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For a semisimple Lie group $G$ satisfying the equal rank condition, the most basic family of unitary irreducible representations is the Discrete Series found by Harish-Chandra. In this paper, we continue our study of the branching laws for Discrete Series when restricted to a subgroup $H$ of the same type by use of integral and differential operators in combination with our previous duality principle. Many results are presented in generality, others are shown in detail for Holomorphic Discrete Series., Comment: Comments are welcome

The first paper in this series introduced a new approach to strong convergence of random matrices that is based primarily on soft arguments. This method was applied to achieve a refined qualitative and quantitative understanding of strong convergence of random permutation matrices and of more general representations of the symmetric group. In this paper, we introduce new ideas that make it possible to achieve stronger quantitative results and that facilitate the application of the method to new models. When applied to the Gaussian GUE/GOE/GSE ensembles of dimension $N$, these methods achieve strong convergence for noncommutative polynomials with matrix coefficients of dimension $\exp(o(N))$. This provides a sharp form of a result of Pisier on strong convergence with coefficients in a subexponential operator space. Analogous results up to logarithmic factors are obtained for Haar-distributed random matrices in $\mathrm{U}(N)/\mathrm{O}(N)/\mathrm{Sp}(N)$. We further illustrate the methods of this paper in the following applications. 1. We obtain improved rates for strong convergence of random permutations. 2. We obtain a quantitative form of strong convergence of the model introduced by Hayes for the solution of the Peterson-Thom conjecture. 3. We prove strong convergence of tensor GUE models of $\Gamma$-independence. 4. We prove strong convergence of all nontrivial representations of $\mathrm{SU}(N)$ of dimension up to $\exp(N^{1/3-\delta})$, improving a result of Magee and de la Salle., Comment: 52 pages; added a new application, and minor revisions

A family of random matrices $\boldsymbol{X}^N=(X_1^N,\ldots,X_d^N)$ is said to converge strongly to a family of bounded operators $\boldsymbol{x}=(x_1,\ldots,x_d)$ when $\|P(\boldsymbol{X}^N,\boldsymbol{X}^{N*})\|\to\|P(\boldsymbol{x}, \boldsymbol{x}^*)\|$ for every noncommutative polynomial $P$. This phenomenon plays a key role in several recent breakthroughs on random graphs, geometry, and operator algebras. However, proofs of strong convergence are notoriously delicate and have relied largely on problem-specific methods. In this paper, we develop a new approach to strong convergence that uses only soft arguments. Our method exploits the fact that for many natural models, the expected trace of $P(\boldsymbol{X}^N,\boldsymbol{X}^{N*})$ is a rational function of $\frac{1}{N}$ whose lowest order asymptotics are easily understood. We develop a general technique to deduce strong convergence directly from these inputs using the inequality of A. and V. Markov for univariate polynomials and elementary Fourier analysis. To illustrate the method, we develop the following applications. 1. We give a short proof of the result of Friedman that random regular graphs have a near-optimal spectral gap, and obtain a sharp understanding of the large deviations probabilities of the second eigenvalue. 2. We prove a strong quantitative form of the strong convergence property of random permutation matrices due to Bordenave and Collins. 3. We extend the above to any stable representation of the symmetric group, providing many new examples of the strong convergence phenomenon., Comment: 37 pages, 1 figure; improved exposition and minor corrections

We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it. This allows new, streamlined proofs of the gap labeling and Aomoto index theorems. We prove a version of this new formula for the Anderson model on trees.

For a semisimple Lie group $G$, we study Discrete Series representations with admissible branching to a symmetric subgroup $H$. This is done using a canonical associated symmetric subgroup $H_0$, forming a pseudo-dual pair with $H$, and a corresponding branching law for this group with respect to its maximal compact subgroup. This is in analogy with either Blattner's or Kostant-Heckmann multiplicity formulas, and has some resemblance to Frobenius reciprocity. We give several explicit examples and links to Kobayashi-Pevzner theory of symmetry breaking and holographic operators. Our method is well adapted to computer algorithms, such as for example the Atlas program., Comment: To appear in "Symmetry in Geometry and Analysis -- Festschrift for Toshiyuki Kobayashi''. M. Pevzner und H. Sekiguchi eds., Progress in Mathematics, Springer 2024

