Vladimir Al. Osipov——Integrability in random matrix theory and its applications

Release time：2022-08-26Views：565

Title:Integrability in random matrix theory and its applications

Speaker:Vladimir Al. Osipov（Faculty of Sciences, Holon Institute of Technology）

Time:Wednesday, August 31, 2022, 16:00-17:30 (UTC+8).

Online access: Zoom Meeting ID: 882 8540 7533 , Password: 028422,

Link:https://zoom.us/j/88285407533?pwd=NFlsWlZlb2F3c3VmMHBqOXVud1NpQT09

Abstract：Random matrices are widely used to model quantum systems with chaos and disorder. In such models,the observable is expressed as a quantum operator averaged over an ensemble of random matrices with agiven probability measure. In my talk, I demonstrate a general approach, “the random matrix integrabletheory”, to the nonperturbative calculation of the random-matrix integrals. With this approach, the internalsymmetries of the integration measure, expressed in terms of highly non-trivial nonlinear relations for theoriginal integral (the Toda lattice hierarchy, the Kadomtsev-Petviashvili hierarchy) and the relations followingfrom the deformation of the integration measure (Virasoro constraints), are used to represent the integral asa solution of differential equations, where the differentials are taken over the internal (physical) parameters ofthe model [1,2]. This method represents a particular implementation of results obtained within a more generaltheory ofτ-functions. In particular, the central theorem of this theory states the existence of the Toda latticeand Kadomtsev-Petviashvili hierarchies for the typical random-matrix integrals.The particular implementation of the integrable theory will be discussed in the example of the physicalproblem of quantum transport in chaotic cavities [3,4]. A brief introduction to the physics of the problemand the advantage of the integrable theory method for calculation of the conductance cumulants, and ofthe shot-noise-conductance joint cumulants are going to be presented. In particular, we demonstrate howthe conductance cumulant generation function can be expressed in terms of the solution of the PainleveV transcendent equation. In addition, the results of the integrable theory implementation to the averagedrandom-matrix characteristic polynomials [1], and also for the problem of the power spectrum of the eigenlevelsequences in the quantum chaotic system [2,5] will be discussed.

[1] V.Al.Osipov, E.Kanzieper, “Correlations of RMT characteristic polynomials and integrability: Random Hermitian matrices”, Annals of Physics

325 (2010) 2251

[2] R.Riser, V.Al.Osipov, E.Kanzieper, “Nonperturbative theory of power spectrum in complex systems”, Annals of Physics 413 (2020) 168065

[3] V.Al.Osipov, E.Kanzieper, “Integrable theory of quantum transport in chaotic cavities”, Phys.Rev.Let. 101 (2008) 176804

[4] V.Al.Osipov, E.Kanzieper, “Statistics of thermal to shot noise crossover in chaotic cavities”, J.Phys.A:Math.Theor. 42 (2009) 475101

[5] R.Riser, V.Al.Osipov, E.Kanzieper, “Power-spectrum of long eigenlevel sequences in quantum chaology”, Phys.Rev.Let. 118 (2017) 204101

Integrability in random matrix theory and its applications.pdf

More information about the Functional Analysis Seminar can be foundhere.