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 longtime behaviour of branching processes in random fields of spacedependent 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, variational analysis, stochastic homogenisation, potential theory, and paracontrolled calculus.
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 timedependent 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.
Besides the treatment of the random Schrödinger operator in the discrete euclidean space, also the Schrödinger operator in the two dimensional continuous euclidean space is investigated, where the potential is considered to be white noise. This lies in the institute's longterm research area on the modern analysis of SPDEs using particularly innovative methods. Due to the low regularity of white noise in two dimensions, the definition of the operator a priori does not make sense. But with a renormalisation procedure using paracontrolled calculus the operator is properly defined on boxes. Then, it is proved that the leading eigenvalues scaled by one over the logarithm of the volume of the box converge almost surely with an explicit limit. The institute's midterm goals comprise a proof of a similar intermittency picture as described above for the discrete random Schrödinger operator, prospectively also in dimension three.
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] = a^{γ}. The figure shows the principal Dirichlet eigenvector ψ_{1}^{(n)} in the box B_{n}=(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

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. 19171964, DOI 10.1214/20AOP1497 .
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 unboundedrange jumps, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 12261257, DOI 10.1214/18AIHP917 .
Abstract
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almostsure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearestneighbor 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 longrange 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 twoscale convergence 
F. Flegel, M. Heida, The fractional pLaplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unboundedrange jumps, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 8/18/39 (published online on 28.11.2019), DOI 10.1007/s0052601916634 .
Abstract
We study a general class of discrete pLaplace operators in the random conductance model with longrange 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 pLaplace 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 heavytailed random conductance model, Electronic Journal of Probability, 23 (2018), pp. 68/168/43, DOI doi:10.1214/18EJP160 .
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, BorelCantelli arguments, the RayleighRitz formula, results from percolation theory, and path arguments. 
M. Biskup, W. König, Eigenvalue order statistics for random Schrödinger operators with doublyexponential tails, Communications in Mathematical Physics, 341 (2016), pp. 179218.

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. 591638.
Abstract
We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuoustime random walk among random conductances (RWRC) in a timedependent, growing box in Z^{d}. 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 timedependent 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 pth power of the pnorm of the gradient of the square root for some 2d/(d+2) < p < 2. This extends the DonskerVaradhanGärtner 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 nonexit 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. 10351054.
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 offdiagonal blocks the density of states typically exhibits an inverse squareroot 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 nonmonotone 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. 840863.

W. König, H. Lacoin, P. Mörters, N. Sidorova, A two cities theorem for the parabolic Anderson model, The Annals of Probability, 37 (2009), pp. 347392.
Contributions to Collected Editions

F. DEN Hollander, W. König, R. Soares Dos Santos, The parabolic Anderson model on a GaltonWatson 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. XXIII, 820, DOI 10.1007/9783030607548 .
Abstract
We study the longtime asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical GaltonWatson random tree with bounded degrees. We identify the secondorder 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 treelike 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.
Talks, 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, Massasymptotics for the parabolic Anderson model in 2D, BerlinLeipzig Workshop in Analysis and Stochastics, January 16  18, 2019, MaxPlanckInstitut 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, Massasymptotics for the parabolic Anderson model in 2D, 10th OxfordBerlin 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 StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität 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 MathematikerVereinigung (DMV), ParisLodron University of Salzburg, Austria, September 12, 2017.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, BerlinLeipzig Workshop in Analysis and Stochastics, November 29  December 1, 2017, MaxPlanckInstitut 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 Dirichleteigenvector in the heavytailed random conductance model, Summer School 2015 of the RTG 1845 BerlinPotsdam ``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 manyparticle systems in the Saha regime, BerlinLeipzig 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, BerlinLeipzig 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 AndersonModell, 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.