357 research outputs found
Integrable structure of Ginibre's ensemble of real random matrices and a Pfaffian integration theorem
No full textIn the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95, 230201 (2005)], an exact solution was reported for the probability p_{n,k} to find exactly k real eigenvalues in the spectrum of an nxn real asymmetric matrix drawn at
random from Ginibre's Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the
key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a
GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined
Statistical QCD with non-positive measure
No full textTalk given at the Fine Theoretical Physics Institute (FTPI) workshop, Continuous Advances in QCD (CAQCD-08), held May 15-18, 2008.Splittorff, Kim; Verbaarschot, Jac; Osborne, James; Akemann, Gernot. (2008). Statistical QCD with non-positive measure. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/42205
The solution of a chiral random matrix model with complex eigenvalues
No full textWe describe in detail the solution of the extension of the chiral Gaussian unitary ensemble (chGUE) into the complex plane. The correlation functions of the model are first calculated for a finite number of N complex eigenvalues, where we exploit the existence of orthogonal Laguerre polynomials in the complex plane. When taking the large-N limit we derive new correlation functions in the case of weak and strong non-Hermiticity, thus describing the transition from the chGUE to a generalized Ginibre ensemble. We briefly discuss applications to the Dirac operator eigenvalue spectrum in quantum chromodynamics with non-vanishing chemical potential
Effective theories of finite volume QCD
No full textFinite volume QCD close to the chiral limit cannot be described by chiral Perturbation Theory using the usual p-expansion when the correlation length of pions becomes larger than the size of the box.
An alternative approach to this problem was proposed by Gasser and Leutwyler in 1987, it is referred to as -expansion. In 1993 Shuryak and Verbaarschot conjectured that the spectral properties of the leading order of this alternative expansion were shared with a simpler theory called chiral Random Matrix Theory. In the following years this equivalence was widely used.
In the first part of this work we prove this equivalence for any value of masses and for both zero and non-zero chemical potential. In particular the equivalence of all the low energy spectral properties imply the equivalence of the individual eigenvalue distributions, which are particularly useful to determine low energy constants from Lattice QCD with chiral fermions.
In the second part, working in -expansion with an accuracy up to the next to the leading order, we determine the volume and mass dependence of scalar and pseudoscalar two-point functions in -flavour QCD, in the presence of an isospin chemical potential. Thanks to the non-vanishing chemical potential these correlation functions show a dependence on both chiral condensate and pion decay constant already at leading order
Microscopic universality of complex matrix model correlation functions at weak non-Hermiticity
No full textThe microscopic correlation functions of non-chiral random matrix models with complex eigenvalues are analyzed for a wide class of non-Gaussian measures. In the large-N limit of weak non-Hermiticity, where N is the size of the complex matrices, we can prove that all k-point correlation functions including an arbitrary number of Dirac mass terms are universal close to the origin. To this aim we establish the universality of the asymptotics of orthogonal polynomials in the complex plane. The universality of the correlation functions then follows from that of the kernel of orthogonal polynomials and a mapping of massive to massless correlators
Chiral random two-matrix theory and QCD with imaginary chemical potential
No full textWe summarise recent results for the chiral Random Two-Matrix Theory constructed to describe QCD in the epsilon-regime with imaginary chemical potential. The virtue of this theory is that unquenched Lattice simulations can be used to determine both low energy constants Sigma and F in the leading order chiral Lagrangian, due to their respective coupling to quark mass and chemical potential. We briefly recall the analytic formulas for all density and individual eigenvalue correlations and then illustrate them in detail in the simplest, quenched case with imaginary isospin chemical potential. Some peculiarities are pointed out for this example: i) the factorisation of density and individual eigenvalue correlation functions for large chemical potential and ii) the factorisation of the non-Gaussian weight function of bi-orthogonal polynomials into Gaussian weights with ordinary orthogonal polynomials
