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We study the infimum of the spectrum, or ground state energy (g.s.e.), of a discrete Schr\"odinger operator on $\theta\mathbb{Z}^d$ parameterized by a potential $V:\mathbb{R}^d\rightarrow\mathbb{R}_{\ge 0}$ and a frequency parameter $\theta\in (0,1)$. We relate this g.s.e. to that of a corresponding continuous semiclassical Schr\"odinger operator on $\mathbb{R}^d$ with parameter $\theta$, arising from the same choice of potential. We show that: the discrete g.s.e. is at most the continuous one for continuous periodic $V$ and irrational $\theta$; the opposite inequality holds up to a factor of $1-o(1)$ as $\theta\rightarrow 0$ for sufficiently regular smooth periodic $V$; and the opposite inequality holds up to a constant factor for every bounded $V$ and $\theta$ with the property that discrete and continuous averages of $V$ on fundamental domains of $\theta \mathbb{Z}^d$ are comparable. Our proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on $\theta \mathbb{Z}^d$ to low-energy functions for the continuous operator on $\mathbb{R}^d$, and vice versa.

We survey recent progress on efficient algorithms for approximately diagonalizing a square complex matrix in the models of rational (variable precision) and finite (floating point) arithmetic. This question has been studied across several research communities for decades, but many mysteries remain. We present several open problems which we hope will be of broad interest., Comment: 11pp. Invited survey for ISSAC 2023. Comments welcome

Given an $n\times n$ matrix with integer entries in the range $[-h,h]$, how close can two of its distinct eigenvalues be? The best previously known examples have a minimum gap of $h^{-O(n)}$. Here we give an explicit construction of matrices with entries in $[0,h]$ with two eigenvalues separated by at most $h^{-n^2/16+o(n^2)}$. Up to a constant in the exponent, this agrees with the known lower bound of $\Omega((2\sqrt{n})^{-n^2}h^{-n^2})$ \cite{mahler1964inequality}. Bounds on the minimum gap are relevant to the worst case analysis of algorithms for diagonalization and computing canonical forms of integer matrices. In addition to our explicit construction, we show there are many matrices with a slightly larger gap of roughly $h^{-n^2/32}$. We also construct 0-1 matrices which have two eigenvalues separated by at most $2^{-n^2/64+o(n^2)}$., Comment: 9pp

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

We study the bit complexity of two related fundamental computational problems in linear algebra and control theory. Our results are: (1) An $\tilde{O}(n^{\omega+3}a+n^4a^2+n^\omega\log(1/\epsilon))$ time algorithm for finding an $\epsilon-$approximation to the Jordan Normal form of an integer matrix with $a-$bit entries, where $\omega$ is the exponent of matrix multiplication. (2) An $\tilde{O}(n^6d^6a+n^4d^4a^2+n^3d^3\log(1/\epsilon))$ time algorithm for $\epsilon$-approximately computing the spectral factorization $P(x)=Q^*(x)Q(x)$ of a given monic $n\times n$ rational matrix polynomial of degree $2d$ with rational $a-$bit coefficients having $a-$bit common denominators, which satisfies $P(x)\succeq 0$ for all real $x$. The first algorithm is used as a subroutine in the second one. Despite its being of central importance, polynomial complexity bounds were not previously known for spectral factorization, and for Jordan form the best previous best running time was an unspecified polynomial in $n$ of degree at least twelve \cite{cai1994computing}. Our algorithms are simple and judiciously combine techniques from numerical and symbolic computation, yielding significant advantages over either approach by itself., Comment: 19pp

Let $G$ be a random $d$-regular graph. We prove that for every constant $\alpha > 0$, with high probability every eigenvector of the adjacency matrix of $G$ with eigenvalue less than $-2\sqrt{d-2}-\alpha$ has $\Omega(n/$polylog$(n))$ nodal domains., Comment: 18 pages. Minor changes to the introduction

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint matrix, which in some cases improves previously known results. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure $1-o(1)$ and polynomial diameter. Both bounds rely on spectral gaps -- of a certain Schr\"odinger operator in the first case, and a certain continuous time Markov chain in the second -- which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables., Comment: Refined the statement + proof of Theorem 1.1, comparison with related work. Fixed some minor mistakes, added references