We develop a framework for proving rapid convergence of shifted QR algorithms which use Ritz values as shifts, in finite arithmetic. Our key contribution is a dichotomy result which addresses the known forward-instability issues surrounding the shifted QR iteration [Parlett and Le 1993]: we give a procedure which provably either computes a set of approximate Ritz values of a Hessenberg matrix with good forward stability properties, or leads to early decoupling of the matrix via a small number of QR steps. Using this framework, we show that the shifting strategy introduced in Part I of this series [Banks, Garza-Vargas, and Srivastava 2021] converges rapidly in finite arithmetic with a polylogarithmic bound on the number of bits of precision required, when invoked on matrices of controlled eigenvector condition number and minimum eigenvalue gap.

We give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $\delta\|M\|$ in $O(n^4+n^3\log^2(n/\delta)+\log(n/\delta)^2\log\log(n/\delta))$ floating point operations using $O(\log^2(n/\delta))$ bits of precision. While the $O(n^4)$ complexity is prohibitive for large matrices, the algorithm is simple and may be useful for provably computing the eigenvalues of small matrices using controlled precision, in particular for computing Ritz values in shifted QR algorithms as in (Banks, Garza-Vargas, Srivastava, 2022)., Comment: Comments welcome

Rapid convergence of the shifted QR algorithm on symmetric matrices was shown more than fifty years ago. Since then, despite significant interest and its practical relevance, an understanding of the dynamics and convergence properties of the shifted QR algorithm on nonsymmetric matrices has remained elusive. We introduce a new family of shifting strategies for the Hessenberg shifted QR algorithm. We prove that when the input is a diagonalizable Hessenberg matrix $H$ of bounded eigenvector condition number $\kappa_V(H)$ -- defined as the minimum condition number of $V$ over all diagonalizations $VDV^{-1}$ of $H$ -- then the shifted QR algorithm with a certain strategy from our family is guaranteed to converge rapidly to a Hessenberg matrix with a zero subdiagonal entry, in exact arithmetic. Our convergence result is nonasymptotic, showing that the geometric mean of certain subdiagonal entries of $H$ decays by a fixed constant in every $QR$ iteration. The arithmetic cost of implementing each iteration of our strategy scales roughly logarithmically in the eigenvector condition number $\kappa_V(H)$, which is a measure of the nonnormality of $H$. The key ideas in the design and analysis of our strategy are: (1) We are able to precisely characterize when a certain shifting strategy based on Ritz values stagnates. We use this information to design certain ``exceptional shifts'' which are guaranteed to escape stagnation whenever it occurs. (2) We use higher degree shifts (of degree roughly $\log \kappa_V(H)$) to dampen transient effects due to nonnormality, allowing us to treat nonnormal matrices in a manner similar to normal matrices., Comment: 31pp. Comments welcome. v3: slight changes in notation, exposition. v4: to motivate the proof of the general result, a section discussing in detail the case of normal inputs has been added. A discussion on the scope of this research has been added to the first section. Many remarks and footnotes with clarifications have been added throughout the paper. The title has been modified

Let $\mathbf{T}_{d, N}: \Omega \to \mathbb{R}^{N^d}$ be a random real symmetric Wigner-type tensor. For unit vectors $(u_N^{(i, j)})_{i \in I, j \in [d-2]} \subset \mathbb{S}^{N-1}$, we study the contracted tensor ensemble \[ \left(\frac{1}{\sqrt{N}}\mathbf{T}_{d, N}\left[u_N^{(i, 1)} \otimes \cdots \otimes u_N^{(i, d-2)}\right]\right)_{i \in I}. \] For large $N$, we show that the joint spectral distribution of this ensemble is well-approximated by a semicircular family $(s_i)_{i \in I}$ whose covariance $(\mathbf{K}_{i, i'}^{(N)})_{i, i'\in I}$ is given by the rescaled overlaps of the corresponding symmetrized contractions \[ \mathbf{K}_{i, i'}^{(N)} = \frac{1}{d(d-1)}\langle u_N^{(i, 1)} \odot \cdots \odot u_N^{(i, d-2)}, u_N^{(i', 1)} \odot \cdots \odot u_N^{(i', d-2)} \rangle, \] which is the true covariance of the ensemble up to a $O_d(N^{-1})$ correction. We further characterize the extreme cases of the variance $\mathbf{K}_{i, i}^{(N)} \in [\frac{1}{d!}, \frac{1}{d(d-1)}]$. Our analysis relies on a tensorial extension of the usual graphical calculus for moment method calculations in random matrix theory, allowing us to access the independence in our random tensor ensemble., Comment: v3: updated to incorporate feedback from referees, including a shorter proof of Proposition 4.2; 31 pages, 7 figures

Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that $\boxtimes_d$ converges to $\boxtimes$ as $d$ goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building off our results we give a new short and conceptual proof of a recent result [Steinerberger (2020), Hoskins and Kabluchko (2020)] that connects root distributions of polynomial derivatives with free fractional convolution powers., Comment: 42 pages, 4 figures, 3 apendices

Background: Limited research exists on the association between changes in physical activity levels and injury in children. Objective: To assess how well different variations of the acute:chronic workload ratio (ACWR), a measure of change in activity, predict injury in children. Methods: We conducted a prospective cohort study using data from 1670 Danish schoolchildren measured over 5.5 years (2008 to 2014). Coupled 4-week, uncoupled 4-week, and uncoupled 5-week ACWRs were calculated using activity frequency in the past week as the acute load (numerator), and average weekly activity frequency in the past 4 or 5 weeks as the chronic load (denominator). We modelled the relationship between different ACWR variations and injury using generalized linear and generalized additive models, with and without accounting for repeated measures. Results: The prognostic relationship between the ACWR and injury risk was best represented using a generalized additive mixed model for the uncoupled 5-week ACWR. It predicted an injury risk of ~3% for ACWRs between 0.8 (activity level decreased by 20%) and 1.5 (activity level increased by 50%). When activity decreased by more than 20% (ACWR< 0.8), injury risk was lower (minimum of 1.5% at ACWR=0). When activity increased by more than 50% (ACWR > 1.5), injury risk was higher (maximum of 6% at ACWR = 5). Girls were at significantly higher risk of injury than boys. Conclusion: Increases in physical activity in children are associated with much lower injury risks compared to previous results in adults.

For any multi-graph $G$ with edge weights and vertex potential, and its universal covering tree $\mathcal{T}$, we completely characterize the point spectrum of operators $A_{\mathcal{T}}$ on $\mathcal{T}$ arising as pull-backs of local, self-adjoint operators $A_{G}$ on $G$. This builds on work of Aomoto, and includes an alternative proof of the necessary condition for point spectrum he derived in (Aomoto, 1991). Our result gives a finite time algorithm to compute the point spectrum of $A_{\mathcal{T}}$ from the graph $G$, and additionally allows us to show that this point spectrum is contained in the spectrum of $A_{G}$. Finally, we prove that typical pull-back operators have a spectral delocalization property: the set of edge weight and vertex potential parameters of $A_{G}$ giving rise to $A_{\mathcal{T}}$ with purely absolutely continuous spectrum is open and its complement has large codimension., Comment: 22 pages, 4 figures, comments welcome; v2: Theorem 3.4 has been strengthened, a more detailed discussion with new examples has been added to Section 3

Let $G_n$ be an $n \times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n \times n$ matrix with $\Vert A \Vert \le 1$, and let $\gamma \in (0,1)$. We show that with probability $0.99$, $A + \gamma G_n$ has all of its eigenvalue condition numbers bounded by $O\left(n^{5/2}/\gamma^{3/2}\right)$ and eigenvector condition number bounded by $O\left(n^3 /\gamma^{3/2}\right)$. Furthermore, we show that for any $s > 0$, the probability that $A + \gamma G_n$ has two eigenvalues within distance at most $s$ of each other is $O\left(n^4 s^{1/3}/\gamma^{5/2}\right).$ In fact, we show the above statements hold in the more general setting of non-Gaussian perturbations with real, independent, absolutely continuous entries with a finite moment assumption and appropriate normalization. This extends the previous work [Banks et al. 2019] which proved an eigenvector condition number bound of $O\left(n^{3/2} / \gamma\right)$ for the simpler case of {\em complex} i.i.d. Gaussian matrix perturbations. The case of real perturbations introduces several challenges stemming from the weaker anticoncentration properties of real vs. complex random variables. A key ingredient in our proof is new lower tail bounds on the small singular values of the complex shifts $z-(A+\gamma G_n)$ which recover the tail behavior of the complex Ginibre ensemble when $\Im z\neq 0$. This yields sharp control on the area of the pseudospectrum $\Lambda_\epsilon(A+\gamma G_n)$ in terms of the pseudospectral parameter $\epsilon>0$, which is sufficient to bound the overlaps and eigenvector condition number via a limiting argument., Comment: Concurrent independent work by Jain et al. is mentioned in Remark 1.1

