Towards a full solution of the large N double-scaled SYK model

We compute the exact, all energy scale, 4-point function of the large $N$ double-scaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL$(2)$, that generalizes the Schwarzian to the complete range of energies.Comment: 52+28 pages, 14 figures; v2: references revised, typos corrected, changed normalization of SL(2)_q 6j symbo

Nonintersecting Brownian excursions

We consider the process of $n$ Brownian excursions conditioned to be nonintersecting. We show the distribution functions for the top curve and the bottom curve are equal to Fredholm determinants whose kernel we give explicitly. In the simplest case, these determinants are expressible in terms of Painlev\'{e} V functions. We prove that as $n\to \infty$, the distributional limit of the bottom curve is the Bessel process with parameter 1/2. (This is the Bessel process associated with Dyson's Brownian motion.) We apply these results to study the expected area under the bottom and top curves.Comment: Published at http://dx.doi.org/10.1214/105051607000000041 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Integrability in Random Two-Matrix Models under Finite-Rank Perturbations

Checinski T. Integrability in Random Two-Matrix Models under Finite-Rank Perturbations. Bielefeld: Universität Bielefeld; 2019.In Quantum Chromodynamics low energy spectral properties of the Dirac operator can be described by random matrix ensembles. In time-series analysis strong statistical fluctuations coincide with eigenvalue statistics of random matrices. These two completely different fields share the same type of random matrix ensembles: chiral symmetric random matrices. The analysis of two random-matrix models of this type is presented: the product of two coupled Wishart matrices and the sum of two independent Wishart matrices. Here, we expose the integrability of these models and compute quantities being of interest in Quantum Chromodynamics and in time- series analysis, respectively

Chiral Random Matrix Theory: Generalizations and Applications

Kieburg M. Chiral Random Matrix Theory: Generalizations and Applications. Bielefeld: Fakultät für Physik; 2015

Non-intersecting squared Bessel paths at a hard-edge tacnode

The squared Bessel process is a 1-dimensional diffusion process related to the squared norm of a higher dimensional Brownian motion. We study a model of $n$ non-intersecting squared Bessel paths, with all paths starting at the same point $a>0$ at time $t=0$ and ending at the same point $b>0$ at time $t=1$. Our interest lies in the critical regime $ab=1/4$, for which the paths are tangent to the hard edge at the origin at a critical time $t^*\in (0,1)$. The critical behavior of the paths for $n\to\infty$ is studied in a scaling limit with time $t=t^*+O(n^{-1/3})$ and temperature $T=1+O(n^{-2/3})$. This leads to a critical correlation kernel that is defined via a new Riemann-Hilbert problem of size $4\times 4$. The Riemann-Hilbert problem gives rise to a new Lax pair representation for the Hastings-McLeod solution to the inhomogeneous Painlev\'e II equation $q"(x) = xq(x)+2q^3(x)-\nu,$ where $\nu=\alpha+1/2$ with $\alpha>-1$ the parameter of the squared Bessel process. These results extend our recent work with Kuijlaars and Zhang \cite{DKZ} for the homogeneous case $\nu = 0$.Comment: 54 pages, 13 figures. Corrected error in Theorem 2.

WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations

WavePacket is an open-source program package for the numerical simulation of quantum-mechanical dynamics. It can be used to solve time-independent or time-dependent linear Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semiclassical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry.The graphical capabilities allow visualization of quantum dynamics 'on the fly', including Wigner phase space representations. Being easy to use and highly versatile, WavePacket is well suited for the teaching of quantum mechanics as well as for research projects in atomic, molecular and optical physics or in physical or theoretical chemistry.The present Part I deals with the description of closed quantum systems in terms of Schr\"odinger equations. The emphasis is on discrete variable representations for spatial discretization as well as various techniques for temporal discretization.The upcoming Part II will focus on open quantum systems and dimension reduction; it also describes the codes for optimal control of quantum dynamics.The present work introduces the MATLAB version of WavePacket 5.2.1 which is hosted at the Sourceforge platform, where extensive Wiki-documentation as well as worked-out demonstration examples can be found