We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree $\Delta$ is bounded by $O(n \Delta^{7/5}/\log^{1/5-o(1)}n)$ for any $\Delta$, and by $O(n\log^{1/2}d/\log^{1/4-o(1)}n)$ for simple $d$-regular graphs when $d\ge \log^{1/4}n$. In fact, the same bounds hold for the number of eigenvalues in any interval of width $\lambda_2/\log_\Delta^{1-o(1)}n$ containing the second eigenvalue $\lambda_2$. The main ingredient in the proof is a polynomial (in $k$) lower bound on the typical support of a closed random walk of length $2k$ in any connected graph, which in turn relies on new lower bounds for the entries of the Perron eigenvector of submatrices of the normalized adjacency matrix., Comment: A previous version of this paper proved the main result for d-regular graphs. The current version proves a more general result for the normalized adjacency matrix of bounded degree graphs. New version fixes an incorrect citation in the proof of Proposition 5.2. 24pp, 3 figures

We prove that every reversible Markov semigroup which satisfies a Poincar\'e inequality satisfies a matrix-valued Poincar\'e inequality for Hermitian $d\times d$ matrix valued functions, with the same Poincar\'e constant. This generalizes recent results [Aoun et al. 2019, Kathuria 2019] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators., Comment: fixed a reference

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

Consider $n$ complex random matrices $X_1,\ldots,X_n$ of size $d\times d$ sampled i.i.d. from a distribution with mean $E[X]=\mu$. While the concentration of averages of these matrices is well-studied, the concentration of other functions of such matrices is less clear. One function which arises in the context of stochastic iterative algorithms, like Oja's algorithm for Principal Component Analysis, is the normalized matrix product defined as $\prod\limits_{i=1}^{n}\left(I + \frac{X_i}{n}\right).$ Concentration properties of this normalized matrix product were recently studied by \cite{HW19}. However, their result is suboptimal in terms of the dependence on the dimension of the matrices as well as the number of samples. In this paper, we present a stronger concentration result for such matrix products which is optimal in $n$ and $d$ up to constant factors. Our proof is based on considering a matrix Doob martingale, controlling the quadratic variation of that martingale, and applying the Matrix Freedman inequality of Tropp \cite{TroppIntro15}.

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

We show that for every prime $d$ and $\alpha\in (0,1/6)$, there is an infinite sequence of $(d+1)$-regular graphs $G=(V,E)$ with girth at least $2\alpha \log_{d}(|V|)(1-o_d(1))$, second adjacency matrix eigenvalue bounded by $(3/\sqrt{2})\sqrt{d}$, and many eigenvectors fully localized on small sets of size $O(|V|^\alpha)$. This strengthens the results of Ganguly-Srivastava, who constructed high girth (but not expanding) graphs with similar properties, and may be viewed as a discrete analogue of the "scarring" phenomenon observed in the study of quantum ergodicity on manifolds. Key ingredients in the proof are a technique of Kahale for bounding the growth rate of eigenfunctions of graphs, discovered in the context of vertex expansion and a method of Erd\H{o}s and Sachs for constructing high girth regular graphs.

A matrix $A\in\mathbb{C}^{n\times n}$ is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every $A\in \mathbb{C}^{n\times n}$ is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E.B. Davies: for each $\delta\in (0,1)$, every matrix $A\in \mathbb{C}^{n\times n}$ is at least $\delta\|A\|$-close to one whose eigenvectors have condition number at worst $c_n/\delta$, for some constants $c_n$ dependent only on $n$. Our proof uses tools from random matrix theory to show that the pseudospectrum of $A$ can be regularized with the addition of a complex Gaussian perturbation. Along the way, we explain how a variant of a theorem of \'Sniady implies a conjecture of Sankar, Spielman and Teng on the optimal constant for smoothed analysis of condition numbers., Comment: 17pp. Fixed a mistake in the appendix, all results are unchanged. Accepted version, to appear in CPAM

We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp. This study was initiated by Brooks and Lindenstrauss (2009) who relied on the observation that certain suitably normalized averaging operators on high girth graphs are hyper-contractive and can be used to approximate projectors onto the eigenspaces of such graphs. Informally, their delocalization result in the contrapositive states that for any $\varepsilon \in (0,1)$ and positive integer $k,$ if a $(d+1)-$regular graph has an eigenvector which supports $\varepsilon$ fraction of the $\ell_2^2$ mass on a subset of $k$ vertices, then the graph must have a cycle of size $\tilde{O}(\log_{d}(k)/\varepsilon^2)$, suppressing logarithmic terms in $1/\varepsilon$. In this paper, we improve the upper bound to $\tilde{O}(\log_{d}(k)/\varepsilon)$ and present a construction showing a lower bound of $\Omega(\log_d(k)/\varepsilon)$. Our construction is probabilistic and involves gluing together a pair of trees while maintaining high girth as well as control on the eigenvectors and could be of independent interest., Comment: 20 pages, 4 figures

We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+\epsilon$ error must use $\Omega(n\log n/\epsilon^2)$ bits of space in the worst case, improving the $\Omega(n/\epsilon^2)$ bound of Andoni et al. and matching the best known upper bound achieved by spectral sparsifiers. Our proof is based on a rigidity phenomenon for cut (and spectral) approximation which may be of independent interest: any two $d-$regular graphs which approximate each other's cuts significantly better than a random graph approximates the complete graph must overlap in a constant fraction of their edges.

