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Wie verhält sich ein mikroskopisch komplexes Medium auf großen Skalen, d.h. wenn die mikroskopische Skala ε klein wird?

Viele physikalische Vorgänge oder Phänomene werden durch zufällige Operatoren beschrieben, insbesondere durch ihr Spektrum. Die meist studierten solchen Operatoren sind der zufällige Schrödinger-Operator (der Anderson-Operator) im euklidischen Raum und der Laplace-Operator im diskreten euklidischen Raum mit zufälligen Gewichten (Leitfähigkeiten) auf den Kanten. Die Motivationen sind vielfältig und reichen von der Beschreibung von optischen Eigenschaften verunreinigter Medien über Leitfähigkeiten in Legierungen und anderen ungeordneten Materialien zum Langzeitverhalten von Verzweigungsprozesen bei zufälligen raumabhängigen Verzweigungsraten. Etliche der Fragen kann man zurückführen auf die Asymptotik der Haupteigenwerte dieser Operatoren und der Lücken zwischen ihnen und das Verhalten der zugehörigen Eigenfunktionen in großen Boxen, insbesondere der Unterscheidung, ob sie sich in kleinen Gebieten konzentrieren (Lokalisierung) oder gleichmäßig über die gesamte Box ausbreiten (Homogenisierung). Die Untersuchung solcher Fragen erfordert eine Kombination von wahrscheinlichkeitstheoretischen mit analytischen Methoden wie große Abweichungen, räumliche Extremwertstatistik, Punktprozesskonvergenz, Ergodentheorie, Martingalkonvergenz bzw. Variationsrechnung, stochastische Homogenisierung und Potentialtheorie und parakontrollierte Analysis.

Beitrag des Instituts

Am WIAS wird das Spektrum des Anderson-Operators im diskreten euklidischen Raum mit einem sehr interessanten Potential untersucht, das auf eine nichttriviale räumliche Gestalt der Eigenfunktion führt, die durch ein deterministisches Variationsproblem gegeben ist. Ein Höhepunkt ist der Beweis von Lokalisierung der Eigenfunktionen sowie eine Extremwertstatistik für die führenden Eigenwerte in großen Boxen, ein anderer ist der Beweis, dass der Hauptbeitrag zur Lösung der zugehörigen zeitabhängigen Gleichung (des parabolischen Anderson-Modells) in nur einer einzigen Insel konzentriert ist. Zu dieser zeitabhängigen Gleichung (Wärmeleitungsgleichung mit zufälligem Potential) wurde auch eine umfangreiche Übersicht in Buchform verfasst.

Neben der Untersuchung des zufälligen Schrödingeroperators im diskreten euklidischen Raum wird am Institut auch die räumlich kontinuierliche Variante untersucht, wobei das zufällige Potential ein Gaußsches weißes Rauschen ist, und man muss sich zunächst auf Dimension Zwei einschränken. Diese Aufgabe liegt in der langfristigen Forschung des Instituts über moderne Behandlung von stochastischen partiellen Differentialgleichungen mit Hilfe innovativer Methoden. Wegen der geringen Regularität des Potentials ist die Definition des Operators a priori nicht sinnvoll, aber mit Hilfe einer Renormierung mit parakontrollierter Analysis kann man ihn in Boxen konstruieren. Dann wurde gezeigt, dass die führenden Eigenwerte, geteilt durch den Logarithmus des Volumens der Box, fast sicher gegen einen expliziten Grenzwert konvergieren. Mittelfristige Ziele des Instituts sehen auch einen Beweis einer geometrischen Intermittenzaussage vor, die ähnlich zu dem Bild ist, das oben für den diskreten zufälligen Schrödingeroperator beschrieben wurde, auch in Dimension Drei.

Eine andere Linie verfolgt die Frage, unter welchen Bedingungen an das zufällige Potential Lokalisierung beziehungsweise Homogenisierung auftritt in den Haupteigenfunktionen des diskreten Laplace-Operators mit zufälligen Leitfähigkeiten.

In einem Doktorandenprojekt wird gezeigt, dass spektrale Lokalisierung bzw. Homogenisierung im Wesentlichen an Hand von expliziten Momentenbedingungen an das zufällige Potential entschieden wird.


Dichotomie in dem Verhalten des Spektrums des Laplacians mit zufälligen i.i.d. Leitfähigkeiten ω. Zur Vereinfachung nehmen wir an, dass P[ω ≤ a] = aγ. Die Abbildung zeigt den Haupteigenvektor ψ1(n) in der Box Bn=(-n,n)d mit verschwindenden Dirichlet-Randbedingungen. Diagramm (a) zeigt den Vektor für kleine n, während (b) und (c) das asymptotische Verhalten für große n darstellen. Abhängig davon, ob γ größer oder kleiner als 1/4 ist, lokalisiert (b) oder homogenisiert (c) der Haupteigenvektor fast sicher wenn die Boxgröße gegen unendlich strebt.


Publikationen

 Monografien

  • W. König, The Parabolic Anderson Model -- Random Walks in Random Potential, Pathways in Mathematics, Birkhäuser, Basel, 2016, xi+192 pages, (Monograph Published).

