@article{6086,
  abstract     = {We show that linear analytic cocycles where all Lyapunov exponents are negative infinite are nilpotent. For such one-frequency cocycles we show that they can be analytically conjugated to an upper triangular cocycle or a Jordan normal form. As a consequence, an arbitrarily small analytic perturbation leads to distinct Lyapunov exponents. Moreover, in the one-frequency case where the th Lyapunov exponent is finite and the st negative infinite, we obtain a simple criterion for domination in which case there is a splitting into a nilpotent part and an invertible part.},
  author       = {Sadel, Christian and Xu, Disheng},
  journal      = {Ergodic Theory and Dynamical Systems},
  number       = {4},
  pages        = {1082--1098},
  publisher    = {Cambridge University Press},
  title        = {{Singular analytic linear cocycles with negative infinite Lyapunov exponents}},
  doi          = {10.1017/etds.2017.52},
  volume       = {39},
  year         = {2019},
}

@inproceedings{8175,
  abstract     = {We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik--Deift--Johansson and Baik--Rains in resolving Ulam's problem on longest increasing subsequences of random permutations and the last passage percolation (corner growth) discrete versions thereof. Moreover they interpolate between said measures and the uniform measure on partitions. In the new KPZ-like 1/3 exponent edge scaling limit with logarithmic corrections, we find new probability distributions generalizing the classical Tracy--Widom GUE, GOE and GSE distributions from the theory of random matrices.},
  author       = {Betea, Dan and Bouttier, Jérémie and Nejjar, Peter and Vuletíc, Mirjana},
  booktitle    = {Proceedings on the 31st International Conference on Formal Power Series and Algebraic Combinatorics},
  location     = {Ljubljana, Slovenia},
  publisher    = {Formal Power Series and Algebraic Combinatorics},
  title        = {{New edge asymptotics of skew Young diagrams via free boundaries}},
  year         = {2019},
}

@article{6511,
  abstract     = {Let U and V be two independent N by N random matrices that are distributed according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly, in the limit of large N, to a deterministic measure which is supported on a single ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior of the single ring, we establish the convergence of the empirical eigenvalue distribution on the optimal local scale of order N−1/2+ε and establish the optimal convergence rate. The same results hold true when U and V are Haar distributed on O(N).},
  author       = {Bao, Zhigang and Erdös, László and Schnelli, Kevin},
  issn         = {0091-1798},
  journal      = {Annals of Probability},
  number       = {3},
  pages        = {1270--1334},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Local single ring theorem on optimal scale}},
  doi          = {10.1214/18-AOP1284},
  volume       = {47},
  year         = {2019},
}

@article{6843,
  abstract     = {The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space Wp(X), where S is a countable discrete metric space and 0<p<∞ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of X×(0,1]-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of Wp(X) splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that Wp(X) is isometrically rigid for all 0<p<∞.},
  author       = {Gehér, György Pál and Titkos, Tamás and Virosztek, Daniel},
  issn         = {1096-0813},
  journal      = {Journal of Mathematical Analysis and Applications},
  number       = {2},
  publisher    = {Elsevier},
  title        = {{On isometric embeddings of Wasserstein spaces – the discrete case}},
  doi          = {10.1016/j.jmaa.2019.123435},
  volume       = {480},
  year         = {2019},
}

@inproceedings{7035,
  abstract     = {The aim of this short note is to expound one particular issue that was discussed during the talk [10] given at the symposium ”Researches on isometries as preserver problems and related topics” at Kyoto RIMS. That is,  the role of Dirac masses by  describing  the  isometry group of various metric spaces  of probability  measures.   This  article  is  of  survey  character,  and  it  does  not  contain  any  essentially  new results.From an isometric point of view, in some cases, metric spaces of measures are similar to C(K)-type function  spaces.   Similarity  means  here  that  their  isometries  are  driven  by  some  nice  transformations of  the  underlying  space.   Of  course,  it  depends  on  the  particular  choice  of  the  metric  how  nice  these transformations should be.  Sometimes, as we will see, being a homeomorphism is enough to generate an isometry.  But sometimes we need more:  the transformation must preserve the underlying distance as well.  Statements claiming that isometries in questions are necessarily induced by homeomorphisms are called Banach-Stone-type results, while results asserting that the underlying transformation is necessarily an isometry are termed as isometric rigidity results.As  Dirac  masses  can  be  considered  as  building  bricks  of  the  set  of  all  Borel  measures,  a  natural question arises:Is it enough to understand how an isometry acts on the set of Dirac masses?  Does this action extend uniquely to all measures?In what follows, we will thoroughly investigate this question.},
  author       = {Geher, Gyorgy Pal and Titkos, Tamas and Virosztek, Daniel},
  booktitle    = {Kyoto RIMS Kôkyûroku},
  location     = {Kyoto, Japan},
  pages        = {34--41},
  publisher    = {Research Institute for Mathematical Sciences, Kyoto University},
  title        = {{Dirac masses and isometric rigidity}},
  volume       = {2125},
  year         = {2019},
}

