Bulk Universality and Related Properties of Hermitian Matrix Models

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \ref{t:U.t1}). The proof as our previous proof in \cite{Pa-Sh:97} is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper \cite{BPS:95} on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest

Tall tales from de Sitter space II: Field theory dualities

We consider the evolution of massive scalar fields in (asymptotically) de Sitter spacetimes of arbitrary dimension. Through the proposed dS/CFT correspondence, our analysis points to the existence of new nonlocal dualities for the Euclidean conformal field theory. A massless conformally coupled scalar field provides an example where the analysis is easily explicitly extended to 'tall' background spacetimes.Comment: 31 pages, 2 figure

Level-Spacing Distributions and the Bessel Kernel

The level spacing distributions which arise when one rescales the Laguerre or Jacobi ensembles of hermitian matrices is studied. These distributions are expressible in terms of a Fredholm determinant of an integral operator whose kernel is expressible in terms of Bessel functions of order $\alpha$. We derive a system of partial differential equations associated with the logarithmic derivative of this Fredholm determinant when the underlying domain is a union of intervals. In the case of a single interval this Fredholm determinant is a Painleve tau function.Comment: 18 pages, resubmitted to make postscript compatible, no changes to manuscript conten

Optimal estimation of qubit states with continuous time measurements

We propose an adaptive, two steps strategy, for the estimation of mixed qubit states. We show that the strategy is optimal in a local minimax sense for the trace norm distance as well as other locally quadratic figures of merit. Local minimax optimality means that given $n$ identical qubits, there exists no estimator which can perform better than the proposed estimator on a neighborhood of size $n^{-1/2}$ of an arbitrary state. In particular, it is asymptotically Bayesian optimal for a large class of prior distributions. We present a physical implementation of the optimal estimation strategy based on continuous time measurements in a field that couples with the qubits. The crucial ingredient of the result is the concept of local asymptotic normality (or LAN) for qubits. This means that, for large $n$, the statistical model described by $n$ identically prepared qubits is locally equivalent to a model with only a classical Gaussian distribution and a Gaussian state of a quantum harmonic oscillator. The term `local' refers to a shrinking neighborhood around a fixed state $\rho_{0}$. An essential result is that the neighborhood radius can be chosen arbitrarily close to $n^{-1/4}$. This allows us to use a two steps procedure by which we first localize the state within a smaller neighborhood of radius $n^{-1/2+\epsilon}$, and then use LAN to perform optimal estimation.Comment: 32 pages, 3 figures, to appear in Commun. Math. Phy

Effective Action for the Quark-Meson Model

The scale dependence of an effective average action for mesons and quarks is described by a nonperturbative flow equation. The running couplings lead to spontaneous chiral symmetry breaking. We argue that for strong Yukawa coupling between quarks and mesons the low momentum physics is essentially determined by infrared fixed points. This allows us to establish relations between various parameters related to the meson potential. The results for $f_\pi$ and \VEV{\olpsi\psi} are not very sensitive to the poorly known details of the quark--meson effective action at scales where the mesonic bound states form. For realistic constituent quark masses we find $f_\pi$ around 100\MeV.Comment: 56 pages (including 10 figures and 1 table), uses epsf.st

Crises and collective socio-economic phenomena: simple models and challenges

Financial and economic history is strewn with bubbles and crashes, booms and busts, crises and upheavals of all sorts. Understanding the origin of these events is arguably one of the most important problems in economic theory. In this paper, we review recent efforts to include heterogeneities and interactions in models of decision. We argue that the Random Field Ising model (RFIM) indeed provides a unifying framework to account for many collective socio-economic phenomena that lead to sudden ruptures and crises. We discuss different models that can capture potentially destabilising self-referential feedback loops, induced either by herding, i.e. reference to peers, or trending, i.e. reference to the past, and account for some of the phenomenology missing in the standard models. We discuss some empirically testable predictions of these models, for example robust signatures of RFIM-like herding effects, or the logarithmic decay of spatial correlations of voting patterns. One of the most striking result, inspired by statistical physics methods, is that Adam Smith's invisible hand can badly fail at solving simple coordination problems. We also insist on the issue of time-scales, that can be extremely long in some cases, and prevent socially optimal equilibria to be reached. As a theoretical challenge, the study of so-called "detailed-balance" violating decision rules is needed to decide whether conclusions based on current models (that all assume detailed-balance) are indeed robust and generic.Comment: Review paper accepted for a special issue of J Stat Phys; several minor improvements along reviewers' comment

Chirality and Symmetry Breaking in a discrete internal Space

In previous papers the permutation group S_4 has been suggested as an ordering scheme for elementary particles, and the appearance of this finite symmetry group was taken as indication for the existence of a discrete inner symmetry space underlying elementary particle interactions. Here it is pointed out that a more suitable choice than the tetrahedral group S_4 is the pyritohedral group A_4 x Z_2 because its vibrational spectrum exhibits exactly the mass multiplet structure of the 3 fermion generations. Furthermore it is noted that the same structure can also be obtained from a primordial symmetry breaking S_4 --> A_4. Since A_4 is a chiral group, while S_4 is achiral, an argument can be given why the chirality of the inner pyritohedral symmetry leads to parity violation of the weak interactions.Comment: 42 pages, 3 table