These lecture notes are meant to accompany two lectures given at the CDM 2016 conference, about the Kadison-Singer Problem. They are meant to complement the survey by the same authors (along with Spielman) which appeared at the 2014 ICM. In the first part of this survey we will introduce the Kadison-Singer problem from two perspectives ($C^*$ algebras and spectral graph theory) and present some examples showing where the difficulties in solving it lie. In the second part we will develop the framework of interlacing families of polynomials, and show how it is used to solve the problem. None of the results are new, but we have added annotations and examples which we hope are of pedagogical value.

We use the method of interlacing families of polynomials to derive a simple proof of Bourgain and Tzafriri's Restricted Invertibility Principle, and then to sharpen the result in two ways. We show that the stable rank can be replaced by the Schatten 4-norm stable rank and that tighter bounds hold when the number of columns in the matrix under consideration does not greatly exceed its number of rows. Our bounds are derived from an analysis of smallest zeros of Jacobi and associated Laguerre polynomials.

The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree $d$ in $n$ variables contains $(n/d)^{\Omega(d)}$ pairwise distant cones in a certain metric, and therefore that any semidefinite representation of such cones must have dimension at least $(n/d)^{\Omega(d)}$ (even if a small approximation is allowed). The proof contains several ingredients of independent interest, including the identification of a large subspace in which the elementary symmetric polynomials lie in the relative interior of the set of hyperbolic polynomials, and quantitative versions of several basic facts about real rooted polynomials., Comment: Fixed a mistake in the proof of Lemma 6. The statement is unchanged except for constant factors, and the main theorem is unaffected. Wrote a slightly stronger statement for the main theorem, emphasizing approximate representations (the proof is the same). Added one figure

We show that in any graph, the average length of a flow path in an electrical flow between the endpoints of a random edge is $O(\log^2 n)$. This is a consequence of a more general result which shows that the spectral norm of the entrywise absolute value of the transfer impedance matrix of a graph is $O(\log^2 n)$. This result implies a simple oblivious routing scheme based on electrical flows in the case of transitive graphs.

We prove the following Alon-Boppana type theorem for general (not necessarily regular) weighted graphs: if $G$ is an $n$-node weighted undirected graph of average combinatorial degree $d$ (that is, $G$ has $dn/2$ edges) and girth $g> 2d^{1/8}+1$, and if $\lambda_1 \leq \lambda_2 \leq \cdots \lambda_n$ are the eigenvalues of the (non-normalized) Laplacian of $G$, then \[ \frac {\lambda_n}{\lambda_2} \geq 1 + \frac 4{\sqrt d} - O \left( \frac 1{d^{\frac 58} }\right) \] (The Alon-Boppana theorem implies that if $G$ is unweighted and $d$-regular, then $\frac {\lambda_n}{\lambda_2} \geq 1 + \frac 4{\sqrt d} - O\left( \frac 1 d \right)$ if the diameter is at least $d^{1.5}$.) Our result implies a lower bound for spectral sparsifiers. A graph $H$ is a spectral $\epsilon$-sparsifier of a graph $G$ if \[ L(G) \preceq L(H) \preceq (1+\epsilon) L(G) \] where $L(G)$ is the Laplacian matrix of $G$ and $L(H)$ is the Laplacian matrix of $H$. Batson, Spielman and Srivastava proved that for every $G$ there is an $\epsilon$-sparsifier $H$ of average degree $d$ where $\epsilon \approx \frac {4\sqrt 2}{\sqrt d}$ and the edges of $H$ are a (weighted) subset of the edges of $G$. Batson, Spielman and Srivastava also show that the bound on $\epsilon$ cannot be reduced below $\approx \frac 2{\sqrt d}$ when $G$ is a clique; our Alon-Boppana-type result implies that $\epsilon$ cannot be reduced below $\approx \frac 4{\sqrt d}$ when $G$ comes from a family of expanders of super-constant degree and super-constant girth. The method of Batson, Spielman and Srivastava proves a more general result, about sparsifying sums of rank-one matrices, and their method applies to an "online" setting. We show that for the online matrix setting the $4\sqrt 2 / \sqrt d$ bound is tight, up to lower order terms.

