arXiv subjects

Wolfgang Koenig

Publications and source records attributed to Wolfgang Koenig.

16 recordsLinked to original sources

arXiv2025

Self-repellent Brownian Bridges in an Interacting Bose Gas

We consider a model of $d$-dimensional interacting quantum Bose gas, expressed in terms of an ensemble of interacting Brownian bridges in a large box and undergoing the influence of all the interactions between the legs of each of the Brownian bridges. We study the thermodynamic limit of the system and give an explicit formula for the limiting free energy and a necessary and sufficient criterion for the occurrence of a condensation phase transition. For $d\geq 5$ and sufficiently small interaction, we prove that the condensate phase is not empty. The ideas of proof rely on the similarity of the interaction to that of the self-repellent random walk, and build on a lace expansion method conducive to treating {\it paths} undergoing mutual repellence within each bridge.

math.PR↗

arXiv2017

Mean-field interaction of Brownian occupation measures, I: uniform tube property of the Coulomb functional

We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, derived in [BKM15]. Our methods rely on deriving H{ö}lder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures.

math.PR↗

arXiv2017

Mean-Field interacton of Brownian occupation measures. II: A rigorous construction of the Pekar process

We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [6] in terms of the {\it{Pekar variational formula}}, which coincides with the behavior of the partition function of the {\it{polaron problem}} under strong coupling. Based on this, in 1986 Spohn [14] made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the {\it{Pekar process}}, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the "mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [11], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [8], as well as an idea inspired by a {\it{partial path exchange}} argument appearing in [1].

math.PR↗

arXiv2017

Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials

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.

math.PR↗

arXiv2017

Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

We study the solutions $u=u(x,t)$ to the Cauchy problem on $\mathbb Z^d\times(0,\infty)$ for the parabolic equation $\partial_t u=Δu+ξu$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $Δ$ is the discrete Laplacian on $\mathbb Z^d$ and $ξ=(ξ(z))_{z\in\mathbb Z^d}$ is an i.i.d.\ random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):=\sum_x u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian $Δ+ξ$ and the distance to the origin. The processes $t\mapsto Z_t$ and $t \mapsto \tfrac1t \log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for $Δ+ξ$ in large sets recently proved by the first two authors.

math.PR↗

arXiv2016

Eigenvalue fluctuations for lattice Anderson Hamiltonians

We study the statistics of Dirichlet eigenvalues of the random Schrödinger operator $-ε^{-2}Δ^{(\text{d})}+ξ^{(ε)}(x)$, with $Δ^{(\text{d})}$ the discrete Laplacian on $\mathbb Z^d$ and $ξ^{(ε)}(x)$ uniformly bounded independent random variables, on sets of the form $D_ε:=\{x\in \mathbb Z^d\colon xε\in D\}$ for $D\subset \mathbb R^d$ bounded, open and with a smooth boundary. If $\mathbb Eξ^{(ε)}(x)=U(xε)$ holds for some bounded and continuous $U\colon D\to \mathbb R$, we show that, as $ε\downarrow0$, the $k$-th eigenvalue converges to the $k$-th Dirichlet eigenvalue of the homogenized operator $-Δ+U(x)$, where $Δ$ is the continuum Dirichlet Laplacian on $D$. Assuming further that $\text{Var}(ξ^{(ε)}(x))=V(xε)$ for some positive and continuous $V\colon D\to \mathbb R$, we establish a multivariate central limit theorem for simple eigenvalues centered by their expectation. The limiting covariance for a given pair of simple eigenvalues is expressed as an integral of $V$ against the product of squares of the corresponding eigenfunctions of $-Δ+U(x)$.

math.PR↗

arXiv2013

Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails

We consider random Schrödinger operators of the form $Δ+ξ$, where $Δ$ is the lattice Laplacian on $\mathbb Z^d$ and $ξ$ 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 $\mathbb Z^d$. We show that for $ξ$ 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 $ξ$ 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.

math.PR↗

arXiv2012

Large Deviations for Brownian Intersection Measures

We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq 2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any measurable set $A\subset \R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen \cite{Ch09} derived the logarithmic asymptotics of the upper tails of the total mass $\ell_t(\R^d)$ as $t\to\infty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^{-p}\ell_t$ on the set of positive measures on some open bounded set $B\subset\R^d$ as $t\to\infty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $U\subset B$. This extends earlier studies on the intersection measure by König and Mörters \cite{KM01,KM05}.

math.PR↗

arXiv2009

Random walks conditioned to stay in Weyl chambers of type C and D

We construct the conditional versions of a multidimensional random walk given that it does not leave the Weyl chambers of type C and of type D, respectively, in terms of a Doob h-transform. Furthermore, we prove functional limit theorems for the rescaled random walks. This is an extension of recent work by Eichelsbacher and Koenig who studied the analogous conditioning for the Weyl chamber of type A. Our proof follows recent work by Denisov and Wachtel who used martingale properties and a strong approximation of random walks by Brownian motion. Therefore, we are able to keep minimal moment assumptions. Finally, we present an alternate function that is amenable to an h-transform in the Weyl chamber of type C.