@article{72,
  abstract     = {We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition having density ρ on ℤ− and λ on ℤ+, and a second class particle initially at the origin. For ρ&lt;λ, there is a shock and the second class particle moves with speed 1−λ−ρ. For large time t, we show that the position of the second class particle fluctuates on a t1/3 scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time t.},
  author       = {Ferrari, Patrick and Ghosal, Promit and Nejjar, Peter},
  issn         = {0246-0203},
  journal      = {Annales de l'institut Henri Poincare (B) Probability and Statistics},
  number       = {3},
  pages        = {1203--1225},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Limit law of a second class particle in TASEP with non-random initial condition}},
  doi          = {10.1214/18-AIHP916},
  volume       = {55},
  year         = {2019},
}

@article{7423,
  abstract     = {We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices introduced by Borodin and Péché, second, the product of two independent random matrices where one has correlated entries, and third, the case when the two random matrices become also coupled through a fixed matrix. The singular value statistics of all three ensembles is shown to be determinantal and we derive double contour integral representations for their respective kernels. Three different kernels are found in the limit of infinite matrix dimension at the origin of the spectrum. They depend on finite rank perturbations of the correlation and coupling matrices and are shown to be integrable. The first kernel (I) is found for two independent matrices from the second, and two weakly coupled matrices from the third ensemble. It generalises the Meijer G-kernel for two independent and uncorrelated matrices. The third kernel (III) is obtained for the generalised Wishart ensemble and for two strongly coupled matrices. It further generalises the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices, provides an interpolation between the kernels (I) and (III), generalising previous findings of part of the authors.},
  author       = {Akemann, Gernot and Checinski, Tomasz and Liu, Dangzheng and Strahov, Eugene},
  issn         = {0246-0203},
  journal      = {Annales de l'Institut Henri Poincaré, Probabilités et Statistiques},
  number       = {1},
  pages        = {441--479},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles}},
  doi          = {10.1214/18-aihp888},
  volume       = {55},
  year         = {2019},
}

@article{405,
  abstract     = {We investigate the quantum Jensen divergences from the viewpoint of joint convexity. It turns out that the set of the functions which generate jointly convex quantum Jensen divergences on positive matrices coincides with the Matrix Entropy Class which has been introduced by Chen and Tropp quite recently.},
  author       = {Virosztek, Daniel},
  journal      = {Linear Algebra and Its Applications},
  pages        = {67--78},
  publisher    = {Elsevier},
  title        = {{Jointly convex quantum Jensen divergences}},
  doi          = {10.1016/j.laa.2018.03.002},
  volume       = {576},
  year         = {2019},
}

@article{429,
  abstract     = {We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.},
  author       = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
  issn         = {1432-2064},
  journal      = {Probability Theory and Related Fields},
  number       = {1-2},
  pages        = {293–373},
  publisher    = {Springer},
  title        = {{Stability of the matrix Dyson equation and random matrices with correlations}},
  doi          = {10.1007/s00440-018-0835-z},
  volume       = {173},
  year         = {2019},
}

@article{10879,
  abstract     = {We study effects of a bounded and compactly supported perturbation on multidimensional continuum random Schrödinger operators in the region of complete localisation. Our main emphasis is on Anderson orthogonality for random Schrödinger operators. Among others, we prove that Anderson orthogonality does occur for Fermi energies in the region of complete localisation with a non-zero probability. This partially confirms recent non-rigorous findings [V. Khemani et al., Nature Phys. 11 (2015), 560–565]. The spectral shift function plays an important role in our analysis of Anderson orthogonality. We identify it with the index of the corresponding pair of spectral projections and explore the consequences thereof. All our results rely on the main technical estimate of this paper which guarantees separate exponential decay of the disorder-averaged Schatten p-norm of χa(f(H)−f(Hτ))χb in a and b. Here, Hτ is a perturbation of the random Schrödinger operator H, χa is the multiplication operator corresponding to the indicator function of a unit cube centred about a∈Rd, and f is in a suitable class of functions of bounded variation with distributional derivative supported in the region of complete localisation for H.},
  author       = {Dietlein, Adrian M and Gebert, Martin and Müller, Peter},
  issn         = {1664-039X},
  journal      = {Journal of Spectral Theory},
  keywords     = {Random Schrödinger operators, spectral shift function, Anderson orthogonality},
  number       = {3},
  pages        = {921--965},
  publisher    = {EMS Press},
  title        = {{Perturbations of continuum random Schrödinger operators with applications to Anderson orthogonality and the spectral shift function}},
  doi          = {10.4171/jst/267},
  volume       = {9},
  year         = {2019},
}

