Theorie zufälliger Matrizen: invariante Ensembles und Matrizen mit abhängigen Einträgen, repulsive Teilchensysteme
Phasenübergänge in Spinsystemen: Ising Modelle mit Blockstruktur und/oder zufälligen Interaktionen
The survey is dedicated to a celebrated series of quantitave results, developed by the Lithuanian school of probability, on the normal approximation for a real-valued random variable. The key ingredient is a bound on cumulants of the type , which is weaker than Cramér’s condition of finite exponential moments. We give a self-contained proof of some of the "main lemmas" in a book by Saulis and Statulevičius (1989), and an accessible introduction to the Cramér-Petrov series. In addition, we explain relations with heavy-tailed Weibull variables, moderate deviations, and mod-phi convergence. We discuss some methods for bounding cumulants such as summability of mixed cumulants and dependency graphs, and briefly review a few recent applications of the method of cumulants for the normal approximation.
@article{cumulants,
title = {The Method of Cumulants for the Normal Approximation},
author = {Döring, Hanna and Jansen, Sabine and Schubert, Kristina},
journal = {Probab. Surv.},
fjournal = {Probability Surveys},
volume = {19},
year = {2022},
pages = {185--270},
mrclass = {60F05 (60F10 60G70)},
mrnumber = {4408127},
doi = {10.1214/22-ps7},
eprint = {2102.01459},
archiveprefix = {arXiv}
}
We continue our analysis of Ising models on the (directed) Erdős-Rényi random graph . We prove a quenched Central Limit Theorem for the magnetization and describe the fluctuations of the log-partition function. In the current note we consider the low temperature regime and the case when an external magnetic field is present. In both cases, we assume that satisfies .
@article{kabluchko2021fluctuations,
title = {Fluctuations of the Magnetization for Ising models on Erdös-Rényi Random Graphs -- the Regimes of Low Temperature and External Magnetic Field},
author = {Kabluchko, Zakhar and Löwe, Matthias and Schubert, Kristina},
journal = {ALEA Lat. Am. J. Probab. Math. Stat.},
fjournal = {ALEA. Latin American Journal of Probability and Mathematical Statistics},
volume = {19},
year = {2022},
number = {1},
pages = {537--563},
mrclass = {82B44 (82B20)},
mrnumber = {4394308},
doi = {10.30757/alea.v19-21},
eprint = {2012.08204},
archiveprefix = {arXiv}
}
We analyze Ising/Curie-Weiss models on the Erdős-Rényi graph with N vertices and edge probability p=p(N) that were introduced by Bovier and Gayrard [J. Statist. Phys., 72(3-4):643–664, 1993] and investigated in two previous articles by the authors. We prove Central Limit Theorems for the partition function of the model and – at other decay regimes of – for the logarithmic partition function. We find critical regimes for at which the behavior of the fluctuations of the partition function changes.
@article{kabluchko2020fluctuations,
title = {Fluctuations for the partition function of Ising models on Erdös-Rényi random graphs},
author = {Kabluchko, Zakhar and Löwe, Matthias and Schubert, Kristina},
journal = {Ann. Inst. H. Poincaré Probab. Statist.},
fjournal = {Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et Statistiques},
publisher = {Institut Henri Poincaré},
volume = {57},
year = {2021},
number = {4},
pages = {2017--2042},
issn = {0246-0203},
mrclass = {60F05 (82B20)},
mrnumber = {4328559},
doi = {10.1214/20-aihp1137},
eprint = {2002.11372},
archiveprefix = {arXiv}
}
We construct and analyze a random graph model for discrete choice with social interaction and several groups of equal size. We concentrate on the case of two groups of equal sizes and we allow the interaction strength within a group to differ from the interaction strength between the two groups. Given that the resulting graph is sufficiently dense we show that, with probability one, the average decision in each of the two groups is the same as in the fully connected model. In particular, we show that there is a phase transition: If the interaction among a group and between the groups is strong enough the average decision per group will either be positive or negative and the decision of the two groups will be correlated. We also compute the free energy per particle in our model.
