Coexistence and survival in conservative lotka-volterra networks - Knebel, Johannes; Krueger, Torben; Weber, Markus F.; Frey, Erwin (Physical Review Letters, 2013)
Analyzing coexistence and survival scenarios of Lotka-Volterra (LV) networks in which the total biomass is conserved is of vital importance for the characterization of long-term dynamics of ecological communities. Here, we introduce a classification scheme for coexistence scenarios in these conservative LV models and quantify the extinction process by employing the Pfaffian of the network's interaction matrix. We illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time. © 2013 American Physical Society.
CRIS-Team Scopus Importer:2021-09-24

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case - Erdos, Laszlo; Krueger, Torben; Schroeder, Dominik (Communications in Mathematical Physics, 2020)
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).
CRIS-Team Scopus Importer:2021-09-24

Emergence of coexisting ordered states in active matter systems - Huber, L.; Suzuki, Ryo; Frey, Erwin; Bausch, A. R.; Krueger, Torben (Science, 2018)
Active systems can produce a far greater variety of ordered patterns than conventional equilibrium systems. In particular, transitions between disorder and either polar- or nematically ordered phases have been predicted and observed in two-dimensional active systems. However, coexistence between phases of different types of order has not been reported. We demonstrate the emergence of dynamic coexistence of ordered states with fluctuating nematic and polar symmetry in an actomyosin motility assay. Combining experiments with agent-based simulations, we identify sufficiently weak interactions that lack a clear alignment symmetry as a prerequisite for coexistence. Thus, the symmetry of macroscopic order becomes an emergent and dynamic property of the active system. These results provide a pathway by which living systems can express different types of order by using identical building blocks.
CRIS-Team Scopus Importer:2021-09-24

Evolutionary games of condensates in coupled birth-death processes - Knebel, Johannes; Weber, Markus F.; Krueger, Torben; Frey, Erwin (Nature Communications, 2015)
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.
CRIS-Team Scopus Importer:2021-09-24

Inhomogeneous circular law for correlated matrices - Alt, Johannes; Krueger, Torben (Journal of Functional Analysis, 2021)
We consider non-Hermitian random matrices X∈Cn×n with general decaying correlations between their entries. For large n, the empirical spectral distribution is well approximated by a deterministic density, expressed in terms of the solution to a system of two coupled non-linear n×n matrix equations. This density is interpreted as the Brown measure of a linear combination of free circular elements with matrix coefficients on a non-commutative probability space. It is radially symmetric, real analytic in the radial variable and strictly positive on a disk around the origin in the complex plane with a discontinuous drop to zero at the edge. The radius of the disk is given explicitly in terms of the covariances of the entries of X. We show convergence down to local spectral scales just slightly above the typical eigenvalue spacing with an optimal rate of convergence.
CRIS-Team Scopus Importer:2021-09-24
Keywords: Brown measure; Delocalisation; Local law; Non-Hermitian random matrix

Local inhomogeneous circular law - Alt, Johannes; Erdos, Laszlo; Krueger, Torben (Annals of Applied Probability, 2018)
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. [Probab. Theory Related Fields 159 (2014) 545–595; Probab. Theory Related Fields 159 (2014) 619–660] 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.
CRIS-Team Scopus Importer:2021-09-24
Keywords: Circular law; Local law; Variance profile

Local law for random gram matrices - Alt, Johannes; Erdos, Laszlo; Krueger, Torben (Electronic Journal of Probability, 2017)
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*.
CRIS-Team Scopus Importer:2021-09-24
Keywords: Capacity of MIMO channels; General variance profile; Hard edge; Marchenko-Pastur law; Soft edge

Local laws for polynomials of Wigner matrices - Erdos, Laszlo; Krueger, Torben; Nemish, Yuriy (Journal of Functional Analysis, 2020)
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.
CRIS-Team Scopus Importer:2021-09-24
Keywords: Dyson equation; Generalized resolvent; Linearization; Polynomials of random matrices