Finite volume corrections to LECs in Wilson and staggered ChPT
No full textAkemann G, Pucci F. Finite volume corrections to LECs in Wilson and staggered ChPT. In: PoS(Lattice 2012) 264. 2012.We study the simultaneous effect of finite volume and finite lattice spacingcorrections in the framework of chiral perturbation theory (ChPT) in theepsilon regime, for both the Wilson and staggered formulations. In particularthe finite volume corrections to the low energy constants (LECs) in Wilson andstaggered ChPT are computed to next-to-leading order (NLO) in the\epsilon-expansion. For Wilson with N_f = 2 flavours and staggered with genericN_f the partition function at NLO can be rewritten as the LO partition functionwith renormalized effective LECs
Hole probabilities and overcrowding estimates for products of complex Gaussian matrices
No full textAkemann G, Strahov E. Hole probabilities and overcrowding estimates for products of complex Gaussian matrices. Journal of Statistical Physics. 2013;151(6):987-1003.We consider eigenvalues of a product of n non-Hermitian, independent randommatrices. Each matrix in this product is of size N\times N with independentstandard complex Gaussian variables. The eigenvalues of such a product form adeterminantal point process on the complex plane (Akemann and Burda J. Phys A:Math. Theor. 45 (2012) 465201), which can be understood as a generalization ofthe finite Ginibre ensemble. As N\rightarrow\infty, a generalized infiniteGinibre ensemble arises. We show that the set of absolute values of the pointsof this determinantal process has the same distribution as{R_1^{(n)},R_2^{(n)},...}, where R_k^{(n)} are independent, and (R_k^{(n)})^2is distributed as the product of n independent Gamma variables Gamma(k,1). Thisenables us to find the asymptotics for the hole probabilities, i.e. for theprobabilities of the events that there are no points of the process in a discof radius r with its center at 0, as r\rightarrow\infty. In addition, we solvethe relevant overcrowding problem: we derive an asymptotic formula for theprobability that there are more than m points of the process in a fixed disk ofradius r with its center at 0, as m\rightarrow\infty
Non-Hermitian extensions of Wishart random matrix ensembles
No full textAkemann G. Non-Hermitian extensions of Wishart random matrix ensembles. Acta Physica Polonica B. 2011;42(5): 901.We briefly review the solution of three ensembles of non-Hermitian randommatrices generalizing the Wishart-Laguerre (also called chiral) ensembles.These generalizations are realized as Gaussian two-matrix models, where thecomplex eigenvalues of the product of the two independent rectangular matricesare sought, with the matrix elements of both matrices being either real,complex or quaternion real. We also present the more general case depending ona non-Hermiticity parameter, that allows us to interpolate between thecorresponding three Hermitian Wishart ensembles with real eigenvalues and themaximally non-Hermitian case. All three symmetry classes are explicitly solvedfor finite matrix size NxM for all complex eigenvalue correlations functions(and real or mixed correlations for real matrix elements). These are given interms of the corresponding kernels built from orthogonal or skew-orthogonalLaguerre polynomials in the complex plane. We then present the correspondingthree Bessel kernels in the complex plane in the microscopic large-N scalinglimit at the origin, both at weak and strong non-Hermiticity with M-N greateror equal to 0 fixed
Higher Order Analogues of Tracy-Widom Distributions via the Lax Method
No full textAkemann G, Atkin M. Higher Order Analogues of Tracy-Widom Distributions via the Lax Method. J.Phys. A. 2013;46(1): 15202.We study the distribution of the largest eigenvalue in formal Hermitianone-matrix models at multicriticality, where the spectral density acquires anextra number of k-1 zeros at the edge. The distributions are directly expressedthrough the norms of orthogonal polynomials on a semi-infinite interval, as analternative to using Fredholm determinants. They satisfy non-linear recurrencerelations which we show form a Lax pair, making contact to the stringliterature in the early 1990's. The technique of pseudo-differential operatorsallows us to give compact expressions for the logarithm of the gap probabilityin terms of the Painleve XXXIV hierarchy. These are the higher order analoguesof the Tracy-Widom distribution which has k=1. Using known Backlundtransformations we show how to simplify earlier equivalent results that arederived from Fredholm determinant theory, valid for even k in terms of thePainleve II hierarchy
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