@article{6240,
  abstract     = {For a general class of large non-Hermitian random block matrices X we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of X as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.},
  author       = {Alt, Johannes and Erdös, László and Krüger, Torben H and Nemish, Yuriy},
  issn         = {0246-0203},
  journal      = {Annales de l'Institut Henri Poincaré, Probabilités et Statistiques},
  number       = {2},
  pages        = {661--696},
  publisher    = {Institut Henri Poincaré},
  title        = {{Location of the spectrum of Kronecker random matrices}},
  doi          = {10.1214/18-AIHP894},
  volume       = {55},
  year         = {2019},
}

@phdthesis{6179,
  abstract     = {In the first part of this thesis we consider large random matrices with arbitrary expectation and a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.
In the second part we consider Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are uni- versal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner- Dyson-Mehta universality conjecture for the last remaining universality type. Our analysis holds not only for exact cusps, but approximate cusps as well, where an ex- tended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow- nian motion to the cusp regime.
In the third and final part we explore the entrywise linear statistics of Wigner ma- trices and identify the fluctuations for a large class of test functions with little regularity. This enables us to study the rectangular Young diagram obtained from the interlacing eigenvalues of the random matrix and its minor, and we find that, despite having the same limit, the fluctuations differ from those of the algebraic Young tableaux equipped with the Plancharel measure.},
  author       = {Schröder, Dominik J},
  issn         = {2663-337X},
  pages        = {375},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{From Dyson to Pearcey: Universal statistics in random matrix theory}},
  doi          = {10.15479/AT:ISTA:th6179},
  year         = {2019},
}

@article{6182,
  abstract     = {We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields 173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is
a systematic diagrammatic control of a multivariate cumulant expansion.},
  author       = {Erdös, László and Krüger, Torben H and Schröder, Dominik J},
  issn         = {2050-5094},
  journal      = {Forum of Mathematics, Sigma},
  publisher    = {Cambridge University Press},
  title        = {{Random matrices with slow correlation decay}},
  doi          = {10.1017/fms.2019.2},
  volume       = {7},
  year         = {2019},
}

@article{6186,
  abstract     = {We prove that the local eigenvalue statistics of real symmetric Wigner-type
matrices near the cusp points of the eigenvalue density are universal. Together
with the companion paper [arXiv:1809.03971], which proves the same result for
the complex Hermitian symmetry class, this completes the last remaining case of
the Wigner-Dyson-Mehta universality conjecture after bulk and edge
universalities have been established in the last years. We extend the recent
Dyson Brownian motion analysis at the edge [arXiv:1712.03881] to the cusp
regime using the optimal local law from [arXiv:1809.03971] and the accurate
local shape analysis of the density from [arXiv:1506.05095, arXiv:1804.07752].
We also present a PDE-based method to improve the estimate on eigenvalue
rigidity via the maximum principle of the heat flow related to the Dyson
Brownian motion.},
  author       = {Cipolloni, Giorgio and Erdös, László and Krüger, Torben H and Schröder, Dominik J},
  issn         = {2578-5885},
  journal      = {Pure and Applied Analysis },
  number       = {4},
  pages        = {615–707},
  publisher    = {MSP},
  title        = {{Cusp universality for random matrices, II: The real symmetric case}},
  doi          = {10.2140/paa.2019.1.615},
  volume       = {1},
  year         = {2019},
}

@article{284,
  abstract     = {Borel probability measures living on metric spaces are fundamental
mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are interested in the description of the structure of the isometries of such metric spaces. We overview some of the recent results of the topic and we also provide some new ones concerning the Wasserstein distance. More specifically, we consider the space of all Borel probability measures on the unit sphere of a Euclidean space endowed with the Wasserstein metric W_p for arbitrary p &gt;= 1, and we show that the action of a Wasserstein isometry on the set of Dirac measures is induced by an isometry of the underlying unit sphere.},
  author       = {Virosztek, Daniel},
  issn         = {2064-8316},
  journal      = {Acta Scientiarum Mathematicarum},
  number       = {1-2},
  pages        = {65 -- 80},
  publisher    = {Springer Nature},
  title        = {{Maps on probability measures preserving certain distances - a survey and some new results}},
  doi          = {10.14232/actasm-018-753-y},
  volume       = {84},
  year         = {2018},
}