Accurate description of the excess charge in water cluster anions is challenging for standard semi-local and (global) hybrid density functional approximations (DFAs). Using the recent unitary invariant implementation of the Perdew-Zunger self-interaction correction (SIC) method using Fermi-L\"owdin orbitals, we assess the effect of self-interaction error on the vertical detachment energies of water clusters anions with the local spin density approximation (LSDA), Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation, and the strongly constrained and appropriately normed (SCAN) meta-GGA functionals. Our results show that for the relative energies of isomers with respect to reference CCSD(T) values, the uncorrected SCAN functional has the smallest deviation of 21 meV, better than that for the MP2 method. The performance of SIC-SCAN is comparable to that of MP2 and is better than SIC-LSDA and SIC-PBE, but it reverses the ordering of the two lowest isomers for water hexamer anions. Removing self interaction error (SIE) corrects the tendency of LSDA, PBE, and SCAN to over-bind the extra electron. The vertical detachment energies (VDEs) of water cluster anions, obtained from the total energy differences of corresponding anion and neutral clusters, are significantly improved by removing self-interaction and are better than the hybrid B3LYP functional, but fall short of MP2 accuracy. Removing SIE results in substantial improvement in the position of the eigenvalue of the extra electron. The negative of the highest occupied eigenvalue after SIC provides an excellent approximation to the VDE, especially for SIC-PBE where the mean absolute error with respect to CCSD(T) is only 17 meV, the best among all approximations compared in this work., Comment: 32 pages, 5 figures, Accepted to Physical Chemistry Chemical Physics

We use tools from free probability to study the spectra of Hermitian operators on infinite graphs. Special attention is devoted to universal covering trees of finite graphs. For operators on these graphs we derive a new variational formula for the spectral radius and provide new proofs of results due to Sunada and Aomoto using free probability. With the goal of extending the applicability of free probability techniques beyond universal covering trees, we introduce a new combinatorial product operation on graphs and show that, in the non-commutative probability context, it corresponds to the notion of freeness with amalgamation. We show that Cayley graphs of amalgamated free products of groups, as well as universal covering trees, can be constructed using our graph product., Comment: v2: Key references and new results added. Important bibliographic note (before Sec 1.1) has been added; v3: added discussions, generalized Thm 1.6 from adjacency matrices to graph Jacobi matrices; v5: small update on the definition of the graph product, strengthening of main theorem of Section 5, clarifications added, explanations expanded, figures updated, minor errors fixed