math.PR↗

arXiv2005

Large systems of path-repellent Brownian motions in a trap at positive temperature

We study a model of $ N $ mutually repellent Brownian motions under confinement to stay in some bounded region of space. Our model is defined in terms of a transformed path measure under a trap Hamiltonian, which prevents the motions from escaping to infinity, and a pair-interaction Hamiltonian, which imposes a repellency of the $N$ paths. In fact, this interaction is an $N$-dependent regularisation of the Brownian intersection local times, an object which is of independent interest in the theory of stochastic processes. The time horizon (interpreted as the inverse temperature) is kept fixed. We analyse the model for diverging number of Brownian motions in terms of a large deviation principle. The resulting variational formula is the positive-temperature analogue of the well-known Gross-Pitaevskii formula, which approximates the ground state of a certain dilute large quantum system; the kinetic energy term of that formula is replaced by a probabilistic energy functional. This study is a continuation of the analysis in \cite{ABK04} where we considered the limit of diverging time (i.e., the zero-temperature limit) with fixed number of Brownian motions, followed by the limit for diverging number of motions. \bibitem[ABK04]{ABK04} {\sc S.~Adams, J.-B.~Bru} and {\sc W.~König}, \newblock Large deviations for trapped interacting Brownian particles and paths, \newblock {\it Ann. Probab.}, to appear (2004).

math.PR↗

arXiv2005

Orthogonal polynomial ensembles in probability theory

We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther and also comprise the zeros of the Riemann zeta function. The existing proofs require a substantial technical machinery and heavy tools from various parts of mathematics, in particular complex analysis, combinatorics and variational analysis. Particularly in the last decade, a number of fine results have been achieved, but it is obvious that a comprehensive and thorough understanding of the matter is still lacking. Hence, it seems an appropriate time to provide a surveying text on this research area.

math.PR↗

arXiv2005

The universality classes in the parabolic Anderson model

We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on $\Z^d$. We consider general i.i.d. potentials and show that exactly \emph{four} qualitatively different types of intermittent behaviour can occur. These four universality classes depend on the upper tail of the potential distribution: (1) tails at $\infty$ that are thicker than the double-exponential tails, (2) double-exponential tails at $\infty$ studied by Gärtner and Molchanov, (3) a new class called \emph{almost bounded potentials}, and (4) potentials bounded from above studied by Biskup and König. The new class (3), which contains both unbounded and bounded potentials, is studied in both the annealed and the quenched setting. We show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time. The characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas, respectively, Gaussian densities.

math.PR↗

arXiv2004

The parabolic Anderson model

This is a survey on the intermittent behavior of the parabolic {Anderson} model, which is the Cauchy problem for the heat equation with random potential on the lattice $\Z^d$. We first introduce the model and give heuristic explanations of the long-time behavior of the solution, both in the annealed and the quenched setting for time-independent potentials. We thereby consider examples of potentials studied in the literature. In the particularly important case of an i.i.d. potential with double-exponential tails we formulate the asymptotic results in detail. Furthermore, we explain that, under mild regularity assumptions, there are only four different universality classes of asymptotic behaviors. Finally, we study the moment Lyapunov exponents for space-time homogeneous catalytic potentials generated by a {Poisson} field of random walks.

math.PR↗

arXiv2003

Long-time tails in the parabolic Anderson model with bounded potential

We consider the parabolic Anderson problem $\partial_t u=κΔu+ξu$ on $(0,\infty)\times \Z^d$ with random i.i.d. potential $ξ=(ξ(z))_{z\in\Z^d}$ and the initial condition $u(0,\cdot)\equiv1$. Our main assumption is that $\esssupξ(0)=0$. Depending on the thickness of the distribution $\prob(ξ(0)\in\cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t,0)$ and the almost-sure asymptotics of $u(t,0)$ as $t\to\infty$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schrödinger operator $-κΔ-ξ$ at the bottom of its spectrum. In our class of $ξ$ distributions, the Lifshitz exponent ranges from $d/2$ to $\infty$; the power law is typically accompanied by lower-order corrections.

math-ph↗

arXiv2003

Brownian intersection local times: Exponential moments and law of large masses

Consider p independent Brownian motions in R^d, each running up to its first exit time from an open domain B, and their intersection local time l as a measure on B. We give a sharp criterion for the finiteness of exponential moments, E[exp(\sum_{i=1}^n (int_B f_i(x) l(dx))^{1/p})], where f_1, ...,f_n are nonnegative, bounded functions with compact support in B. We also derive a law of large numbers for intersection local time conditioned to have large total mass.

math.PR↗

arXiv2000

Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model

We consider the large-time behavior of the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$ to the parabolic Anderson problem $\partial_t u=κΔu+ξu$ with initial data $u(0,\cdot)=1$ and non-positive finite i.i.d. potentials $(ξ(z))_{z\in\Z}$. Unlike in dimensions $d\ge2$, the almost-sure decay rate of $u(t,0)$ as $t\to\infty$ is not determined solely by the upper tails of $ξ(0)$; too heavy lower tails of $ξ(0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $ξ(x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to $d=1$. The result answers an open question from our previous study \cite{BK00} of this model in general dimension.

math-ph↗