@article{556,
  abstract     = {We investigate the free boundary Schur process, a variant of the Schur process introduced by Okounkov and Reshetikhin, where we allow the first and the last partitions to be arbitrary (instead of empty in the original setting). The pfaffian Schur process, previously studied by several authors, is recovered when just one of the boundary partitions is left free. We compute the correlation functions of the process in all generality via the free fermion formalism, which we extend with the thorough treatment of “free boundary states.” For the case of one free boundary, our approach yields a new proof that the process is pfaffian. For the case of two free boundaries, we find that the process is not pfaffian, but a closely related process is. We also study three different applications of the Schur process with one free boundary: fluctuations of symmetrized last passage percolation models, limit shapes and processes for symmetric plane partitions and for plane overpartitions.},
  author       = {Betea, Dan and Bouttier, Jeremie and Nejjar, Peter and Vuletic, Mirjana},
  issn         = {1424-0637},
  journal      = {Annales Henri Poincare},
  number       = {12},
  pages        = {3663--3742},
  publisher    = {Springer Nature},
  title        = {{The free boundary Schur process and applications I}},
  doi          = {10.1007/s00023-018-0723-1},
  volume       = {19},
  year         = {2018},
}

@unpublished{6183,
  abstract     = {We study the unique solution $m$ of the Dyson equation \[ -m(z)^{-1} = z - a
+ S[m(z)] \] on a von Neumann algebra $\mathcal{A}$ with the constraint
$\mathrm{Im}\,m\geq 0$. Here, $z$ lies in the complex upper half-plane, $a$ is
a self-adjoint element of $\mathcal{A}$ and $S$ is a positivity-preserving
linear operator on $\mathcal{A}$. We show that $m$ is the Stieltjes transform
of a compactly supported $\mathcal{A}$-valued measure on $\mathbb{R}$. Under
suitable assumptions, we establish that this measure has a uniformly
$1/3$-H\"{o}lder continuous density with respect to the Lebesgue measure, which
is supported on finitely many intervals, called bands. In fact, the density is
analytic inside the bands with a square-root growth at the edges and internal
cubic root cusps whenever the gap between two bands vanishes. The shape of
these singularities is universal and no other singularity may occur. We give a
precise asymptotic description of $m$ near the singular points. These
asymptotics generalize the analysis at the regular edges given in the companion
paper on the Tracy-Widom universality for the edge eigenvalue statistics for
correlated random matrices [arXiv:1804.07744] and they play a key role in the
proof of the Pearcey universality at the cusp for Wigner-type matrices
[arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band
mass formula from [arXiv:1804.07744] to the von Neumann algebra setting by
showing that the spectral mass of the bands is topologically rigid under
deformations and we conclude that these masses are quantized in some important
cases.},
  author       = {Alt, Johannes and Erdös, László and Krüger, Torben H},
  booktitle    = {arXiv},
  title        = {{The Dyson equation with linear self-energy: Spectral bands, edges and  cusps}},
  doi          = {10.48550/arXiv.1804.07752},
  year         = {2018},
}

@article{690,
  abstract     = {We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than N−2/3 with a deterministic shift of order (Np)−1.},
  author       = {Lee, Jii and Schnelli, Kevin},
  journal      = {Probability Theory and Related Fields},
  number       = {1-2},
  publisher    = {Springer},
  title        = {{Local law and Tracy–Widom limit for sparse random matrices}},
  doi          = {10.1007/s00440-017-0787-8},
  volume       = {171},
  year         = {2018},
}

@article{70,
  abstract     = {We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT−1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima→∞limT→∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several airy processes.},
  author       = {Nejjar, Peter},
  issn         = {1980-0436},
  journal      = {Latin American Journal of Probability and Mathematical Statistics},
  number       = {2},
  pages        = {1311--1334},
  publisher    = {Instituto Nacional de Matematica Pura e Aplicada},
  title        = {{Transition to shocks in TASEP and decoupling of last passage times}},
  doi          = {10.30757/ALEA.v15-49},
  volume       = {15},
  year         = {2018},
}

@article{181,
  abstract     = {We consider large random matrices X with centered, independent entries but possibly di erent variances. We compute the normalized trace of f(X)g(X∗) for f, g functions analytic on the spectrum of X. We use these results to compute the long time asymptotics for systems of coupled di erential equations with random coe cients. We show that when the coupling is critical, the norm squared of the solution decays like t−1/2.},
  author       = {Erdös, László and Krüger, Torben H and Renfrew, David T},
  journal      = {SIAM Journal on Mathematical Analysis},
  number       = {3},
  pages        = {3271 -- 3290},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Power law decay for systems of randomly coupled differential equations}},
  doi          = {10.1137/17M1143125},
  volume       = {50},
  year         = {2018},
}