TY - JOUR AB - In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via 𝑁â‰Ș𝑀 channels, the density 𝜌 of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio 𝜙:=𝑁/đ‘€â‰€1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit 𝜙→0, we recover the formula for the density 𝜌 that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any 𝜙<1 but in the borderline case 𝜙=1 an anomalous 𝜆−2/3 singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries. AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H AU - Nemish, Yuriy ID - 9912 JF - Annales Henri PoincarĂ© SN - 1424-0637 TI - Scattering in quantum dots via noncommutative rational functions VL - 22 ER - TY - JOUR AB - We consider random n×n matrices X with independent and centered entries and a general variance profile. We show that the spectral radius of X converges with very high probability to the square root of the spectral radius of the variance matrix of X when n tends to infinity. We also establish the optimal rate of convergence, that is a new result even for general i.i.d. matrices beyond the explicitly solvable Gaussian cases. The main ingredient is the proof of the local inhomogeneous circular law [arXiv:1612.07776] at the spectral edge. AU - Alt, Johannes AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H ID - 15013 IS - 2 JF - Probability and Mathematical Physics SN - 2690-0998 TI - Spectral radius of random matrices with independent entries VL - 2 ER - TY - JOUR AB - We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically. AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H AU - Nemish, Yuriy ID - 7512 IS - 12 JF - Journal of Functional Analysis SN - 00221236 TI - Local laws for polynomials of Wigner matrices VL - 278 ER - TY - JOUR AB - For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner–Dyson–Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also the key input in the companion paper (Cipolloni et al. in Pure Appl Anal, 2018. arXiv:1811.04055) where the cusp universality for real symmetric Wigner-type matrices is proven. The novel cusp fluctuation mechanism is also essential for the recent results on the spectral radius of non-Hermitian random matrices (Alt et al. in Spectral radius of random matrices with independent entries, 2019. arXiv:1907.13631), and the non-Hermitian edge universality (Cipolloni et al. in Edge universality for non-Hermitian random matrices, 2019. arXiv:1908.00969). AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H AU - Schröder, Dominik J ID - 6185 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Cusp universality for random matrices I: Local law and the complex Hermitian case VL - 378 ER - TY - JOUR AB - We study the unique solution m of the Dyson equation \( -m(z)^{-1} = z\1 - a + S[m(z)] \) on a von Neumann algebra A with the constraint Imm≄0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of A and S is a positivity-preserving linear operator on A. We show that m is the Stieltjes transform of a compactly supported A-valued measure on R. Under suitable assumptions, we establish that this measure has a uniformly 1/3-Hölder continuous density with respect to the Lebesgue measure, which is supported on finitely many intervals, called bands. In fact, the density is analytic inside the bands with a square-root growth at the edges and internal cubic root cusps whenever the gap between two bands vanishes. The shape of these singularities is universal and no other singularity may occur. We give a precise asymptotic description of m near the singular points. These asymptotics generalize the analysis at the regular edges given in the companion paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated random matrices [the first author et al., Ann. Probab. 48, No. 2, 963--1001 (2020; Zbl 1434.60017)] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [G. Cipolloni et al., Pure Appl. Anal. 1, No. 4, 615--707 (2019; Zbl 07142203); the second author et al., Commun. Math. Phys. 378, No. 2, 1203--1278 (2020; Zbl 07236118)]. We also extend the finite dimensional band mass formula from [the first author et al., loc. cit.] to the von Neumann algebra setting by showing that the spectral mass of the bands is topologically rigid under deformations and we conclude that these masses are quantized in some important cases. AU - Alt, Johannes AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H ID - 14694 JF - Documenta Mathematica KW - General Mathematics SN - 1431-0635 TI - The Dyson equation with linear self-energy: Spectral bands, edges and cusps VL - 25 ER - TY - JOUR AB - We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of the self-consistent density of states. In particular, we establish a strong form of band rigidity which excludes mismatches between location and label of eigenvalues close to internal edges in these general models. AU - Alt, Johannes AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H AU - Schröder, Dominik J ID - 6184 IS - 2 JF - Annals of Probability SN - 0091-1798 TI - Correlated random matrices: Band rigidity and edge universality VL - 48 ER - TY - JOUR AB - We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent. AU - Ajanki, Oskari H AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H ID - 429 IS - 1-2 JF - Probability Theory and Related Fields SN - 01788051 TI - Stability of the matrix Dyson equation and random matrices with correlations VL - 173 ER - TY - JOUR AB - We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields 173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion. AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H AU - Schröder, Dominik J ID - 6182 JF - Forum of Mathematics, Sigma TI - Random matrices with slow correlation decay VL - 7 ER - TY - JOUR AB - We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paper [arXiv:1809.03971], which proves the same result for the complex Hermitian symmetry class, this completes the last remaining case of the Wigner-Dyson-Mehta universality conjecture after bulk and edge universalities have been established in the last years. We extend the recent Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp regime using the optimal local law from [arXiv:1809.03971] and the accurate local shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752]. We also present a PDE-based method to improve the estimate on eigenvalue rigidity via the maximum principle of the heat flow related to the Dyson Brownian motion. AU - Cipolloni, Giorgio AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H AU - Schröder, Dominik J ID - 6186 IS - 4 JF - Pure and Applied Analysis SN - 2578-5893 TI - Cusp universality for random matrices, II: The real symmetric case VL - 1 ER - TY - JOUR AB - For a general class of large non-Hermitian random block matrices X we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of X as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles. AU - Alt, Johannes AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H AU - Nemish, Yuriy ID - 6240 IS - 2 JF - Annales de l'institut Henri Poincare SN - 0246-0203 TI - Location of the spectrum of Kronecker random matrices VL - 55 ER - TY - JOUR AB - We consider large random matrices X with centered, independent entries which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et. al. [11,12] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X. AU - Alt, Johannes AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H ID - 566 IS - 1 JF - Annals Applied Probability TI - Local inhomogeneous circular law VL - 28 ER - TY - JOUR AB - We consider large random matrices X with centered, independent entries but possibly di erent variances. We compute the normalized trace of f(X)g(X∗) for f, g functions analytic on the spectrum of X. We use these results to compute the long time asymptotics for systems of coupled di erential equations with random coe cients. We show that when the coupling is critical, the norm squared of the solution decays like t−1/2. AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H AU - Renfrew, David T ID - 181 IS - 3 JF - SIAM Journal on Mathematical Analysis TI - Power law decay for systems of randomly coupled differential equations VL - 50 ER - TY - GEN AB - We study the unique solution $m$ of the Dyson equation \[ -m(z)^{-1} = z - a + S[m(z)] \] on a von Neumann algebra $\mathcal{A}$ with the constraint $\mathrm{Im}\,m\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is a self-adjoint element of $\mathcal{A}$ and $S$ is a positivity-preserving linear operator on $\mathcal{A}$. We show that $m$ is the Stieltjes transform of a compactly supported $\mathcal{A}$-valued measure on $\mathbb{R}$. Under suitable assumptions, we establish that this measure has a uniformly $1/3$-H\"{o}lder continuous density with respect to the Lebesgue measure, which is supported on finitely many intervals, called bands. In fact, the density is analytic inside the bands with a square-root growth at the edges and internal cubic root cusps whenever the gap between two bands vanishes. The shape of these singularities is universal and no other singularity may occur. We give a precise asymptotic description of $m$ near the singular points. These asymptotics generalize the analysis at the regular edges given in the companion paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated random matrices [arXiv:1804.07744] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band mass formula from [arXiv:1804.07744] to the von Neumann algebra setting by showing that the spectral mass of the bands is topologically rigid under deformations and we conclude that these masses are quantized in some important cases. AU - Alt, Johannes AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H ID - 6183 T2 - arXiv TI - The Dyson equation with linear self-energy: Spectral bands, edges and cusps ER - TY - JOUR AB - Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur. AU - Ajanki, Oskari H AU - KrĂŒger, Torben H AU - Erdös, LĂĄszlĂł ID - 721 IS - 9 JF - Communications on Pure and Applied Mathematics SN - 00103640 TI - Singularities of solutions to quadratic vector equations on the complex upper half plane VL - 70 ER - TY - JOUR AB - We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),
,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes. AU - Ajanki, Oskari H AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H ID - 1337 IS - 3-4 JF - Probability Theory and Related Fields SN - 01788051 TI - Universality for general Wigner-type matrices VL - 169 ER - TY - JOUR AB - We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗. AU - Alt, Johannes AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H ID - 1010 JF - Electronic Journal of Probability SN - 10836489 TI - Local law for random Gram matrices VL - 22 ER - TY - JOUR AB - We prove optimal local law, bulk universality and non-trivial decay for the off-diagonal elements of the resolvent for a class of translation invariant Gaussian random matrix ensembles with correlated entries. AU - Ajanki, Oskari H AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H ID - 1489 IS - 2 JF - Journal of Statistical Physics TI - Local spectral statistics of Gaussian matrices with correlated entries VL - 163 ER - TY - JOUR AB - Condensation phenomena arise through a collective behaviour of particles. They are observed in both classical and quantum systems, ranging from the formation of traffic jams in mass transport models to the macroscopic occupation of the energetic ground state in ultra-cold bosonic gases (Bose-Einstein condensation). Recently, it has been shown that a driven and dissipative system of bosons may form multiple condensates. Which states become the condensates has, however, remained elusive thus far. The dynamics of this condensation are described by coupled birth-death processes, which also occur in evolutionary game theory. Here we apply concepts from evolutionary game theory to explain the formation of multiple condensates in such driven-dissipative bosonic systems. We show that the vanishing of relative entropy production determines their selection. The condensation proceeds exponentially fast, but the system never comes to rest. Instead, the occupation numbers of condensates may oscillate, as we demonstrate for a rock-paper-scissors game of condensates. AU - Knebel, Johannes AU - Weber, Markus AU - KrĂŒger, Torben H AU - Frey, Erwin ID - 1824 JF - Nature Communications TI - Evolutionary games of condensates in coupled birth-death processes VL - 6 ER - TY - JOUR AB - We extend the proof of the local semicircle law for generalized Wigner matrices given in MR3068390 to the case when the matrix of variances has an eigenvalue -1. In particular, this result provides a short proof of the optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for sample covariance matrices X*X, where the variances of the entries of X may vary. AU - Ajanki, Oskari H AU - Erdös, LĂĄszlĂł AU - KrĂŒger, Torben H ID - 2179 JF - Electronic Communications in Probability TI - Local semicircle law with imprimitive variance matrix VL - 19 ER -