@article{Ising_binary_choice,
author = {Löwe, Matthias and Schubert, Kristina and Vermet, Franck},
title = {Multi-group Binary Choice with Social Interaction and a Random Communication Structure - a Random Graph Approach},
volume = {556},
pages = {124735},
year = {2020},
month = oct,
day = {15},
issn = {0378-4371},
doi = {10.1016/j.physa.2020.124735},
journal = {Physica A: Statistical Mechanics and its Applications},
archiveprefix = {arXiv},
eprint = {1904.11890}
}
We continue our analysis of Ising models on the (directed) Erdős-Rényi random graph. This graph is constructed on vertices and every edge has probability to be present. These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993] and analyzed by the authors in a previous note, in which we consider the case of satisfying and . In the current note we prove a quenched Central Limit Theorem for the magnetization for satisfying in the high-temperature regime . We also show a non-standard Central Limit Theorem for at the critical temperature . For we obtain a Gaussian limiting distribution for the magnetization. Finally, on the critical line the limiting distribution for the magnetization contains a quadratic component as well as a -term. Hence, at we observe a phase transition in for the fluctuations of the magnetization.
@article{kabluchko2019fluctuations,
title = {Fluctuations of the Magnetization for Ising Models on Erdős-Rényi Random Graphs -- the Regimes of Small p and the Critical Temperature},
author = {Kabluchko, Zakhar and Löwe, Matthias and Schubert, Kristina},
doi = {10.1088/1751-8121/aba05f},
year = {2020},
month = aug,
publisher = {{IOP} Publishing},
volume = {53},
number = {35},
pages = {355004},
journal = {Journal of Physics A: Mathematical and Theoretical},
archiveprefix = {arXiv},
eprint = {1911.10624}
}
We show how to exactly reconstruct the block structure at the critical line in the so-called Ising block model. This model was re-introduced by Berthet, Rigollet and Srivastava in a recent paper. There the authors show how to exactly reconstruct blocks away from the critical line and they give an upper and a lower bound on the number of observations one needs; thereby they establish a minimax optimal rate (up to constants). Our technique relies on a combination of their methods with fluctuation results for block spin Ising models. The latter are extended to the full critical regime. We find that the number of necessary observations depends on whether the interaction parameter between two blocks is positive or negative: In the first case, there are about observations required to exactly recover the block structure, while in the latter observations suffice.
@article{Ising_exact_recovery,
author = {Löwe, Matthias and Schubert, Kristina},
doi = {10.1214/20-EJS1703},
fjournal = {Electronic Journal of Statistics},
journal = {Electron. J. Statist.},
number = {1},
pages = {1796--1815},
publisher = {The Institute of Mathematical Statistics and the Bernoulli Society},
title = {Exact recovery in block spin Ising models at the critical line},
volume = {14},
year = {2020},
eprinttype = {arxiv},
eprint = {1906.00021}
}
We study a block spin mean-field Ising model, i.e. a model of spins in which the vertices are divided into a finite number of blocks with each block having a fixed proportion of vertices, and where pair interactions are given according to their blocks. For the vector of block magnetizations we prove Large Deviation Principles and Central Limit Theorems under general assumptions for the block interaction matrix. Using the exchangeable pair approach of Stein’s method we establish a rate of convergence in the Central Limit Theorem for the block magnetization vector in the high temperature regime.
@article{loeweSinulis,
author = {Knöpfel, Holger and Löwe, Matthias and Schubert, Kristina and Sinulis, Arthur},
title = {Fluctuation Results for General Block Spin Ising Models},
journal = {J. Stat. Phys.},
fjournal = {Journal of Statistical Physics},
volume = {178},
year = {2020},
number = {5},
pages = {1175--1200},
issn = {0022-4715},
mrclass = {60F05 (60F10 82B20)},
mrnumber = {4081224},
doi = {10.1007/s10955-020-02489-0},
eprinttype = {arxiv},
eprint = {1902.02080}
}
We analyze Ising/Curie-Weiss models on the (directed) Erdős-Rényi random graph on vertices in which every edge is present with probability . These models were introduced by Bovier and Gayrard [J. Stat. Phys., 1993]. We prove a quenched Central Limit Theorem for the magnetization in the high-temperature regime when satisfies .
@article{Ising_random_interaction1,
author = {Kabluchko, Zakhar and Löwe, Matthias and Schubert, Kristina},
year = {2019},
title = {Fluctuations of the Magnetization for Ising Models on Dense Erdős–Rényi Random Graphs},
journal = {Journal of Statistical Physics},
issn = {1572-9613},
doi = {10.1007/s10955-019-02358-5},
eprinttype = {arxiv},
eprint = {1905.12326}
}
We discuss the limiting spectral density of real symmetric random matrices. Other than in standard random matrix theory the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic process. Under assumptions on this process, which are satisfied, e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures on finite state spaces, or by Gaussian Markov processes, we show that the limiting spectral distribution depends on the way the matrix is filled with the stochastic process. If the filling is in a certain way compatible with the symmetry condition on the matrix, the limiting law of the empirical eigenvalue distribution is the well known semi-circle law. For other fillings we show that the semi-circle law cannot be the limiting spectral density.