Group synchronization requires to estimate unknown elements $({\theta}_v)_{v\in V}$ of a compact group ${\mathfrak G}$ associated to the vertices of a graph $G=(V,E)$, using noisy observations of the group differences associated to the edges. This model is relevant to a variety of applications ranging from structure from motion in computer vision to graph localization and positioning, to certain families of community detection problems. We focus on the case in which the graph $G$ is the $d$-dimensional grid. Since the unknowns ${\boldsymbol \theta}_v$ are only determined up to a global action of the group, we consider the following weak recovery question. Can we determine the group difference ${\theta}_u^{-1}{\theta}_v$ between far apart vertices $u, v$ better than by random guessing? We prove that weak recovery is possible (provided the noise is small enough) for $d\ge 3$ and, for certain finite groups, for $d\ge 2$. Viceversa, for some continuous groups, we prove that weak recovery is impossible for $d=2$. Finally, for strong enough noise, weak recovery is always impossible., Comment: 21 pages

Anderson's paving conjecture, now known to hold due to the resolution of the Kadison-Singer problem asserts that every zero diagonal Hermitian matrix admits non-trivial pavings with dimension independent bounds. In this paper, we develop a technique extending the arguments of Marcus, Spielman and Srivastava in their solution of the Kadison-Singer problem to show the existence of non-trivial pavings for collections of matrices. We show that given zero diagonal Hermitian contractions $A^{(1)}, \cdots, A^{(k)} \in M_n(\mathbb{C})$ and $\epsilon > 0$, one may find a paving $X_1 \amalg \cdots \amalg X_r = [n]$ where $r \leq 18k\epsilon^{-2}$ such that, \[\lambda_{max} (P_{X_i} A^{(j)} P_{X_i}) < \epsilon, \quad i \in [r], \, j \in [k].\] As a consequence, we get the correct asymptotic estimates for paving general zero diagonal matrices; zero diagonal contractions can be $(O(\epsilon^{-2}),\epsilon)$ paved. As an application, we give a simplified proof wth slightly better estimates of a theorem of Johnson, Ozawa and Schechtman concerning commutator representations of zero trace matrices., Comment: 23 pages. In the previous version, we had erroneously claimed that the main theorem in this paper implies a polylogarithmic bound in the commutator theorem of Johnson, Ozawa and Schechtman. This has been corrected with a weaker bound. The main results in the paper are unchanged

We study the problem of approximating the largest root of a real-rooted polynomial of degree $n$ using its top $k$ coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in $k$ that use the top $k$ coefficients to approximate the maximum root within a factor of $n^{1/k}$ and $1+O(\tfrac{\log n}{k})^2$ when $k\leq \log n$ and $k>\log n$ respectively. We also prove corresponding information-theoretic lower bounds of $n^{\Omega(1/k)}$ and $1+\Omega\left(\frac{\log \frac{2n}{k}}{k}\right)^2$, and show strong lower bounds for noisy version of the problem in which one is given access to approximate coefficients. This problem has applications in the context of the method of interlacing families of polynomials, which was used for proving the existence of Ramanujan graphs of all degrees, the solution of the Kadison-Singer problem, and bounding the integrality gap of the asymmetric traveling salesman problem. All of these involve computing the maximum root of certain real-rooted polynomials for which the top few coefficients are accessible in subexponential time. Our results yield an algorithm with the running time of $2^{\tilde O(\sqrt[3]n)}$ for all of them.

We prove a Chernoff-type bound for sums of matrix-valued random variables sampled via a random walk on an expander, confirming a conjecture due to Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the Golden-Thompson inequality which improves in some ways the inequality of Sutter, Berta, and Tomamichel, and may be of independent interest, as well as an adaptation of an argument for the scalar case due to Healy. Secondarily, we also provide a generic reduction showing that any concentration inequality for vector-valued martingales implies a concentration inequality for the corresponding expander walk, with a weakening of parameters proportional to the squared mixing time., Comment: Fixed a minor bug in the proof of Theorem 3.4

We give a strongly polynomial time algorithm which determines whether or not a bivariate polynomial is real stable. As a corollary, this implies an algorithm for testing whether a given linear transformation on univariate polynomials preserves real-rootedness. The proof exploits properties of hyperbolic polynomials to reduce real stability testing to testing nonnegativity of a finite number of polynomials on an interval.

We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three ingredients: (1) a formula for the expected characteristic polynomial of the sum of a regular graph with a random permutation of another regular graph, (2) a proof that this expected polynomial is real rooted and that the family of polynomials considered in this sum is an interlacing family, and (3) strong bounds on the roots of the expected characteristic polynomial of a union of random perfect matchings, established using the framework of finite free convolutions we recently introduced.

We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szeg\"o in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.