Non-linear PDEs for gap probabilities in random matrices and KP theory

Airy and Pearcey-like kernels and generalizations arising in random matrix theory are expressed as double integrals of ratios of exponentials, possibly multiplied with a rational function. In this work it is shown that such kernels are intimately related to wave functions for polynomial (Gelfand–Dickey) reductions or rational reductions of the KP-hierarchy; their Fredholm determinant also satisfies linear PDEs (Virasoro constraints), yielding, in a systematic way, non-linear PDEs for the Fredholm determinant of such kernels. Examples include Fredholm determinants giving the gap probability of some infinite-dimensional diffusions, like the Airy process, with or without outliers, and the Pearcey process, with or without inliers.     

Differential Equations for Dyson Processes

We call a Dyson process any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n×n Hermitian matrices, and the eigenvalues describe n curves. Given sets X₁,...,X_m the probability that for each k no curve passes through X_k at time _k is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the Laguerre process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite process at the edge leads to the Airy process (which was introduced by Prähofer and Spohn as the limiting stationary process for a polynuclear growth model) and in the bulk to the sine process; scaling the Laguerre process at the edge leads to the Bessel process.In earlier work the authors found a system of ordinary differential equations with independent variable whose solution determined the probabilitieswhere A() denotes the top curve of the Airy process. Our first result is a generalization and strengthening of this. We assume that each X_k is a finite union of intervals and find a system of partial differential equations, with the end-points of the intervals of the X_k as independent variables, whose solution determines the probability that for each k no curve passes through X_k at time _k. Then we find the analogous systems for the Hermite process (which is more complicated) and also for the sine process. Finally we find an analogous system of PDEs for the Bessel process, which is the most difficult.Dedicated to Freeman Dyson on the occasion of his eightieth birthdayAcknowledgement We thank Kurt Johansson for sending us his unpublished notes on the extended Hermite kernel. This work was supported by National Science Foundation under grants DMS-0304414 (first author) and DMS-0243982 (second author).     

Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models

We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function h_t(x) with corner initialization. We prove, with one exception, that the limiting distribution function of h_t(x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin–Okounkov identity and a novel, rigorous saddle point analysis. In the fixed x, large t regime, we find a Brownian motion representation. This model is equilvalent to the Seppäläinen–Johansson model. Hence some of our results are not new, but the proofs are.     

Scale Invariance of the PNG Droplet and the Airy Process

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single time (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y ^–2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.     

Gaussian modulated Ai- and Bi-based solutions of the 2D PWE: a comparison

We present a comparison of the evolution features, in terms of the intensity moments up to the second order, of what are here referred to as Ai-Gauss and Bi-Gauss wave functions, which originate from source functions consisting of Gaussian-like modulated Airy patterns (of the first and second kind). Both have already been considered in the literature, the former being in particular analysed in detail. A paraxial-optics oriented view of the cos-like Airy–Hardy integrals, which stand out as a generalization of the well-known Airy integral, is also developed.     

Stieltjes Integrals of Hölder Continuous Functions with Applications to Fractional Brownian Motion

We give a new estimate on Stieltjes integrals of Hölder continuous functions and use it to prove an existence-uniqueness theorem for solutions of ordinary differential equations with Hölder continuous forcing. We construct stochastic integrals with respect to fractional Brownian motion, and establish sufficient conditions for its existence. We prove that stochastic differential equations with fractional Brownian motion have a unique solution with probability 1 in certain classes of Hölder-continuous functions. We give tail estimates of the maximum of stochastic integrals from tail estimates of the Hölder coefficient of fractional Brownian motion. In addition we apply the techniques used for ordinary Brownian motion to construct stochastic integrals of deterministic functions with respect to fractional Brownian motion and give tail estimates of its maximum.     

Random Skew Plane Partitions and the Pearcey Process

We study random skew 3D partitions weighted by q ^vol and, specifically, the q → 1 asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel asymptotics for correlations in the bulk of the disordered region, Airy kernel asymptotics near a general point of the frozen boundary, and a Pearcey kernel asymptotics near a cusp of the frozen boundary.     

Large <Emphasis Type="Italic">n</Emphasis> Limit of Gaussian Random Matrices with External Source,Part III: Double Scaling Limit

We consider the double scaling limit in the random matrix ensemble with an external source

Airy Functions Over Local Fields

Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated with compact p-adic Lie groups also have these properties.     

