Theory of Random Determinants

The chiral condensate in QCD at zero temperature does not depend on the quark chemical potential (up to one third the nucleon mass), whereas the spectral density of the Dirac operator shows a strong dependence on the chemical potential. The cancellations which make this possible also occur on the microscopic scale, where they can be investigated by means of a random matrix model. We show that they can be understood in terms of orthogonality properties of orthogonal polynomials. In the strong non-Hermiticity limit they are related to integrability properties of the spectral density. As a by-product we find exact analytical expressions for the partially quenched chiral condensate in the microscopic domain at nonzero chemical potential.

show abstract

“…Inserting the finite N one flavor partition function (23) in (1) we immediately find that the chiral condensate is given by…”

Section: The µ-Independence Of the Microscopic Condensatementioning

confidence: 99%

“…Such a representation of the eigenvalue density in the complex plane is well known [23,24]; the function F is known as the partially quenched condensate…”

Section: The Spectral Density As a Total Derivativementioning

confidence: 99%

Chiral condensate at nonzero chemical potential in the microscopic limit of QCD

Osborn

Splittorff²,

Verbaarschot

2008

Phys. Rev. D

View full text Add to dashboard Cite

show abstract

“…The quenched spectral density is given by the replica limit [80,81,82] 13) with generating function given by…”

Section: 31mentioning

confidence: 99%

Qcd, Chiral Random Matrix Theoryand Integrability

Verbaarschot

2006

NATO Science Series II: Mathematics, Physics and Chemistry

View full text Add to dashboard Cite

SummaryRandom Matrix Theory has been a unifying approach in physics and mathematics. In these lectures we discuss applications of Random Matrix Theory to QCD and emphasize underlying integrable structures. In the first lecture we give an overview of QCD, its low-energy limit and the microscopic limit of the Dirac spectrum which, as we will see in the second lecture, can be described by chiral Random Matrix Theory. The main topic of the third lecture is the recent developments on the relation between the QCD partition function and integrable hierarchies (in our case the Toda lattice hierarchy). This is an efficient way to obtain the QCD Dirac spectrum from the low energy limit of the QCD partition function. Finally, we will discuss the QCD Dirac spectrum at nonzero chemical potential. We will show that the microscopic spectral density is given by the replica limit of the Toda lattice equation. Recent results by Osborn on the Dirac spectrum of full QCD will be discussed.

show abstract

“…14 In general, for a random matrix with i.i.d. standard normal distribution, Girko's circle law [62] tells us that all eigenvalues are uniformly distributed on the unit disk in the complex plane. To combine Girko's circle law with results presented here to obtain a sharper estimate of the stability regions would be interesting.…”

Section: Synchronization In Time-delayed Networkmentioning

confidence: 99%

Synchronization in networks with random interactions: Theory and applications

Feng

Jirsa²,

Ding³

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

Synchronization is an emergent property in networks of interacting dynamical elements. Here we review some recent results on synchronization in randomly coupled networks. Asymptotical behavior of random matrices is summarized and its impact on the synchronization of network dynamics is presented. Robert May's results on the stability of equilibrium points in linear dynamics are first extended to nonlinear systems where the synchronized dynamics can be periodic or chaotic and then to systems with time delayed coupling. Finally, applications of our results to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included.Behaviour and structures of interacting dynamical systems are complex. Most research on complex networks has been focused so far on networks with deterministic interactions. In this paper we study the behaviour of networks with random interactions. We extend Robert May's results on the stability of equilibrium points in linear dynamics to nonlinear systems where the synchronized dynamics can be periodic or chaotic and then to systems with time delayed coupling. Results are then applied to networks in Neuroscience.

show abstract

Theory of Random Determinants

Cited by 276 publications

References 0 publications

Chiral condensate at nonzero chemical potential in the microscopic limit of QCD

Chiral condensate at nonzero chemical potential in the microscopic limit of QCD

Qcd, Chiral Random Matrix Theoryand Integrability

Synchronization in networks with random interactions: Theory and applications

Contact Info

Product

Resources

About