28 research outputs found
Free Energy of the Two-Matrix Model/dToda Tau-Function
We provide an integral formula for the free energy of the two-matrix model
with polynomial potentials of arbitrary degree (or formal power series). This
is known to coincide with the tau-function of the dispersionless
two--dimensional Toda hierarchy. The formula generalizes the case studied by
Kostov, Krichever, Mineev-Weinstein, Wiegmann, Zabrodin and separately
Takhtajan in the case of conformal maps of Jordan curves. Finally we generalize
the formula found in genus zero to the case of spectral curves of arbitrary
genus with certain fixed data.Comment: Ver 2: 18 pages added important formulas for higher genus spectral
curves, few typos removed (and few added). Ver 3: 19 pages (minor changes).
Typos removed, added appendix and improved exposition Ver 4: 19 pages, minor
corrections. Version submitted Ver 4; corrections prompted by referee and
accepted in Nuclear Phys.
Macroscopic and microscopic (non-)universality of compact support random matrix theory
A random matrix model with a σ-model like constraint, the restricted trace ensemble (RTE), is solved in the large-n limit. In the macroscopic limit the smooth connected two-point resolvent G(z,w) is found to be non-universal, extending previous results from monomial to arbitrary polynomial potentials. Using loop equation techniques we give a closed though non-universal expression for G(z,w), which extends recursively to all higher k-point resolvents. These findings are in contrast to the usual unconstrained one-matrix model. However, in the microscopic large-n limit, which probes only correlations at distance of the mean level spacing, we are able to show that the constraint does not modify the universal sine-law. In the case of monomial potentials V(M)=M2p, we provide a relation valid for finite-n between the k-point correlation function of the RTE and the unconstrained model. In the microscopic large-n limit they coincide which proves the microscopic universality of RTEs
Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds
We reconsider the problem of calculating a general spectral correlation
function containing an arbitrary number of products and ratios of
characteristic polynomials for a N x N random matrix taken from the Gaussian
Unitary Ensemble (GUE).
Deviating from the standard "supersymmetry" approach, we integrate out
Grassmann variables at the early stage and circumvent the use of the
Hubbard-Stratonovich transformation in the "bosonic" sector. The method,
suggested recently by one of us, is shown to be capable of calculation when
reinforced with a generalization of the Itzykson-Zuber integral to a
non-compact integration manifold. We arrive to such a generalisation by
discussing the Duistermaat-Heckman localization principle for integrals over
non-compact homogeneous Kaehler manifolds.
In the limit of large the asymptotic expression for the correlation
function reproduces the result outlined earlier by Andreev and Simons.Comment: 34 page, no figures. In this version we added a few references and
modified the introduction accordingly. We also included a new Appendix on
deriving our Itzykson-Zuber type integral following the diffusion equation
metho
Mandatory Identification Bar Checks: How Bouncers Are Doing Their Job
The behavior of bouncers at on site establishments that served alcohol was observed. Our aim was to better understand how bouncers went about their job when the bar had a mandatory policy to check identification of all customers. Utilizing an ethnographic decision model, we found that bouncers were significantly more likely to card customers that were more casually dressed than others, those who were in their 30s, and those in mixed racial groups. We posit that bouncers who failed to ask for identification did so because they appeared to know customers, they appeared to be of age, or they took a break and no one was checking for identification at the door. We found that bouncers presented a commanding presence by their dress and demeanor. Bouncers, we posit, function in three primary roles: customer relations, state law management, and establishment rule enforcer
Compact support probability distributions in random matrix theory
We consider a generalization of the fixed and bounded trace ensembles introduced by Bronk and Rosenzweig up to an arbitrary polynomial potential. In the large-N limit we prove that the two are equivalent and that their eigenvalue distribution coincides with that of the "canonical" ensemble with measure exp[-Tr V(M)]. The mapping of the corresponding phase boundaries is illuminated in an explicit example. In the case of a Gaussian potential we are able to derive exact expressions for the one- and two-point correlator for finite , having finite support
Breakdown of universality in multi-cut matrix models
We solve the puzzle of the disagreement between orthogonal polynomials
methods and mean field calculations for random NxN matrices with a disconnected
eigenvalue support. We show that the difference does not stem from a Z2
symmetry breaking, but from the discreteness of the number of eigenvalues. This
leads to additional terms (quasiperiodic in N) which must be added to the naive
mean field expressions. Our result invalidates the existence of a smooth
topological large N expansion and some postulated universality properties of
correlators. We derive the large N expansion of the free energy for the general
2-cut case. From it we rederive by a direct and easy mean-field-like method the
2-point correlators and the asymptotic orthogonal polynomials. We extend our
results to any number of cuts and to non-real potentials.Comment: 35 pages, Latex (1 file) + 3 figures (3 .eps files), revised to take
into account a few reference
Second and Third Order Observables of the Two-Matrix Model
In this paper we complement our recent result on the explicit formula for the
planar limit of the free energy of the two-matrix model by computing the second
and third order observables of the model in terms of canonical structures of
the underlying genus g spectral curve. In particular we provide explicit
formulas for any three-loop correlator of the model. Some explicit examples are
worked out.Comment: 22 pages, v2 with added references and minor correction
Random Matrix Theories in Quantum Physics: Common Concepts
We review the development of random-matrix theory (RMT) during the last
decade. We emphasize both the theoretical aspects, and the application of the
theory to a number of fields. These comprise chaotic and disordered systems,
the localization problem, many-body quantum systems, the Calogero-Sutherland
model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions.
