Browsing by Author "Biskup, Marek"

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    A central limit theorem for the effective conductance: I. Linear boundary data and small ellipticity contrasts
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Biskup, Marek; Salvi, Michele; Wolff, Tilman
    We consider resistor networks on Zd where each nearest-neighbor edge is assigned a non-negative random conductance. Given a finite set with a prescribed boundary condition, the effective conductance is the minimum of the Dirichlet energy over functions that agree with the boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box is known to converge to a deterministic limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and arbitrary ellipticity contrasts are to be addressed in a subsequent paper.
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    Eigenvalue fluctuations for lattice Anderson Hamiltonians
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Biskup, Marek; Fukushima, Ryoki; König, Wolfgang
    We consider the random Schrödinger operator on a large box in the lattice with a large prefactor in front of the Laplacian part of the operator, which is proportional to the square of the diameter of the box. The random potential is assumed to be independent and bounded; its expectation function and variance function is given in terms of continuous bounded functions on the rescaled box. Our main result is a multivariate central limit theorem for all the simple eigenvalues of this operator, after centering and rescaling. The limiting covariances are expressed in terms of the limiting homogenized eigenvalue problem; more precisely, they are equal to the integral of the product of the squares of the eigenfunctions of that problem times the variance function.
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    Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Biskup, Marek; Fukushima, Ryoki; König, Wolfgang
    We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors recent work where similar conclusions have been obtained for bounded random potentials.
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    Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Biskup, Marek; König, Wolfgang
    We consider random Schrödinger operators of the form Delta+zeta , where D is the lattice Laplacian on Zd and Delta is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of Zd. We show that for sigma with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where zeta takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem.
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    Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Biskup, Marek; König, Wolfgang; Santos, Renato Soares dos
    We study the solutions to the Cauchy problem on the with random potential and localised initial data. Here we consider the random Schrödinger operator, i.e., the Laplace operator with random field, whose upper tails are doubly exponentially distributed in our case. We prove that, for large times and with large probability, a majority of the total mass of the solution resides in a bounded neighborhood of a site that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian and the distance to the origin. The processes of mass concentration and the rescaled total mass are shown to converge in distribution under suitable scaling of space and time. Aging results are also established. The proof uses the characterization of eigenvalue order statistics for the random Schrödinger operator in large sets recently proved by the first two authors.
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    Mini-Workshop: Mathematical Approaches to Collective Phenomena in Large Quantum Systems
    (Zürich : EMS Publ. House, 2008) Biskup, Marek; Seiringer, Robert
    [no abstract available]
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    New Horizons in Motions in Random Media
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Andres, Sebastian; Biskup, Marek; Faggionato, Alessandra; Slowik, Martin
    The general topic of the mini-workshop "New Horizons in Motions in Random Media" was the study of random walks in random environments, both in their own right and in relation to stochastic homogenization and to models in statistical mechanics, in particular spin system. This is a subject at the intersection of probability, analysis and mathematical physics, and the workshop brought together leading researchers from those areas. While each of these areas has been quite active for decades with many remarkable breakthroughs obtained throughout the years, the workshop provided a unique opportunity to identify principal new objectives and initiate new collaborations.
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    Scaling Limits in Models of Statistical Mechanics
    (Zürich : EMS Publ. House, 2012) Biskup, Marek; van der Hofstad, Remco; Sidoravicius, Vladas
    This has been the third workshop around Statistical Mechanics organized in the last 6 years. The main topic consisted of spatial random processes and their connections to statistical mechanics. The common underlying theme of the subjects discussed at the meeting is the existence of a scaling limit, i.e., a continuum object that approximates the discrete one under study at sufficiently large spatial scales. The specific topics that have been discussed included two-dimensional and high-dimensional critical models, random graphs and various random geometric problems, such as random interlacements, polymers, etc. The workshop bolstered interactions between groups of researchers in these areas and led to interesting and fruitful exchanges of ideas.
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    Scaling Limits in Models of Statistical Mechanics
    (Zürich : EMS Publ. House, 2009) Biskup, Marek; van der Hofstad, Remco; Sidoravicius, Vladas
    The workshop brought together researchers interested in spatial random processes and their connection to statistical mechanics. The principal subjects of interest were scaling limits and, in general, limit laws for various two-dimensional critical models, percolation, random walks in random environment, polymer models, random fields and hierarchical diffusions. The workshop fostered interactions between groups of researchers in these areas and led to interesting and fruitful exchanges of ideas.
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    Spatial Random Processes and Statistical Mechanics
    (Zürich : EMS Publ. House, 2006) Biskup, Marek; van der Hofstad, Remco; Sidoravicius, Vladas
    The workshop focused on the broad area of spatial random processes and their connection to statistical mechanics. The subjects of interest included random walk in random environment, interacting random walks, polymer models, random fields and spin systems, dynamical problems, metastability as well as problems involving two-dimensional conformal geometry. The workshop brought together many leading researchers in these fields who reported to each other on their recent achievements and exchanged ideas for new problems and potential solutions.