What are the large scale properties of a microscopically complex material, that is when the microscopic scale ε becomes small?

Many questions about processes and phenomena are described by random operators, most notably by the random Schrödinger operator (Anderson operator) and the Laplace operator with random weights (conductances) on the edges between nearest neighbours. Examples are electric conductance properties through an alloy or other disordered media, optical properties of diluted materials, and the long-time behaviour of branching processes in random fields of space-dependent branching rates. Many of the questions can be traced back to the properties of the leading eigenvalues and the gaps between them and the corresponding eigenfunctions in large boxes, in particular the question whether the eigenfunctions concentrate on small subareas (localisation) or spread out uniformly over the entire box (homogenisation). The investigation of such questions requires a combination of probabilistic and analytic means like extreme value analysis, martingale convergence, ergodic theory, large deviations, and variational analysis, stochastic homogenisation and potential theory.

### Contribution of the Institute

At WIAS the random Schrödinger operator in the discrete euclidean space with a particular random potential is investigated, whose eigenfunctions show an interesting shape that comes from a deterministic variational formula. One highlight is the derivation of the localisation of all the leading eigenfunctions and an asymptotic for the corresponding eigenvalues and a full description in terms of a Poisson point convergence. Another highlight is the proof that the solution of the corresponding time-dependent equation (the parabolic Anderson model or the heat equation with random potential) is asymptotically concentrated in just one single island. On this subject a comprehensive survey text in monograph form has been published.

Another highlight is the understanding of the interplay between localised and homogenised behaviour of the leading eigenfunctions of the Laplace operator in large boxes of the discrete euclidean space that is gained in a PhD project. In fact, it is proved that this distinction almost entirely depends on a certain explicit moment condition on the random potential.

Dichotomy in the spectral properties of the random conductance Laplacian with i.i.d. weights ω. For simplicity, we assume that P[ω ≤ a] = aγ. The figure shows the principal Dirichlet eigenvector ψ1(n) in the box Bn=(-n,n)d for small n (a) and the asymptotic shape for large n (b,c). Depending on whether γ is smaller or greater than 1/4, the principal Dirichlet eigenvector either almost surely localizes (b) or homogenizes (c) as the box size tends to infinity.

## Publications

### Monographs

• 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).

### Articles in Refereed Journals

• 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.

### Preprints, Reports, Technical Reports

• F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps, Preprint no. 2371, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2371 .
Abstract, PDF (598 kByte)
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 the normalized 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, Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model, Preprint no. 2290, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2290 .
Abstract, PDF (567 kByte)
We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of $Z^d$ ($dgeq 2$) 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 $gamma=sup qgeq 0colon E [w^-q]1/4$ 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.

### Talks, Poster

• 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.