Gaussian Random Matrix Models¶for q-deformed Gaussian Variables

We construct a family of random matrix models for the q-deformed Gaussian random variables G _μ=a _μ+a^*_μ, where the annihilation operators a _μ and creation operators $a\gwia_\nu$ fulfill the $q$-deformed commutation relation a _μ a^*_ν−q a^*_ν a _μ=Γ_μν, Γ_μν is the covariance and 0<q<1 is a given number. An important feature of the considered random matrices is that the joint distribution of their entries is Gaussian. Received: 29 March 2000 / Accepted: 1 August 2000     

The Jancovici–Lebowitz–Manificat Law for Large Fluctuations of Random Complex Zeroes

Consider a Gaussian Entire Function

Essential Self-Adjointness of¶Translation-Invariant Quantum Field Models¶for Arbitrary Coupling Constants

The Hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants. The Hamiltonian has a translation invariant part. By means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self-adjoint extension of the translation invariant part is shown. With a non-perturbative approach it is proved that the Hamiltonian is essentially self-adjoint on a domain. A typical example is the Pauli–Fierz model with spin 1/2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants. Received: 26 May 1999 / Accepted: 9 November 1999     

Random Homogenization and Singular Perturbations¶in Perforated Domains

The paper considers the singularly perturbed Dirichlet problem −ɛΔu ^ɛ+u ^ɛ=f in a randomly perforated domain Ω^ɛ, which is obtained from a bounded open set Ω in R ^N after removing many holes of size ɛ ^q . The perforated domain is described in terms of an ergodic dynamical system acting on a probability space. Imposing certain conditions on the domain, the behaviour of u ^ɛ when ɛ→ 0 in Lebesgue spaces L ⁿ (Ω) is studied. Test functions together with the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution of holes of size ɛ ^p with the intensity λɛ^{− r} is then considered. The above results apply in some cases; other cases are treated by the Wiener sausage approach. Received: 15 December 1999 / Accepted: 14 April 2000     

Bootstrap Multiscale Analysis and Localization¶in Random Media

We introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events. For quantum and classical waves in random media, we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side L with probability higher than 1 − e ^{− L ζ}, for any 0<ζ<1. The starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent, at some sufficiently large scale, with probability bigger than 1 − (d is the dimension). Note that from the same starting hypothesis we get conclusions that are valid for any 0 < ζ < 1. This is achieved by the repeated use of a bootstrap argument. As an application, we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the Hilbert–Schmidt norm, and better estimates on the behavior of the eigenfunctions. Received: 29 November 2000 / Accepted: 21 June 2001     

Recurrence and Transience of Random Walks¶in Random Environments on a Strip

We explain the necessary and sufficient conditions for recurrent and transient behavior of a random walk in a stationary ergodic random environment on a strip in terms of properties of a top Lyapunov exponent. This Lyapunov exponent is defined for a product of a stationary sequence of positive matrices. In the one-dimensional case this approach allows us to treat wider classes of random walks than before. Received: 15 March 2000 / Accepted: 14 April 2000     

Avalanche Dynamics in Quenched Random Directed Sandpile Models

We numerically investigate the quenched random directed sandpile models which are local, conservative and Abelian. A local flow balance between the outflow of grains during a single toppling at a site and the total number of grains flowing into the same site plays an important role when all the nearest-neighbouring sites of the above-mentioned site topple for once. The quenched model has the same critical exponents with the Abelian deterministic directed sandpile model when the local flow balance exists, otherwise the critical exponents of this quenched model and the annealed Abelian random directed sandpile model are the same. These results indicate that the presence or absence of this local flow balance determines the universality class of the Abelian directed sandpile model.     

Boundary Layer Stability¶in Real Vanishing Viscosity Limit

In the previous paper [20], an Evans function machinery for the study of boundary layer stability was developed. There, the analysis was restricted to strongly parabolic perturbations, that is to an approximation of the form u _t +(F(u)) _x =ν(B(u)u _x ) _x $ (ν≪1) with an “elliptic” matrix B. However, real models, like the Navier–Stokes approximation of the Euler equations for a gas flow, involve incompletely parabolic perturbations: B is not invertible in general. We first adapt the Evans function to this realistic framework, assuming that the boundary is not characteristic, neither for the hyperbolic first order system u _t +(F(u)) _x = 0, nor for the perturbed system. We then apply it to the various kinds of boundary layers for a gas flow. We exhibit some examples of unstable boundary layers for a perfect gas, when the viscosity dominates heat conductivity. Received: 27 November 2000/ Accepted: 16 March 2001     

Kardar–Parisi–Zhang Equation and Large Deviations for Random Walks in Weak Random Environments

We consider the transition probabilities for random walks in \(1+1\) dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE.     

