Rank one HCIZ at high temperature: interpolating between classical and free convolutions

We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to cNβ2→c, called the high-temperature regime and show that it can be used to construct a promising one-parameter interpolation family, with parameter c between the classical and the free convolution. This c-convolution has a simple interpretation in terms of another associated family of distribution indexed by c, called the Markov-Krein transform: the c-convolution of two distributions corresponds to the classical convolution of their Markov-Krein transforms. We derive first cumulant-moment relations, a central limit theorem, a Poisson limit theorem and show several numerical examples of c-convoluted distributions.

Reconstruction of Differential Operators with Frozen Argument

We study spectral properties of a wide class of differential operators with frozen arguments by putting them into a general framework of rank-one perturbation theory. In particular, we give a complete characterization of possible eigenvalues for these operators and solve the inverse spectral problem of reconstructing the perturbation from the resulting spectrum. This approach provides a unified treatment of several recent studies and gives a clear explanation and interpretation of the obtained results.

Singularly perturbed rank one linear operators

The basic principles of the theory of singularly perturbed self-adjoint operatorsare generalized to the case of closed linear operators with non-symmetric perturbation of rank one.Namely, firstly linear closed operators are considered that coincide with each other on a dense set in a Hilbert space.The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $\delta$-function.Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. The main facts of the theory include the definition of a singularly perturbed linear operator and the resolvent formula in the cases of ${\mathcal H}_{-1}$-class and ${\mathcal H}_{-2}$-class.The paper additionally describes the possibility of the appearance a point of the point spectrum and the construction of a perturbation with a predetermined point.In comparison with self-adjoint perturbations, the description of perturbations by non-symmetric terms is unexpected.Namely, in some cases, when the perturbed by a vectors from ${\mathcal H}_{-2}$ operator can be conveniently described by methods of class ${\mathcal H}_{-1}$, that is impossible in the case of symmetric perturbations of a self-adjoint operator. The perturbation of self-adjoint operators in a non-symmetric manner fully fits into the proposed studies.Such operators, for example, generalize models with nonlocal interactions, perturbations of the harmonic oscillator by the $\delta$-potentials, and can be used to study perturbations generated by a delay or an anticipation.

Truncated Local Uniformization of Formal Integrable Differential Forms

We prove the existence of Local Uniformization for rational codimension one foliations along rational rank one valuations, in any ambient dimension. This result is consequence of the Truncated Local Uniformization of integrable formal differential 1-forms, that we also state and prove in the paper. Thanks to the truncated approach, we perform a classical inductive procedure, based both in the control of the Newton Polygon and in the possibility of avoiding accumulations of values, given by the existence of suitable Tschirnhausen transformations.

Emergent fractal phase in energy stratified random models

We study the effects of partial correlations in kinetic hopping terms of long-range disordered random matrix models on their localization properties. We consider a set of models interpolating between fully-localized Richardson’s model and the celebrated Rosenzweig-Porter model (with implemented translation-invariant symmetry). In order to do this, we propose the energy-stratified spectral structure of the hopping term allowing one to decrease the range of correlations gradually. We show both analytically and numerically that any deviation from the completely correlated case leads to the emergent non-ergodic delocalization in the system unlike the predictions of localization of cooperative shielding. In order to describe the models with correlated kinetic terms, we develop the generalization of the Dyson Brownian motion and cavity approaches basing on stochastic matrix process with independent rank-one matrix increments and examine its applicability to the above set of models.