Local semicircle law with imprimitive variance matrix - Ajanki, Oskari; Erdos, Laszlo; Krueger, Torben (Electronic Communications in Probability, 2014)
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.
CRIS-Team Scopus Importer:2021-09-24
Keywords: Generalised random sample covariance matrices; Generalised Wigner matrices; Hard edge; Local semicircle law

Local Spectral Statistics of Gaussian Matrices with Correlated Entries - Ajanki, Oskari H.; Erdos, Laszlo; Krueger, Torben (Journal of Statistical Physics, 2016)
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.
CRIS-Team Scopus Importer:2021-09-24
Keywords: Dependent random matrix; Local law; Off-diagonal resolvent decay; Universality

Location of the spectrum of Kronecker random matrices - Alt, Johannes; Erdos, Laszlo; Krueger, Torben; Nemish, Yuriy (Annales de l'Institut Henri Poincaré - Probabilités Et Statistiques, 2019)
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.
CRIS-Team Scopus Importer:2021-09-24
Keywords: Block matrices; Local law; Non-Hermitian random matrix; Outliers; Self-consistent pseudospectrum

Power law decay for systems of randomly coupled differential equations - Erdos, Laszlo; Krueger, Torben; Renfrew, David (SIAM Journal on Mathematical Analysis, 2018)
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
CRIS-Team Scopus Importer:2021-09-24
Keywords: Autocorrelation function; Non-Hermitian random matrix; Time evolution of neural networks

RANDOMLY COUPLED DIFFERENTIAL EQUATIONS WITH ELLIPTIC CORRELATIONS - Erdos, László; Krüger, Torben; Renfrew, David (Annals of Applied Probability, 2023)
We consider the long time asymptotic behavior of a large system of N linear differential equations with random coefficients. We allow for general elliptic correlation structures among the coefficients, thus we substantially generalize our previous work (SIAM J. Math. Anal. 50 (2018) 3271–3290) that was restricted to the independent case. In particular, we analyze a recent model in the theory of neural networks (Phys. Rev. E 97 (2018) 062314) that specifically focused on the effect of the distributional asymmetry in the random connectivity matrix X. We rigorously prove and slightly correct the explicit formula from (J. Math. Phys. 41 (2000) 3233–3256) on the time decay as a function of the asymmetry parameter. Our main tool is an asymptotically precise formula for the normalized trace of f(X)g(X∗), in the large N limit, where f and g are analytic functions.
CRIS-Team Scopus Importer:2023-08-11
Keywords: matrix Dyson equation (MDE); Non-Hermitian random matrix; partially symmetric correlation; time evolution of neural networks

Scattering in Quantum Dots via Noncommutative Rational Functions - Erdos, Laszlo; Krueger, Torben; Nemish, Yuriy (Annales Henri Poincaré, 2021)
In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via N≪ M 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/ M≤ 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.
CRIS-Team Scopus Importer:2021-09-24

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half-Plane - Ajanki, Oskari; Erdos, Laszlo; Krueger, Torben (Communications on Pure and Applied Mathematics, 2017)
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.© 2016 Wiley Periodicals, Inc.
CRIS-Team Scopus Importer:2021-09-24

Stability of the matrix Dyson equation and random matrices with correlations - Ajanki, Oskari H.; Erdos, Laszlo; Krueger, Torben (Probability Theory and Related Fields, 2019)
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.
CRIS-Team Scopus Importer:2021-09-24
Keywords: Bulk universality; Correlated random matrix; Local law

Universality for general Wigner-type matrices - Ajanki, Oskari H.; Erdos, Laszlo; Krueger, Torben (Probability Theory and Related Fields, 2017)
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=E|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.
CRIS-Team Scopus Importer:2021-09-24
Keywords: Anisotropic local law; Eigenvector delocalization; Local spectral statistics; Rigidity