Tracy Widom law for sample covariance matrices under quadratic form concentration

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   Matrix anti-concentration inequalities with applications

   Zipei Nie · 2021 · Probability · cited 22 times · AI summary available

   Matrix anti-concentration inequalities bound minimum singular values of random matrices, rather than edge eigenvalue distributions of sample covariance matrices.

   Abstract

   We provide a polynomial lower bound on the minimum singular value of an $m\times m$ random matrix $M$ with jointly Gaussian entries, under a polynomial bound on the matrix norm and a global small-ball probability bound $$\inf_{x,y\in S^{m-1}}\mathbb{P}\left(\left|x^* M y\right|>m^{-O(1)}\right)\ge \frac{1}{2}.$$ With the additional assumption that $M$ is self-adjoint, the global small-ball probability bound can be replaced with a weaker version. We establish two matrix anti-concentration inequalities, which lower bound the minimum singular values of the sum of independent positive semidefinite self-adjoint matrices and the linear combination of independent random matrices with independent Ga…

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   On the method of typical bounded differences

   Lutz Warnke · 2012 · Combinator. Probab. Comp. 25 (2016) 269-299 · cited 99 times · AI summary available

   Typical bounded differences develops concentration inequalities for functions of independent variables, while this paper assumes concentration of vector quadratic forms.

   Abstract

   Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that random functions are near their means. Of particular importance is the case where f(X) is a function of independent random variables X=(X_1, ..., X_n). Here the well known bounded differences inequality (also called McDiarmid's or Hoeffding-Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |f(X)-f(X')| \leq c_k whenever X,X' differ only in X_k. While this is easy to check, the main disadvant…

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   Restricted Maximum of Non-Intersecting Brownian Bridges

   Yamit Yalanda, Nicolás Zalduendo · 2023 · Probability · AI summary available

   Restricted maxima of non-intersecting Brownian bridges have Tracy–Widom limits, but concern bridge heights rather than sample covariance eigenvalues.

   Abstract

   Consider a system of $N$ non-intersecting Brownian bridges in $[0,1]$, and let $\mathcal M_N(p)$ be the maximal height attained by the top path in the interval $[0,p]$, $p\in[0,1]$. It is known that, under a suitable rescaling, the distribution of $\mathcal M_N(p)$ converges, as $N\to\infty$, to a one-parameter family of distributions interpolating between the Tracy-Widom distributions for the Gaussian Orthogonal and Unitary Ensembles (corresponding, respectively, to $p\to1$ and $p\to0$). It is also known that, for fixed $N$, $\mathcal M_N(1)$ is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. Here we show a version of these results for $\mat…

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   An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

   Matthew Kwan, Lisa Sauermann · 2019 · Combinatorics · cited 7 times · AI summary available

   The quadratic Littlewood–Offord work studies concentration of Bernoulli polynomials and Ramsey graphs, not quadratic-form concentration for sample covariance matrices.

   Abstract

   Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/\sqrt{n}$, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if $f$ has point probabilities much larger…

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   Almost sure convergence of products of $2\times2$ nonnegative matrices

   Alain Thomas · 2013 · Probability · preprint, not peer reviewed · AI summary available

   This paper studies convergence and Lyapunov exponents of products of nonnegative matrices, not edge universality for sample covariance matrices.

   Abstract

   We study the almost sure convergence of the normalized columns in an infinite product of nonnegative matrices, and the almost sure rank one property of its limit points. Given a probability on the set of $2\times2$ nonnegative matrices, with finite support $\mathcal A=\{A(0),\dots,A(s-1)\}$, and assuming that at least one of the $A(k)$ is not diagonal, the normalized columns of the product matrix $P_n=A(\omega_1)\dots A(\omega_n)$ converge almost surely (for the product probability) with an exponential rate of convergence if and only if the Lyapunov exponents are almost surely distinct. If this condition is satisfied, given a nonnegative column vector $V$ the column vector $\frac{P_nV}{\Vert…

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   Universal Constraints for Conformal Line Defects

   Barak Gabai, Amit Sever, De-liang Zhong · 2025 · High Energy Physics - Theory · cited 12 times · AI summary available

   Universal constraints for conformal line-defect correlators concern conformal symmetry, unrelated to sample covariance eigenvalue universality.

   Abstract

   We present a novel framework for deriving integral constraints for correlators on conformal line defects. These constraints emerge from the non-linearly realized ambient-space conformal symmetry. To validate our approach, we examine several examples and compare them against existing data for the four-point function of the displacement operator. Additionally, we provide a few new predictions that extend the current understanding of these correlators.

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   On the smoothed analysis of the smallest singular value with discrete noise

   Vishesh Jain, Ashwin Sah, Mehtaab Sawhney · 2020 · Probability · cited 12 times · AI summary available

   Smoothed analysis bounds the smallest singular value of a matrix with discrete noise, rather than the largest eigenvalue of a sample covariance matrix.

   Abstract

   Let $A$ be an $n\times n$ real matrix, and let $M$ be an $n\times n$ random matrix whose entries are i.i.d sub-Gaussian random variables with mean $0$ and variance $1$. We make two contributions to the study of $s_n(A+M)$, the smallest singular value of $A+M$. (1) We show that for all $\epsilon \geq 0$, $$\mathbb{P}[s_n(A + M) \leq \epsilon] = O(\epsilon \sqrt{n}) + 2e^{-\Omega(n)},$$ provided only that $A$ has $\Omega (n)$ singular values which are $O(\sqrt{n})$. This extends a well-known result of Rudelson and Vershynin, which requires all singular values of $A$ to be $O(\sqrt{n})$. (2) We show that any bound of the form $$\sup_{\|{A}\|\leq n^{C_1}}\mathbb{P}[s_n(A+M)\leq n^{-C_3}] \leq n^…

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   The Gauss-Bonnet Theorem for the noncommutative two torus

   Alain Connes, Paula Tretkoff · 2009 · Quantum Algebra · preprint, not peer reviewed · cited 126 times · AI summary available

   The noncommutative Gauss–Bonnet work studies a Laplacian zeta function on the noncommutative torus, not random matrix edge laws.

   Abstract

   In this paper we show that the value at zero of the zeta function of the Laplacian on the non-commutative two torus, endowed with its canonical conformal structure, is independent of the choice of the volume element (Weyl factor) given by a (non-unimodular) state. We had obtained, in the late eighties, in an unpublished computation, a general formula for this value at zero involving modified logarithms of the modular operator of the state. We give here the detailed computation and prove that the result is independent of the Weyl factor as in the classical case, thus proving the analogue of the Gauss-Bonnet theorem for the noncommutative two torus.