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Werner Kirsch

Publications and source records attributed to Werner Kirsch.

At least 19 recordsLinked to original sources
arXiv2025

Dynamical localization and delocalization for random Schrodinger operators with $\delta$-interactions in $\mathbb{R}^3$

We prove that the random Schrodinger operators on $\mathbb{R}^3$ with independent, identically distributed random variables and single-site potentials given by $\delta$-functions on $\mathbb{Z}^3$, exhibit both dynamical localization and dynamical delocalization with probability one. That is, there are regions in the deterministic spectrum that exhibit dynamical localization, the nonspreading of wave packets, and regions in the deterministic spectrum where the models also exhibit nontrivial quantum transport, almost surely. These models are the first examples of ergodic, random Schrodinger operators exhibiting both dynamical localization and delocalization in dimension three or higher. The nontrivial transport is due to the presence of delocalized generalized eigenfunctions at positive energies $E > \pi^2$. The general idea of the proof follows [Hislop, Kirsch, Krishna (2024)] in which lower bounds on moments of the position operator are constructed using these generalized eigenfunctions. A new result of independent interest is a proof of the Combes-Thomas estimate on exponential decay of the Green's function for Schrodinger operators with infinitely-many $\delta$-potentials.

math-ph↗
arXiv2024

Eigenfunctions and Quantum Transport with Applications to Trimmed Schrodinger Operators

We provide a simple proof of dynamical delocalization, that is, time-increasing lower bounds on quantum transport for discrete, one-particle Schrodinger operators on $\ell^2 (\mathbb{Z}^d)$, provided solutions to the Schrodinger equation satisfy certain growth conditions. The proof is based on basic resolvent identities and the Combes-Thomas estimate on the exponential decay of the Green's function. As a consequence, we prove that generalized eigenfunctions for energies outside the spectrum of $H$ must grow exponentially in some directions. We also prove that if $H$ has any absolutely continuous spectrum, then the Schrodinger operator exhibits dynamical delocalization. We apply the general result to $\Gamma$-trimmed Schrodinger operators, with periodic $\Gamma$, and prove dynamical delocalization for these operators. These results also apply to the $\Gamma$-trimmed Anderson model, providing a random, ergodic model exhibiting both dynamical localization in an energy interval and dynamical delocalization.

math-ph↗
arXiv2022

Proof Methods in Random Matrix Theory

In this survey article, we give an introduction to two methods of proof in random matrix theory: The method of moments and the Stieltjes transform method. We thoroughly develop these methods and apply them to show both the semicircle law and the Marchenko-Pastur law for random matrices with independent entries. The material is presented in a pedagogical manner and is suitable for anyone who has followed a course in measure-theoretic probability theory.

math.PR↗
arXiv2022

Random Band and Block Matrices with Correlated Entries

In this paper, we derive limit laws for the empirical spectral distributions of random band and block matrices with correlated entries. In the first part of the paper, we study band matrices with approximately uncorrelated entries. We strengthen previously obtained results while requiring weaker assumptions, which is made possible by a refined application of the method of moments. In the second part of the paper, we introduce a new two-layered correlation structure we call SSB-HKW correlated, which enables the study of structured random matrices with correlated entries. Our results include semicircle laws in probability and almost surely, but we also obtain other limiting spectral distributions depending on the conditions. Simple necessary and sufficient conditions for the limit law to be the semicircle are provided. Our findings strengthen and extend many results already known.

math.PR↗
arXiv2022

The Central Limit Theorem for Weakly Dependent Random Variables by the Moment Method

In this paper, we derive a central limit theorem for collections of weakly correlated random variables indexed by discrete metric spaces, where the correlation decays in the distance of the indices. The correlation structure we study depends solely on the separability of mixed moments. Our investigation yields a new proof for the CLT for $α$-mixing random variables, but also non-$α$-mixing random variables fit within our framework, such as MA($\infty$) processes. In particular, our results can be applied to ARMA($p,q$) process with independent white noise.

