y. Takahashi} [J. Funct. Anal. 205, No. 2, 414--463 (2003; Zbl 1051.60052)] as a one parameter extension of the determinantal point process. In \textit{H. Osada} and \textit{S. Osada} [J. Stat. Phys. 170, No. 2, 421--435 (2018; Zbl 1390.60035)], the tree representation was introduced for determinantal point processes. In this paper, we prove that the tree representation can be applied to \(\alpha\)-determinantal point processes.Fluctuations of linear statistics for Gaussian perturbations of the lattice \(\mathbb{Z}^d\).https://zbmath.org/1460.600432021-06-15T18:09:00+00:00"Yakir, Oren"https://zbmath.org/authors/?q=ai:yakir.orenSummary: We study the point process \(W\) in \(\mathbb{R}^d\) obtained by adding an independent Gaussian vector to each point in \(\mathbb{Z}^d\). Our main concern is the asymptotic size of fluctuations of the linear statistics in the large volume limit, defined as
\[
N(h,R) = \sum_{w\in W} h\left( \frac{w}{R}\right),
\]
where \(h\in \left( L^1\cap L^2\right) (\mathbb{R}^d)\) is a test function and \(R\rightarrow \infty \). We will also consider the stationary counter-part of the process \(W\), obtained by adding to all perturbations a random vector which is uniformly distributed on \([0,1]^d\) and is independent of all the Gaussians. We focus on two main examples of interest, when the test function \(h\) is either smooth or is an indicator function of a convex set with a smooth boundary whose curvature does not vanish.A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality.https://zbmath.org/1460.600042021-06-15T18:09:00+00:00"Akemann, Gernot"https://zbmath.org/authors/?q=ai:akemann.gernot"Byun, Sung-Soo"https://zbmath.org/authors/?q=ai:byun.sung-soo"Kang, Nam-Gyu"https://zbmath.org/authors/?q=ai:kang.nam-gyuSummary: We consider the complex eigenvalues of a Wishart type random matrix model \(X = X_1 X_2^*\), where two rectangular complex Ginibre matrices \(X_{1,2}\) of size \(N \times (N + \nu)\) are correlated through a non-Hermiticity parameter \(\tau \in [0,1]\). For general \(\nu = O(N)\) and \(\tau \), we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation \(X_1 = X_2\) when \(\tau = 1\). The square root of the complex Wishart eigenvalues, corresponding to the nonzero complex eigenvalues of the Dirac matrix \(\mathcal{D} = \begin{pmatrix} 0 & X_1 \\ X_2^* & 0 \end{pmatrix},\) are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value \(\tau_c,\) where the interior of the spectrum splits into two connected components. At multi-criticality, we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostman's equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices.https://zbmath.org/1460.600052021-06-15T18:09:00+00:00"Cipolloni, Giorgio"https://zbmath.org/authors/?q=ai:cipolloni.giorgio"Erdős, László"https://zbmath.org/authors/?q=ai:erdos.laszloAuthors' abstract: ``We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix \(\widetilde{W}\) and its minor \(W\). We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of \(\widetilde{W}\) and \(W\). Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by \textit{I. Dumitriu} and \textit{E. Paquette} [Random Matrices Theory Appl. 7, No. 2, Article ID 1850003, 21 p. (2018; Zbl 1392.60010)]. Unlike in a similar result for Wigner matrices, for sample covariance matrices, the fluctuation may entirely vanish.''
The paper is structured in 5 chapters: 1. Introduction -- 2. Main results (Theorem 2.1) -- 3. Preliminaries (Lemma 3.1) -- 4. Mean and variance computation (4.1 Leading term: calculation of the mean, 4.2 Fluctuation term) -- 5. Computation of the higher order moments of \(F_N\) -- Appendix A (Proof of Lemma 3.1) -- Acknowledgements -- References (18 references).