@article{LS,
author = {Löwe, Matthias and Schubert, Kristina},
title = {On the Limiting Spectral Density of Random Matrices filled with Stochastic Processes},
journal = {Random Operators and Stochastic Equations},
eprinttype = {arxiv},
eprint = {1512.02498},
volume = {27},
year = {2019},
number = {2},
doi = {10.1515/rose-2019-2008}
}
We analyze the high temperature fluctuations of the magnetization of the so-called Ising block model. This model was recently introduced by Berthet, Rigollet and Srivastava. We prove a Central Limit Theorems (CLT) for the magnetization in the high temperature regime. At the same time we show that this CLT breaks down at a line of critical temperatures. At this line we show the validity of a non-standard Central Limit Theorems for the magnetization.
@article{loewe2018,
author = {Löwe, Matthias and Schubert, Kristina},
doi = {10.1214/18-ECP161},
fjournal = {Electronic Communications in Probability},
journal = {Electron. Commun. Probab.},
pages = {12 pp.},
pno = {53},
publisher = {The Institute of Mathematical Statistics and the Bernoulli Society},
title = {Fluctuations for block spin Ising models},
volume = {23},
year = {2018},
eprinttype = {arxiv},
eprint = {1806.06000}
}
We consider the empirical eigenvalue distribution of random real symmetric matrices with stochastically independent skew-diagonals and study its limit if the matrix size tends to infinity. We allow correlations between entries on the same skew-diagonal and we distinguish between two types of such correlations, a rather weak and a rather strong one. For weak correlations the limiting distribution is Wigner’s semi-circle distribution; for strong correlations it is the free convolution of the semi-circle distribution and the limiting distribution for random Hankel matrices.
@article{Schubert_Hankel,
author = {Schubert, Kristina},
title = {Spectral Density for Random Matrices with Independent Skew-Diagonals},
doi = {10.1214/16-ECP3},
fjournal = {Electronic Communications in Probability},
journal = {Electron. Commun. Probab.},
pages = {12 pp.},
pno = {40},
publisher = {The Institute of Mathematical Statistics and the Bernoulli Society},
volume = {21},
year = {2016},
eprinttype = {arxiv},
eprint = {1510.06448}
}
We consider the universality of the nearest neighbour eigenvalue spacing distribution in invariant random matrix ensembles. Focussing on orthogonal and symplectic invariant ensembles, we show that the empirical spacing distribution converges in a uniform way. More precisely, the main result states that the expected Kolmogorov distance of the empirical spacing distribution from its universal limit converges to zero as the matrix size tends to infinity.
@article{MR3416165,
author = {Schubert, Kristina},
title = {Spacings in orthogonal and symplectic random matrix ensembles},
journal = {Constr. Approx.},
fjournal = {Constructive Approximation. An International Journal for
Approximations and Expansions},
volume = {42},
year = {2015},
number = {3},
pages = {481--518},
issn = {0176-4276},
mrclass = {15B52},
mrnumber = {3416165},
doi = {10.1007/s00365-015-9274-6},
eprinttype = {arxiv},
eprint = {1501.05637}
}
We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the empirical distribution of nearest neighbor spacings. We extend existing results for the spacing distribution in two ways. On the one hand, we believe the empirical distribution to be of more practical relevance than the so far considered expected distribution. On the other hand, we use the unfolding, a non-linear rescaling, which transforms the ensemble such that the density of particles is asymptotically constant. This allows to consider all empirical spacings, where previous results were restricted to a tiny fraction of the particles. Moreover, we prove bounds on the rates of convergence. The main ingredient for the proof, a strong bulk universality result for correlation functions in the unfolded setting including optimal rates, should be of independent interest.