math.PR↗
arXiv2021

Optimal Weights in a Two-Tier Voting System with Mean-Field Voters

We analyse two-tier voting systems with voters described by a multi-group mean-field model that allows for correlated voters both within groups as well as across group boundaries. In this model voters are influenced by voters within their group (constituency, member state, etc.) in a positive way. Across group boundaries positive or negative influence is considered. The objective is to determine the optimal weights each group receives in the council, the upper level of the voting system, to minimise the expected quadratic deviation of the council vote from a hypothetical referendum of the overall population in the large population limit. The mean-field model exhibits different behaviour depending on the intensity of interactions between voters. When interaction is weak, we obtain optimal weights given by the sum of a constant term and a term proportional to the square root of the group's population. When interaction is strong, the optimal weights are in general not uniquely determined. Indeed, when all groups are positively coupled, any assignation of weights is optimal. For two competing clusters of groups, the difference in total weights must be a specific number, but the assignation of weights within each cluster is arbitrary. We also obtain conditions for both interaction regimes under which it is impossible to reach the minimal democracy deficit as some of the weights may be negative.

math.PR↗
arXiv2021

Local Semicircle Law for Curie-Weiss Type Ensembles

We derive and compare various forms of local semicircle laws for random matrices with exchangeable entries which exhibit correlations that decay at a very slow rate. In fact, any $l$-point correlation will decay at a rate of $N^{-l/2}$. We call our ensembles \emph{of Curie-Weiss type}, and Curie-Weiss($β$)-distributed entries are admissible as long as $β\leq 1$.

math.PR↗
arXiv2021

Collective Bias Models in Two-Tier Voting Systems and the Democracy Deficit

We analyse optimal voting weights in two-tier voting systems. In our model, the overall population (or union) is split in groups (or member states) of different sizes. The individuals comprising the overall population constitute the first tier, and the council is the second tier. Each group has a representative in the council that casts votes on their behalf. By "optimal weights", we mean voting weights in the council which minimise the democracy deficit, i.e. the expected deviation of the council vote from a (hypothetical) popular vote. We assume that the voters within each group interact via what we call a local collective bias or common belief (through tradition, common values, strong religious beliefs, etc.). We allow in addition an interaction across group borders via a global bias. Thus, the voting behaviour of each voter depends on the behaviour of all other voters. This correlation may be stronger between voters in the same group, but is in general not zero for voters in different groups. We call the respective voting measure a Collective Bias Model (CBM). The "simple CBM" introduced in [12] and in particular the Impartial Culture and the Impartial Anonymous Culture are special cases of our general model. We compute the optimal weights in the large population limit. Those optimal weights are unique as long as there is no "complete" correlation between the groups. In this case, we obtain optimal weights which are the sum of a common constant equal for all groups and a summand which is proportional to the population of each group. We also analyse the conditions under which the optimal weights are negative, thus making it impossible to reach the theoretical minimum of the democracy deficit. This is a new aspect of the model owed to the correlation between votes belonging to different groups.

math.PR↗
arXiv2021

Limit Theorems for Multi-Group Curie-Weiss Models via the Method of Moments

We study a multi-group version of the mean-field or Curie-Weiss spin model. For this model, we show how, analogously to the classical (single-group) model, the three temperature regimes are defined. Then we use the method of moments to determine for each regime how the vector of the group magnetisations behaves asymptotically. Some possible applications to social or political sciences are discussed.

math.PR↗
arXiv2020

Interval Type Local Limit Theorems for Lattice Type Random Variables and Distributions

In this paper, we propose a new interpretation of local limit theorems for univariate and multivariate distributions on lattices. We show that - given a local limit theorem in the standard sense - the distributions are approximated well by the limit distribution, uniformly on intervals of possibly decaying length. We identify the maximally allowable decay speed of the interval lengths. Further, we show that for continuous distributions, the interval type local law holds without any decay speed restrictions on the interval lengths. We show that various examples fit within this framework, such as standardized sums of i.i.d. random vectors or correlated random vectors induced by multidimensional spin models from statistical mechanics.