On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model

We consider the coupling from the past implementation of the random–cluster heat-bath process, and study its random running time, or coupling time. We focus on hypercubic lattices embedded on tori, in dimensions one to three, with cluster fugacity at least one. We make a number of conjectures regarding the asymptotic behaviour of the coupling time, motivated by rigorous results in one dimension and Monte Carlo simulations in dimensions two and three. Amongst our findings, we observe that, for generic parameter values, the distribution of the appropriately standardized coupling time converges to a Gumbel distribution, and that the standard deviation of the coupling time is asymptotic to an explicit universal constant multiple of the relaxation time. Perhaps surprisingly, we observe these results to hold both off criticality, where the coupling time closely mimics the coupon collector’s problem, and also at the critical point, provided the cluster fugacity is below the value at which the transition becomes discontinuous. Finally, we consider analogous questions for the single-spin Ising heat-bath process.     

Photon Diffusion in Random Media and Anisotropy of Scattering in the Henyey–Greenstein and Rayleigh–Gans Models

Journal of Experimental and Theoretical Physics - Monte Carlo (MC) numerical simulation is carried out for the intensity of multiply backscattered radiation as a function of the...     

Statistical Characterization of Boundary Fluctuations in the HT—7 Tokamak

The statistical properties of plasma fluctuations are characterized in the boundary region of HT-7 tokamak. A non-Gaussian feature is observed in fluctuations of ion saturation current and floating potential in most of the scrape-off layer regions. The statistical properties of fluctuations have a clear radial dependence, showing a near-Gaussian character in the proximity of the velocity shear layer location and another region where the poloidal velocity has a trend to zero. Fluctuations show a bursty character with pulses asymmetric in time and the time asymmetry reaches the minimum around the shear layer. From the results, we can see an obvious coupling of the pulses and the poloidal flow.     

Lifshitz Tail for Schrödinger Operator¶with Random Magnetic Field

We study the behavior of the density states at the lower edge of the spectrum for Schr?dinger operators with random magnetic fields. We use a new estimate on magnetic Schr?dinger operators, which is similar to the Avron–Herbst–Simon estimate but the bound is always nonnegative. Received: 3 January 2000 / Accepted: 18 April 2000     

Modification of Boundary Fluctuations by LHCD in the HT—7 Tokamak

Measurements of boundary fluctuations and fluctuation driven electron fluxes have been performed in ohmic and lower hybrid current drive enhanced confinement plasma using a graphite Langmuir probe array on HT-7 tokamak. The fluctuations are significantly suppressed and the turbulent fluxes are remarkably depressed in the enhanced plasma. We characterized the statistical properties of fluctuations and the particle flux and found a non-Gaussian character in the whole scrape-off layer with minimum deviations from Gaussian in the proximity of the velocity shear layer in ohmic plasma. In the enhanced plasma the deviations in the boundary region are all reduces obviously. The fluctuations and induced electron fluxes show sporadic bursts asymmetric in time and the asymmetry is remarkably weakened in the lower hybrid current driving (LHCD) phase. The results suggest a coupling between the statistical behaviour of fluctuations and the turbulent flow.     

Universality at the Edge of the Spectrum¶in Wigner Random Matrices

We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n→+∞. As a corollary, we show that, after proper rescaling, the 1^th, 2^nd, 3^rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases. Received: 15 May 1999 / Accepted: 18 May 1999     

Localization in One Dimensional Random Media:¶A Scattering Theoretic Approach

We use scattering theoretic methods to prove exponential localization for random displacement models in one dimension. The operators we consider model both quantum and classical wave propagation. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations. We show that randomly displaced, non-reflectionless single sites lead to localization. Received: 23 September 1999 / Accepted: 13 March 2000     

The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d= 2, we obtain a complete characterization of finite-size scaling. In dimensions d>2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d= 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like n ^d π _n , where π _n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale n ^d π _n , and hence that “the” incipient infinite cluster is not unique. Below the window, we show that the size of the largest cluster scales like ξ ^d π_ξ log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like n ^d P _∞, where P _∞ is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimensional analogues of results on the dominant component of the Erdős–Rényi mean-field random graph model. Received: 6 December 2000 / Accepted: 25 May 2001     

KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions

In this paper, one-dimensional (1D) nonlinear wave equations

Fluctuations of Linear Eigenvalue Statistics of β Matrix Models in the Multi-cut Regime

We study the asymptotic expansion in n for the partition function of β matrix models with real analytic potentials in the multi-cut regime up to the O(n ^?1) terms. As a result, we find the limit of the generating functional of linear eigenvalue statistics and the expressions for the expectation and the variance of linear eigenvalue statistics, which in the general case contain the quasi periodic in n terms.     

Lifshitz Tails for Random Schrödinger Operators¶with Negative Singular Poisson Potential

We develop a method of asymptotic study of the integrated density of states (IDS) N(E) of a random Schr?dinger operator with a non-positive (attractive) Poisson potential. The method is based on the periodic approximations of the potential instead of the Dirichlet-Neumann bracketing used before. This allows us to derive more precise bounds for the rate of approximations of the IDS by the IDS of respective periodic operators and to obtain rigorously for the first time the leading term of log N(E) as E→−∞ for the Poisson random potential with a singular single-site (impurity) potential, in particular, for the screened Coulomb impurities, dislocations, etc. Received: 18 November 1998 / Accepted: 9 March 1999