We exhibit a randomized algorithm which given a matrix $A\in \mathbb{C}^{n\times n}$ with $\|A\|\le 1$ and $\delta>0$, computes with high probability an invertible $V$ and diagonal $D$ such that $\|A-VDV^{-1}\|\le \delta$ using $O(T_{MM}(n)\log^2(n/\delta))$ arithmetic operations, in finite arithmetic with $O(\log^4(n/\delta)\log n)$ bits of precision. Here $T_{MM}(n)$ is the number of arithmetic operations required to multiply two $n\times n$ complex matrices numerically stably, known to satisfy $T_{MM}(n)=O(n^{\omega+\eta})$ for every $\eta>0$ where $\omega$ is the exponent of matrix multiplication (Demmel et al., Numer. Math., 2007). Our result significantly improves the previously best known provable running times of $O(n^{10}/\delta^2)$ arithmetic operations for diagonalization of general matrices (Armentano et al., J. Eur. Math. Soc., 2018), and (with regards to the dependence on $n$) $O(n^3)$ arithmetic operations for Hermitian matrices (Dekker and Traub, Lin. Alg. Appl., 1971). It is the first algorithm to achieve nearly matrix multiplication time for diagonalization in any model of computation (real arithmetic, rational arithmetic, or finite arithmetic), thereby matching the complexity of other dense linear algebra operations such as inversion and $QR$ factorization up to polylogarithmic factors. The proof rests on two new ingredients. (1) We show that adding a small complex Gaussian perturbation to any matrix splits its pseudospectrum into $n$ small well-separated components. In particular, this implies that the eigenvalues of the perturbed matrix have a large minimum gap, a property of independent interest in random matrix theory. (2) We give a rigorous analysis of Roberts' Newton iteration method (Roberts, Int. J. Control, 1980) for computing the sign function of a matrix in finite arithmetic, itself an open problem in numerical analysis since at least 1986., Comment: 84 pages, 3 figures, comments welcome. Slightly edited intro from first version + explicit statement of forward error Theorem (Corolary 1.7). Minor corrections, new references and clarifications. Appendix with some new proofs added

Injuries occur when an athlete performs a greater amount of activity (workload) than what their body can absorb. To maximize the positive effects of training while avoiding injuries, athletes and coaches need to determine safe workload levels. The International Olympic Committee has recommended using the acute:chronic workload ratio (ACRatio) to monitor injury risk, and has provided thresholds to minimize risk. However, there are several limitations to the ACRatio which may impact the validity of current recommendations. In this review, we discuss previously published and novel challenges with the ACRatio, and possible strategies to address them. These challenges include 1) formulating the ACRatio as a proportion rather than a measure of change, 2) its use of unweighted averages to measure activity loads, 3) inapplicability of the ACRatio to sports where athletes taper their activity, 4) discretization of the ACRatio prior to model selection, 5) the establishment of the model using sparse data, 6) potential bias in the ACRatio of injured athletes, 7) unmeasured confounding, and 8) application of the ACRatio to subsequent injuries.

For a semisimple Lie group $G$ satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for discrete series when restricted to a subgroup $H$ of the same type by combining classical results with recent work of T. Kobayashi; in particular, we prove discrete decomposability under Harish-Chandra's condition of cusp form on the reproducing kernel. We show a relation between discrete decomposability and representing certain intertwining operators in terms of differential operators.

We study the Lanczos algorithm where the initial vector is sampled uniformly from $\mathbb{S}^{n-1}$. Let $A$ be an $n \times n$ Hermitian matrix. We show that when run for few iterations, the output of Lanczos on $A$ is almost deterministic. More precisely, we show that for any $ \varepsilon \in (0, 1)$ there exists $c >0$ depending only on $\varepsilon$ and a certain global property of the spectrum of $A$ (in particular, not depending on $n$) such that when Lanczos is run for at most $c \log n$ iterations, the output Jacobi coefficients deviate from their medians by $t$ with probability at most $\exp(-n^\varepsilon t^2)$ for $t<\Vert A \Vert$. We directly obtain a similar result for the Ritz values and vectors. Our techniques also yield asymptotic results: Suppose one runs Lanczos on a sequence of Hermitian matrices $A_n \in M_n(\mathbb{C})$ whose spectral distributions converge in Kolmogorov distance with rate $O(n^{-\varepsilon})$ to a density $\mu$ for some $\varepsilon > 0$. Then we show that for large enough $n$, and for $k=O(\sqrt{\log n})$, the Jacobi coefficients output after $k$ iterations concentrate around those for $\mu$. The asymptotic setting is relevant since Lanczos is often used to approximate the spectral density of an infinite-dimensional operator by way of the Jacobi coefficients; our result provides some theoretical justification for this approach. In a different direction, we show that Lanczos fails with high probability to identify outliers of the spectrum when run for at most $c' \log n$ iterations, where again $c'$ depends only on the same global property of the spectrum of $A$. Classical results imply that the bound $c' \log n$ is tight up to a constant factor., Comment: v2: A detailed discussion of the motivation and relevance of the main results and definitions has been added. Minor corrections have been made. 38 pages, 3 figures