  • P. Exner, W. König, H. Neidhardt, eds., Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, World Scientific Publishing, Singapore, 2015, xii+383 pages, (Collection Published).

 Artikel in Referierten Journalen

  • K. Chouk, W. van Zuijlen, Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions, The Annals of Probability, 49 (2021), pp. 1917--1964, DOI 10.1214/20-AOP1497 .
    Abstract
    In this paper we consider the Anderson Hamiltonian with white noise potential on the box [0,L]² with Dirichlet boundary conditions. We show that all the eigenvalues divided by log L converge as L → ∞ almost surely to the same deterministic constant, which is given by a variational formula.

  • F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 1226--1257, DOI 10.1214/18-AIHP917 .
    Abstract
    We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for thenormalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence

  • F. Flegel, M. Heida, The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 8/1--8/39 (published online on 28.11.2019), DOI 10.1007/s00526-019-1663-4 .
    Abstract
    We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Lapalace operator to the continuous fractional Laplace operator.

  • F. Flegel, Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model, Electronic Journal of Probability, 23 (2018), pp. 68/1--68/43, DOI doi:10.1214/18-EJP160 .
    Abstract
    We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of Zd (d ≥ 2) with zero Dirichlet condition. We assume that the conductances w are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If γ = sup q ≥ 0; E [w^-q]<∞ <¼, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold γrm c = ¼ is sharp. Indeed, other recent results imply that for γ>¼ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.

  • M. Biskup, W. König, Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails, Communications in Mathematical Physics, 341 (2016), pp. 179--218.

  • W. König, T. Wolff, Large deviations for the local times of a random walk among random conductances in a growing box, Special issue for Pastur's 75th birthday, Markov Processes and Related Fields, 21 (2015), pp. 591--638.
    Abstract
    We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in Zd. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent size of the box.
    An interesting phase transition occurs if the thickness parameter of the conductance tails exceeds a certain threshold: for thicker tails, the random walk spreads out over the entire growing box, for thinner tails it stays confined to some bounded region. In fact, in the first case, the rate function turns out to be equal to the p-th power of the p-norm of the gradient of the square root for some 2d/(d+2) < p < 2. This extends the Donsker-Varadhan-Gärtner rate function for the local times of Brownian motion (with deterministic environment) from p=2 to these values.
    As corollaries of our LDP, we derive the logarithmic asymptotics of the non-exit probability of the RWRC from the growing box, and the Lifshitz tails of the generator of the RWRC, the randomised Laplace operator. To contrast with the annealed, not uniformly elliptic case, we also provide an LDP in the quenched setting for conductances that are bounded and bounded away from zero. The main tool here is a spectral homogenisation result, based on a quenched invariance principle for the RWRC.

  • W. Kirsch, B. Metzger, P. Müller, Random block operators, Journal of Statistical Physics, 143 (2011), pp. 1035--1054.
    Abstract
    We study fundamental spectral properties of random block operators that are common in the physical modelling of mesoscopic disordered systems such as dirty superconductors. Our results include ergodic properties, the location of the spectrum, existence and regularity of the integrated density of states, as well as Lifshits tails. Special attention is paid to the peculiarities arising from the block structure such as the occurrence of a robust gap in the middle of the spectrum. Without randomness in the off-diagonal blocks the density of states typically exhibits an inverse square-root singularity at the edges of the gap. In the presence of randomness we establish a Wegner estimate that is valid at all energies. It implies that the singularities are smeared out by randomness, and the density of states is bounded. We also show Lifshits tails at these band edges. Technically, one has to cope with a non-monotone dependence on the random couplings.

  • G. Grüninger, W. König, Potential confinement property in the parabolic Anderson model, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 45 (2009), pp. 840--863.

  • W. König, H. Lacoin, P. Mörters, N. Sidorova, A two cities theorem for the parabolic Anderson model, The Annals of Probability, 37 (2009), pp. 347--392.

 Beiträge zu Sammelwerken

  • F. DEN Hollander, W. König, R. Soares Dos Santos, The parabolic Anderson model on a Galton--Watson tree, in: In and Out of Equilibrium 3: Celebrating Vladas Sidovaricius, M.E. Vares, R. Fernandez, L.R. Fontes, C.M. Newman, eds., 77 of Progress in Probability, Birkhäuser, 2021, pp. 591--635, DOI 10.1007/978-3-030-60754-8_25 .
    Abstract
    We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.

  • W. König, Branching random walks in random environment, in: Probabilistic Structures in Evolution, E. Baake, A. Wakolbinger, eds., Probabilistic Structures in Evolution, EMS Series of Congress Reports, European Mathematical Society Publishing House, 2021, pp. 23--41, DOI 10.4171/ECR/17-1/2 .