Polar orbits in the Kerr space-time

The motion of test particles in polar orbit about the source of the Kerr field of gravity is studied, using Carter's first integrals for timelike geodesies in the Kerr space-time. Expressions giving the angular coordinates of such particles as functions of the radial one are derived, both for the case of a rotating black hole as well as for that of a naked singularity.     

What have we learned from two-dimensional models of quantum black holes?

The two-dimensional black hole provides a theoretical laboratory in which the quantum nature of black holes may be probed without the complications of four-dimensional dynamics. It is therefore natural to ask, what have we learned from this model? Much recent work has focused on the semi-classical limit where the black hole is similar to the Schwarzschild solution. However, in this essay, we demonstrate that theexact two-dimensional quantum black hole is non-singular. Instead the singularity is replaced by a surface of time reflection symmetry in an extended space-time. The maximally extended space-time thus consists of an infinite sequence of asymptotically flat regions connected by timelike wormholes, rather analogous to the Reissner-Nordström space-time. The implications of this to the apparent loss of quantum information arising from black hole evaporation are also briefly discussed.     

Irreducible kernels and nonperturbative expansions in a theory with purem →m interaction

Recent results on the structure of theS matrix at them-particle threshold (m≧2) in a simplifiedm→m scattering theory with no subchannel interaction are extended to the Green functionF on the basis of off-shell unitarity, through an adequate mathematical extension of some results of Fredholm theory: local two-sheeted or infinite-sheeted structure ofF arounds=(mμ)² depending on the parity of (m?1)(ν?1) (where μ>0 is the mass and ν is the dimension of space-time), off-shell definition of the irreducible kernelU which is the analogue of theK matrix in the two different parity cases (m?1)(ν?1) odd or even, and related local expansion ofF, for (m?1)(ν?1) even, in powers of σ_β ln σ(σ=(mμ)²?s). It is shown that each term in this expansion is the dominant contribution to a Feynman-type integral in which each vertex is a kernelU. The links between the kernelU and Bethe-Salpeter type kernelsG of the theory are exhibited in both parity cases, as also the links between the above expansion ofF and local expansions, in the Bethe-Salpeter type framework, ofF _λ in terms of Feynman-type integrals in which each vertex is a kernelG and which include both dominant and subdominant contributions.     

Chaotic motion of test particles in the Ernst space-time

Trajectories of test particles in the Ernst space-time are studied. The Poincaré surfaces of section are constructed and Lyapunov characteristic exponents are evaluated for a selected set of trajectories. This approach indicates that the number of isolating integrals is not sufficient to separate equations of motion and the particle trajectories are not integrable.     

Tunneling singularities in the open Hubbard chain

We study singularities in the I–V characteristics for sequential tunneling from resonant localized levels (e.g. a quantum dot) into a one-dimensional electron system described by a Hubbard model. Boundary conformal field theory together with the exact solution of the Hubbard model subject to boundary fields allow to compute the exponents describing the singularity arising when the energy of the local level is tuned through the Fermi energy of the wire as a function of electron density and magnetic field. For boundary potentials with bound states a sequence of such singularities can be observed.     

Local Behaviour of Airy Processes

The Airy processes describe limit fluctuations in a wide range of growth models, where each particular Airy process depends on the geometry of the initial profile. We show how the coupling method, developed in the last-passage percolation context, can be used to prove existence of a continuous version and local convergence to Brownian motion. By using similar arguments, we further extend these results to a two parameter limit fluctuation process (Airy sheet).

     

Singularity spectrum of fractal signals from wavelet analysis: Exact results

The multifractal formalism for singular measures is revisited using the wavelet transform. For Bernoulli invariant measures of some expanding Markov maps, the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima. The generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures. It is demonstrated that the Hausdorff dimensionD(h) of the set of singularities of Hölder exponenth can be directly determined from the wavelet transform modulus maxima. The singularity spectrum so obtained is shown to be not disturbed by the presence, in the signal, of a superimposed polynomial behavior of ordern, provided one uses an analyzing wavelet that possesses at leastN>n vanishing moments. However, it is shown that aC behavior generally induces a phase transition in theD(h) singularity spectrum that somewhat masks the weakest singularities. This phase transition actually depends on the numberN of vanishing moments of the analyzing wavelet; its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal. These theoretical results are illustrated with numerical examples. They are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional Brownian motions and turbulent velocity signals.     

The diffusion limit for reversible jump processes onZ d with ergodic random bond conductivities

We consider a reversible jump process on ? ^d whose jump rates themselves are random. We show mean square convergence of this process under diffusion scaling to a limiting Brownian motion with a certain diffusion matrix, characterizing effective conductivity.