Reviewer: Ludwig Paditz (Dresden)Asymptotic performance of port-based teleportation.https://zbmath.org/1460.810092021-06-15T18:09:00+00:00"Christandl, Matthias"https://zbmath.org/authors/?q=ai:christandl.matthias"Leditzky, Felix"https://zbmath.org/authors/?q=ai:leditzky.felix"Majenz, Christian"https://zbmath.org/authors/?q=ai:majenz.christian"Smith, Graeme"https://zbmath.org/authors/?q=ai:smith.graeme.1"Speelman, Florian"https://zbmath.org/authors/?q=ai:speelman.florian"Walter, Michael"https://zbmath.org/authors/?q=ai:walter.michaelQuantum teleportation is a primitive widely used in quantum information science for transmission of unknown quantum state between systems using shared entanglement, joint measurement, classical communication and correction operation. Port-based teleportation (PBT) is a specific variant of such protocol, when a correction operation of receiver is a choice of one among several subsystems (`ports'). However, PBT may not implement ideal state transfer if number of ports is finite. Fundamental limits of PBT fidelity for arbitrary finite input dimension and a large number of ports are estimated in this work. Authors use methods from representation theory of symmetric and unitary groups for analyse of probability distributions on a set of random matrices necessary to describe the quantum measurement in PBT.
Reviewer: Alexander Yurevich Vlasov (Sankt-Peterburg)Mathematical foundations of infinite-dimensional statistical models. Revised paperback edition.https://zbmath.org/1460.620072021-06-15T18:09:00+00:00"Giné, Evarist"https://zbmath.org/authors/?q=ai:gine.evarist"Nickl, Richard"https://zbmath.org/authors/?q=ai:nickl.richardSee the review of the hardback edition in [Zbl 1358.62014].Circular law for sparse random regular digraphs.https://zbmath.org/1460.051742021-06-15T18:09:00+00:00"Litvak, Alexander Evgen'evich"https://zbmath.org/authors/?q=ai:litvak.alexander-e"Lytova, Anna"https://zbmath.org/authors/?q=ai:lytova.anna"Tikhomirov, Konstantin"https://zbmath.org/authors/?q=ai:tikhomirov.konstantin-e"Tomczak-Jaegermann, Nicole"https://zbmath.org/authors/?q=ai:tomczak-jaegermann.nicole"Youssef, Pierre"https://zbmath.org/authors/?q=ai:youssef.pierreSummary: Fix a constant \(C \geq 1\) and let \(d = d(n)\) satisfy \(d \leq \ln^C n\) for every large integer \(n\). Denote by \(A_n\) the adjacency matrix of a uniform random directed \(d\)-regular graph on \(n\) vertices. We show that if \(d \to \infty\) as \(n \to \infty\), the empirical spectral distribution of the appropriately rescaled matrix \(A_n\) converges weakly in probability to the circular law. This result, together with an earlier work of \textit{N. Cook} [Ann. Inst. Henri Poincaré, Probab. Stat. 55, No. 4, 2111--2167 (2019; Zbl 1432.15035)], completely settles the problem of weak convergence of the empirical distribution in a directed \(d\)-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of \(A_n\) based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between matrix entries.Isometric study of Wasserstein spaces -- the real line.https://zbmath.org/1460.460062021-06-15T18:09:00+00:00"Gehér, György Pál"https://zbmath.org/authors/?q=ai:geher.gyorgy-pal"Titkos, Tamás"https://zbmath.org/authors/?q=ai:titkos.tamas"Virosztek, Dániel"https://zbmath.org/authors/?q=ai:virosztek.danielSummary: Recently \textit{B. Kloeckner} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 9, No. 2, 297--323 (2010; Zbl 1218.53079)] described the structure of the isometry group of the quadratic Wasserstein space \(\mathcal{W}_2(\mathbb{R}^n)\). