The universal coefficient of the exact correlator of a large-NN matrix field theory

Exact expressions have been proposed for correlation functions of the large-$N$ (planar) limit of the $(1+1)$-dimensional ${\rm SU}(N)\times {\rm SU}(N)$ principal chiral sigma model. These were obtained with the form-factor bootstrap. The short-distance form of the two-point function of the scaling field $\Phi(x)$, was found to be $N^{-1}\langle {\rm Tr}\,\Phi(0)^{\dagger} \Phi(x)\rangle=C_{2}\ln^{2}mx$, where $m$ is the mass gap, in agreement with the perturbative renormalization group. Here we point out that the universal coefficient $C_{2}$, is proportional to the mean first-passage time of a L\'{e}vy flight in one dimension. This observation enables us to calculate $C_{2}=1/16\pi$.Comment: Text lengthened from 3 to 6 pages, to include discussion of previous results and directions for further work. Some references added. Accepted for publication in Phys. Rev.

What is the connection between ballistic deposition and the Kardar-Parisi-Zhang equation?

Ballistic deposition (BD) is considered to be a paradigmatic discrete growth model that represents the Kardar-Parisi-Zhang (KPZ) universality class. In this paper we question this connection by rigorously deriving a formal continuum equation from the BD microscopic rules, which deviates from the KPZ equation. In one dimension these deviations are not important in the presence of noise, but for higher dimensions or when considering deterministic evolution they are very relevant.Comment: 16 pages, 3 figure

Yield--Optimized Superoscillations

Superoscillating signals are band--limited signals that oscillate in some region faster their largest Fourier component. While such signals have many scientific and technological applications, their actual use is hampered by the fact that an overwhelming proportion of the energy goes into that part of the signal, which is not superoscillating. In the present article we consider the problem of optimization of such signals. The optimization that we describe here is that of the superoscillation yield, the ratio of the energy in the superoscillations to the total energy of the signal, given the range and frequency of the superoscillations. The constrained optimization leads to a generalized eigenvalue problem, which is solved numerically. It is noteworthy that it is possible to increase further the superoscillation yield at the cost of slightly deforming the oscillatory part of the signal, while keeping the average frequency. We show, how this can be done gradually, which enables a trade-off between the distortion and the yield. We show how to apply this approach to non-trivial domains, and explain how to generalize this to higher dimensions.Comment: 8 pages, 5 figure

The Kardar-Parisi-Zhang Equation with Temporally Correlated Noise - A Self Consistent Approach

In this paper we discuss the well known Kardar Parisi Zhang (KPZ) equation driven by temporally correlated noise. We use a self consistent approach to derive the scaling exponents of this system. We also draw general conclusions about the behavior of the dynamic structure factor $\Phi_q(t)$ as a function of time. The approach we use here generalizes the well known self consistent expansion (SCE) that was used successfully in the case of the KPZ equation driven by white noise, but unlike SCE, it is not based on a Fokker-Planck form of the KPZ equation, but rather on its Langevin form. A comparison to two other analytical methods, as well as to the only numerical study of this problem is made, and a need for an updated extensive numerical study is identified. We also show that a generalization of this method to any spatio-temporal correlations in the noise is possible, and two examples of this kind are considered.Comment: 28 pages, 2 figure

Coupled logistic maps and non-linear differential equations

We study the continuum space-time limit of a periodic one dimensional array of deterministic logistic maps coupled diffusively. First, we analyse this system in connection with a stochastic one dimensional Kardar-Parisi-Zhang (KPZ) equation for confined surface fluctuations. We compare the large-scale and long-time behaviour of space-time correlations in both systems. The dynamic structure factor of the coupled map lattice (CML) of logistic units in its deep chaotic regime and the usual d=1 KPZ equation have a similar temporal stretched exponential relaxation. Conversely, the spatial scaling and, in particular, the size dependence are very different due to the intrinsic confinement of the fluctuations in the CML. We discuss the range of values of the non-linear parameter in the logistic map elements and the elastic coefficient coupling neighbours on the ring for which the connection with the KPZ-like equation holds. In the same spirit, we derive a continuum partial differential equation governing the evolution of the Lyapunov vector and we confirm that its space-time behaviour becomes the one of KPZ. Finally, we briefly discuss the interpretation of the continuum limit of the CML as a Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) non-linear diffusion equation with an additional KPZ non-linearity and the possibility of developing travelling wave configurations.Comment: 23 page