The review is preceded by a brief historical survey of the developments of RMT
and of localization theory since their inception. We emphasize the concepts
common to the above-mentioned fields as well as the great diversity of RMT. In
view of the universality of RMT, we suggest that the current development
signals the emergence of a new "statistical mechanics": Stochasticity and
general symmetry requirements lead to universal laws not based on dynamical
principles.Comment: 178 pages, Revtex, 45 figures, submitted to Physics Report
Roadmap of ultrafast x-ray atomic and molecular physics
X-ray free-electron lasers (XFELs) and table-top sources of x-rays based upon high harmonic generation (HHG) have revolutionized the field of ultrafast x-ray atomic and molecular physics, largely due to an explosive growth in capabilities in the past decade. XFELs now provide unprecedented intensity (1020 W cm−2) of x-rays at wavelengths down to ~1 Angstrom, and HHG provides unprecedented time resolution (∼50 attoseconds) and a correspondingly large coherent bandwidth at longer wavelengths. For context, timescales can be referenced to the Bohr orbital period in hydrogen atom of 150 attoseconds and the hydrogen-molecule vibrational period of 8 femtoseconds; wavelength scales can be referenced to the chemically significant carbon K-edge at a photon energy of ∼280 eV (44 Angstroms) and the bond length in methane of ∼1 Ångstrom. With these modern x-ray sources one now has the ability to focus on individual atoms, even when embedded in a complex molecule, and view electronic and nuclear motion on their intrinsic scales (attoseconds and Ångstroms). These sources have enabled coherent diffractive imaging, where one can image non-crystalline objects in three dimensions on ultrafast timescales, potentially with atomic resolution. The unprecedented intensity available with XFELs has opened new fields of multiphoton and nonlinear x-ray physics where behavior of matter under extreme conditions can be explored. The unprecedented time resolution and pulse synchronization provided by HHG sources has kindled fundamental investigations of time delays in photoionization, charge migration in molecules, and dynamics near conical intersections that are foundational to AMO physics and chemistry. This roadmap coincides with the year when three new XFEL facilities, operating at Ångstrom wavelengths, opened for users (European XFEL, Swiss-FEL and PAL-FEL in Korea) almost doubling the present worldwide number of XFELs, and documents the remarkable progress in HHG capabilities since its discovery roughly 30 years ago, showcasing experiments in AMO physics and other applications. Here we capture the perspectives of 17 leading groups and organize the contributions into four categories: ultrafast molecular dynamics, multidimensional x-ray spectroscopies; high-intensity x-ray phenomena; attosecond x-ray science
Fermionic Quantum Gravity
We study the statistical mechanics of random surfaces generated by NxN
one-matrix integrals over anti-commuting variables. These Grassmann-valued
matrix models are shown to be equivalent to NxN unitary versions of generalized
Penner matrix models. We explicitly solve for the combinatorics of 't Hooft
diagrams of the matrix integral and develop an orthogonal polynomial
formulation of the statistical theory. An examination of the large N and double
scaling limits of the theory shows that the genus expansion is a Borel summable
alternating series which otherwise coincides with two-dimensional quantum
gravity in the continuum limit. We demonstrate that the partition functions of
these matrix models belong to the relativistic Toda chain integrable hierarchy.
The corresponding string equations and Virasoro constraints are derived and
used to analyse the generalized KdV flow structure of the continuum limit.Comment: 59 pages LaTeX, 1 eps figure. Uses epsf. References and
acknowledgments adde