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute \(\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))\), the isometry group of the Wasserstein space \(\mathcal{W}_p(\mathbb{R})\) for all \(p \in [1, \infty )\setminus \{2\} \). We show that \(\mathcal{W}_2(\mathbb{R})\) is also exceptional regarding the parameter \(p\): \(\mathcal{W}_p(\mathbb{R})\) is isometrically rigid if and only if \(p\neq 2\). Regarding the underlying space, we prove that the exceptionality of \(p=2\) disappears if we replace \(\mathbb{R}\) by the compact interval \([0,1]\). Surprisingly, in that case, \( \mathcal{W}_p([0,1])\) is isometrically rigid if and only if \(p\neq 1\). Moreover, \( \mathcal{W}_1([0,1])\) admits isometries that split mass, and \(\mathrm{Isom}(\mathcal{W}_1([0,1]))\) cannot be embedded into \(\mathrm{Isom}(\mathcal{W}_1(\mathbb{R}))\).Mean-field optimal control and optimality conditions in the space of probability measures.https://zbmath.org/1460.490192021-06-15T18:09:00+00:00"Burger, Martin"https://zbmath.org/authors/?q=ai:burger.martin"Pinnau, René"https://zbmath.org/authors/?q=ai:pinnau.rene"Totzeck, Claudia"https://zbmath.org/authors/?q=ai:totzeck.claudia"Tse, Oliver"https://zbmath.org/authors/?q=ai:tse.oliverRandom band matrices in the delocalized phase. III: Averaging fluctuations.https://zbmath.org/1460.600092021-06-15T18:09:00+00:00"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.1|yang.fan.4|yang.fan.2|yang.fan.6"Yin, Jun"https://zbmath.org/authors/?q=ai:yin.jun.1|yin.junSummary: We consider a general class of symmetric or Hermitian random band matrices \(H=(h_{xy})_{x,y \in\lbrack\lbrack 1,N\rbrack\rbrack^d}\) in any dimension \(d\ge 1\), where the entries are independent, centered random variables with variances \(s_{xy}=\mathbb{E}|h_{xy}|^2\). We assume that \(s_{xy}\) vanishes if \(|x-y|\) exceeds the band width \(W\), and we are interested in the mesoscopic scale with \(1\ll W\ll N\). Define the generalized resolvent of \(H\) as \(G(H,Z):=(H-Z)^{-1}\), where \(Z\) is a deterministic diagonal matrix with entries \(Z_{xx}\in\mathbb{C}_+\) for all \(x\). Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a self-contained proof for the delocalization of random band matrices in dimensions \(d\ge 2\). More precisely, for any fixed \(d\ge 2\), we prove that the bulk eigenvectors of \(H\) are delocalized in certain averaged sense if \(N\le W^{1+\frac{d}{2}}\). This improves the corresponding results in [\textit{Y. He} and \textit{M. Marcozzi}, J. Stat. Phys. 177, No. 4, 666--716 (2019; Zbl 1448.60023)] that imposed the assumption \(N\ll W^{1+\frac{d}{d+1}}\), and the results in [\textit{L. Erdős} and \textit{A. Knowles}, Ann. Henri Poincaré 12, No. 7, 1227--1319 (2011; Zbl 1247.15033); Commun. Math. Phys. 303, No. 2, 509--554 (2011; Zbl 1226.15024)] that imposed the assumption \(N\ll W^{1+\frac{d}{6}}\). For 1D random band matrices, our fluctuation averaging result was used in Part II and Part I [\textit{P. Bourgade} et al., J. Stat. Phys. 174, No. 6, 1189--1221 (2019; Zbl 1447.60018); Commun. Pure Appl. Math. 73, No. 7, 1526--1596 (2020; Zbl 1446.60005)] to prove the delocalization conjecture and bulk universality for random band matrices with \(N\ll W^{4/3}\).On the strong law of large numbers for linear combinations of concomitants.https://zbmath.org/1460.620622021-06-15T18:09:00+00:00"Dudkina, O. I."https://zbmath.org/authors/?q=ai:dudkina.o-i"Gribkova, N. V."https://zbmath.org/authors/?q=ai:gribkova.nadezhda-vSummary: A theorem on the strong law of large numbers for linear functions of concomitants (induced order statistics) for sequences of independent identically distributed two-dimensional random vectors is proved in this paper. The result complements previous work by \textit{S.-S. Yang} [Ann. Inst. Stat. Math. 33, 463--470 (1981; Zbl 0478.62036)] and the second author with \textit{R. Zitikis} [Math. Methods Stat. 26, No. 4, 267--281 (2017; Zbl 06845133); Ann. Inst. Stat. Math. 71, No. 4, 811--835 (2019; Zbl 1433.62295)]. The proof is based on the conditional independence property of the concomitants established by \textit{P. K. Bhattacharya} [Ann. Stat. 2, 1034--1039 (1974; Zbl 0307.62036)]; the \textit{W. R. van Zwet} strong law of large numbers for linear functions of order statistics [Ann. Probab. 8, 986--990 (1980; Zbl 0448.60025)] is used and classical inequalities apply, including the \textit{H. P. Rosenthal} inequality [Isr. J. Math. 8, 273--303 (1970; Zbl 0213.19303)].