math.PR↗
arXiv2020

Local Central Limit Theorem for Multi-Group Curie-Weiss Models

We define a multi-group version of the mean-field spin model, also called Curie-Weiss model. It is known that, in the high temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group magnetisations, that is the sum of spins belonging to each group. In this article, we prove a local central limit theorem for the group magnetisations in the high temperature regime.

math.PR↗
arXiv2020

Quantum Lattice Wave Guides with Randomness -- Localisation and Delocalisation

In this paper we consider Schrödinger operators on $M \times \mathbb{Z}^{d_2}$, with $M=\{M_{1}, \ldots, M_{2}\}^{d_1}$ (`quantum wave guides') with a `$Γ$-trimmed' random potential, namely a potential which vanishes outside a subset $Γ$ which is periodic with respect to a sub lattice. We prove that (under appropriate assumptions) for strong disorder these operators have \emph{pure point spectrum } outside the set $Σ_{0}=σ(H_{0,Γ^{c}})$ where $H_{0,Γ^{c}} $ is the free (discrete) Laplacian on the complement $Γ^{c} $ of $Γ$. We also prove that the operators have some \emph{absolutely continuous spectrum} in an energy region $\mathcal{E}\subsetΣ_{0}$. Consequently, there is a mobility edge for such models. We also consider the case $-M_{1}=M_{2}=\infty$, i.~e.~ $Γ$-trimmed operators on $\mathbb{Z}^{d}=\mathbb{Z}^{d_1}\times\mathbb{Z}^{d_2}$. Again, we prove localisation outside $Σ_{0} $ by showing exponential decay of the Green function $G_{E+iη}(x,y) $ uniformly in $η>0 $. For \emph{all} energies $E\in\mathcal{E}$ we prove that the Green's function $G_{E+iη} $ is \emph{not} (uniformly) in $\ell^{1}$ as $η$ approaches $0$. This implies that neither the fractional moment method nor multi scale analysis \emph{can} be applied here.

math-ph↗
arXiv2020

Average Weights and Power in Weighted Voting Games

We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the $k$-th largest player under the uniform distribution. We analyze the average voting power of the $k$-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of $n$ and a general theorem about the functional form of the relation between the average Penrose--Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.

cs.GT↗
arXiv2020

Spectral Statistics for an Anderson Model with sporadic potentials

In this paper we consider an Anderson model with a large number of sites with zero interaction. For such models we study the spectral statistics in the region of complete localization. We show that Poisson statistics holds for such energies, by proving the Minami estimate.

math-ph↗
arXiv2019

The Almost Sure Semicircle Law for Random Band Matrices with Dependent Entries

We analyze the empirical spectral distribution of random periodic band matrices with correlated entries. The correlation structure we study was first introduced in 2015 by Hochstättler, Kirsch and Warzel, who named their setup "almost uncorrelated" and showed convergence to the semicircle distribution in probability. We strengthen their results which turn out to be also valid almost surely. Moreover, we extend them to band matrices. Sufficient conditions for convergence to the semicircle distribution both in probability and almost surely are provided. In contrast to convergence in probability, almost sure convergence seems to require a minimal growth rate for the bandwidth. Examples that fit our general setup include Curie-Weiss distributed, correlated Gaussian, and as a special case, independent entries.

math.PR↗
arXiv2019

Eigenvalue statistics for Schrödinger operators with random point interactions on $\mathbb{R}^d$, $d=1,2,3$

We prove that the local eigenvalue statistics at energy $E$ in the localization regime for Schrödinger operators with random point interactions on $\mathbb{R}^d$, for $d=1,2,3$, is a Poisson point process with the intensity measure given by the density of states at $E$ times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrödinger operators in the continuum. The special structure of resolvent of Schrödinger operators with point interactions facilitates the proof of the Minami estimate for these models.

math-ph↗