 Preprints, Reports, Technical Reports

  • T. Matsuda, W. van Zuijlen, Anderson Hamiltonians with singular potentials, Preprint no. 2976, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2976 .
    Abstract, PDF (732 kByte)
    We construct random Schrödinger operators, called Anderson Hamiltonians, with Dirichlet and Neumann boundary conditions for a fairly general class of singular random potentials on bounded domains. Furthermore, we construct the integrated density of states of these Anderson Hamiltonians, and we relate the Lifschitz tails (the asymptotics of the left tails of the integrated density of states) to the left tails of the principal eigenvalues.

 Vorträge, Poster

  • W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability Meeting (Online Event), University of Oxford, Department of Statistics, UK, February 10, 2021.

  • W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability and Statistical Physics Seminar (Online Event), The University of Chicago, Department of Mathematics, Statistics, and Computer Science, USA, February 12, 2021.

  • W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability Seminar, Universidade Federal da Bahia, Instituto de Matematica Doutorado em Matematica (Online Event), Salvador, Brazil, October 21, 2020.

  • W. van Zuijlen, Spectral asymptotics of the Anderson Hamiltonian, Forschungsseminar ''Functional Analysis``, Karlsruher Institut für Technologie, Fakultät für Mathematik, Institut für Analysis, January 21, 2020.

  • W. van Zuijlen, Mass-asymptotics for the parabolic Anderson model in 2D, Berlin--Leipzig Workshop in Analysis and Stochastics, January 16 - 18, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 18, 2019.

  • W. van Zuijlen, The parabolic Anderson model in 2D, mass- and eigenvalue asymptotics, Stochastic Analysis Seminar, University of Oxford, Mathematical Institute, UK, February 4, 2019.

  • W. van Zuijlen, The parabolic Anderson model in 2D, mass- and eigenvalue asymptotics, Analysis and Probability Seminar, Imperial College London, Department of Mathematics, UK, February 5, 2019.

  • W. van Zuijlen, Mass-asymptotics for the parabolic Anderson model in 2D, 10th Oxford-Berlin Young Researchers Meeting on Applied Stochastic Analysis, November 29 - December 1, 2018, University of Oxford, Mathematical Institute, Oxford, UK, November 29, 2018.

  • W. van Zuijlen, The principal eigenvalue of the Anderson Hamiltonian in continuous space, 13th German Probability and Statistics Days 2018 -- Freiburger Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 28, 2018.

  • F. Flegel, Spectral localization vs. homogenization in the random conductance model, 19th ÖMG Congress and Annual DMV Meeting, Minisymposium M6 ``Spectral and Scattering Problems in Mathematical Physics'', September 11 - 15, 2017, Austrian Mathematical Society (ÖMG) and Deutsche Mathematiker-Vereinigung (DMV), Paris-Lodron University of Salzburg, Austria, September 12, 2017.

  • F. Flegel, Spectral localization vs. homogenization in the random conductance model, Berlin-Leipzig Workshop in Analysis and Stochastics, November 29 - December 1, 2017, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

  • M. Heida, Homogenization of the random conductance model, 7th European Congress of Mathematics (ECM), session ``Probability, Statistics and Financial Mathematics'', July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 20, 2016.

  • M. Heida, Homogenization of the random conductance model, Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 26 - 28, 2016, Technische Universität Dortmund, Fachbereich Mathematik, Dortmund, September 26, 2016.

  • F. Flegel, Spectral localization vs. homogenization in the random conductance model, Summer School 2016, August 21 - 26, 2016, Research Training Group (RTG) 1845 ``Stochastic Analysis with Applications in Biology, Finance and Physics'', Hejnice, Czech Republic, August 22, 2016.

  • F. Flegel, Spectral localization vs. homogenization in the random conductance model, Probability Seminar at UCLA, University of California, Los Angeles, Department of Mathematics, Los Angeles, USA, October 13, 2016.

  • F. Flegel, Localization of the first Dirichlet-eigenvector in the heavy-tailed random conductance model, Summer School 2015 of the RTG 1845 Berlin-Potsdam ``Stochastic Analysis with Applications in Biology, Finance and Physics'', September 28 - October 3, 2015, Levico Terme, Italy, October 1, 2015.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, SFB/TR12 Workshop, November 4 - 8, 2012, Universität zu Köln, SFB TR12 ``Symmetries and Universality in Mesoscopic Systems'', Langeoog, November 7, 2012.

  • S. Jansen, Large deviations for interacting many-particle systems in the Saha regime, Berlin-Leipzig Seminar on Analysis and Probability Theory, July 8, 2011, Technische Universität Clausthal, Institut für Mathematik, July 8, 2011.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, Berlin-Leipzig Seminar on Analysis and Probability Theory, Technische Universität Clausthal, Institut für Mathematik, July 8, 2011.

  • B. Metzger, The parabolic Anderson model: The asymptotics of the statistical moments and Lifshitz tails revisited, EURANDOM, Eindhoven, Netherlands, December 1, 2010.

  • W. König, Die Universalitätsklassen im parabolischen Anderson-Modell, Mathematisches Kolloquium, Technische Universität Darmstadt, Fachbereich Mathematik, July 7, 2010.

  • W. König, The parabolic Anderson model, XIV Escola Brasileira de Probabilidade, August 2 - 7, 2010, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil.

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