@article{MR3425540,
author = {Schubert, Kristina and Venker, Martin},
title = {Empirical spacings of unfolded eigenvalues},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {20},
year = {2015},
pages = {Paper No. 120, 37},
issn = {1083-6489},
mrclass = {60B20 (82C22)},
mrnumber = {3425540},
eprinttype = {arxiv},
eprint = {1505.07664},
doi = {10.1214/EJP.v20-4436}
}
The investigation of universality questions for local eigenvalue statistics continues to be a driving force in the theory of Random Matrices. For Matrix Models [53] the method of orthogonal polynomials can be used and the asymptotics of the Christoffel-Darboux kernel [59] become the key for studying universality. In this paper the existing results on the CD-kernel will be extended in two directions. Firstly, in order to analyze the transition from the universal to the non-universal regime, we provide leading order asymptotics that are global rather than local. This allows e.g. to describe the moderate deviations for the largest eigenvalues of unitary ensembles (), where such a transition occurs. Secondly, our asymptotics will be uniform under perturbations of the probability measure that defines the matrix ensemble. Such information is useful for the analysis of a different type of ensembles [25], which is not known to be determinantal and for which the method of orthogonal polynomials cannot be used directly. The just described applications of our results are formulated in this paper but will be proved elsewhere. As a byproduct of our analysis we derive first order corrections for the 1-point correlation functions of unitary ensembles in the bulk. Our proofs are based on the nonlinear steepest descent method [20]. They follow closely [17] and incorporate improvements introduced in [36, 64]. The presentation is self-contained except for a number of general facts from Random Matrix theory and from the theory of singular integral operators.
@article{KSSV,
author = {Kriecherbauer, Thomas and Schubert, Kristina and Schüler, Katharina and Venker, Martin},
title = {Global Asymptotics for the Christoffel-Darboux Kernel of Random Matrix Theory},
journal = {Markov Process. Related Fields},
eprinttype = {arxiv},
eprint = {1401.6772},
volume = {21},
year = {2015},
pages = {639--694},
url = {http://math-mprf.org/journal/articles/id1379/}
}
Universality of local eigenvalue statistics is one of the most striking phenomena of Random Matrix Theory, that also accounts for a lot of the attention that the field has attracted over the past 15 years. In this paper we focus on the empirical spacing distribution and its Kolmogorov distance from the universal limit. We describe new results, some analytical, some numerical, that are contained in [27]. A large part of the paper is devoted to explain basic definitions and facts of Random Matrix Theory, culminating in a sketch of the proof of a weak version of convergence for the empirical spacing distribution.
@incollection{MR3095193,
author = {Kriecherbauer, Thomas and Schubert, Kristina},
title = {Spacings: an example for universality in random matrix theory},
booktitle = {Random matrices and iterated random functions},
series = {Springer Proc. Math. Stat.},
volume = {53},
pages = {45--71},
publisher = {Springer, Heidelberg},
year = {2013},
mrclass = {60B20},
mrnumber = {3095193},
doi = {10.1007/978-3-642-38806-4_3},
eprinttype = {arxiv},
eprint = {1510.06597}
}
One of the most striking phenomena of Random Matrix Theory is that several of their local eigenvalue statistics define universal distributions in the limit of large matrix dimensions. In the first part of this thesis we focus on invariant matrix ensembles including the classical Gaussian ensembles GOE, GUE and GSE. In the main theorem we consider the empirical spacing distribution and show that the expected Kolmogorov distance from the universal limit converges to zero as the matrix size tends to infinity. In the second part of this thesis we present numerical experiments to determine rates of convergence for general beta Hermite and Wigner ensembles.
@thesis{Diss,
author = {Schubert, Kristina},
title = {On the convergence of the nearest neighbour eigenvalue spacing distribution for orthogonal and symplectic ensembles},
school = {Ruhr-Universität Bochum},
address = {Germany},
type = {Dissertation},
year = {2012},
url = {https://hss-opus.ub.ruhr-uni-bochum.de/opus4/frontdoor/index/index/year/2019/docId/3280}
}
@thesis{Dipl,
author = {Schubert, Kristina},
title = {Über die Abweichung der Abstandsverteilung benachbarter Eigenwerte zufälliger Matrizen vom universellen Gesetz},
school = {Ruhr-Universität Bochum},
address = {Germany},
type = {Diplomarbeit},
year = {2007},
url = {https://www.kristina-schubert.de/dl/dipl.pdf}
}
Fachbereich Mathematik/Informatik
04/2021–09/2021Fachbereich Mathematik/Informatik
10/2020–03/2021Fakultät für Mathematik, Lehrstuhl für Stochastik und Analysis
since 10/2018Institut für Mathematische Stochastik
04/2013–09/2018Mathematisches Institut, Lehrstuhl Nichtlineare Analysis und Mathematische Physik
04/2011–03/2013Fakultät für Mathematik. Dissertation: On the Convergence of the Nearest Neighbour Eigenvalue Spacing Distribution for Orthogonal and Symplectic Ensembles
06/2012Fakultät für Mathematik, Lehrstuhl für Analysis
07/2007–03/2011Fakultät für Mathematik. Diplomarbeit: Über die Abweichung der Abstandsverteilung benachbarter Eigenwerte zufälliger Matrizen vom universellen Gesetz
04/2007