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In a recent paper (J Stat Phys 179:729–747, 2020) time evolution of the telegrapher’s equation subjected to a random velocity field was studied in the context of the effective medium approximation. In addition, the solution for the mean-value of the field was presented as a series expansion in Terwiel’s cumulants. Here we have proved that this expansion cuts if the random velocity field is emulated with a stationary exponential-correlated symmetric binary disorder. The present conclusion can also be used to tackle other wave-like models of propagation with Markovian binary disorder. [ABSTRACT FROM AUTHOR]

We investigated a diffusion-like equation with a bounded speed of signal propagation (the so called telegrapher's equation) in a random media. We discuss some properties of the mean-value solution in a well-defined perturbation theory. The frequency-dependent effective-velocity of propagation is studied in the long and short time regime. We show that due to the wave-like character of telegrapher's equation the effective-velocity is a complex dispersive function in time. Exact results and asymptotic perturbative long-time behaviors (for an exponential space-correlated binary disorder) are presented, showing their agreement and corroborating the goodness of the effective medium approximation in continuous system. [ABSTRACT FROM AUTHOR]

Let $$Z_N$$ be the number of self-avoiding paths of length $$N$$ starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on $${\mathbb Z} ^d$$ with parameter $$p>p_c({\mathbb Z} ^d)$$ . The object of this paper is to study the connective constant of the dilute lattice $$\limsup _{N\rightarrow \infty } Z_N^{1/N}$$ , which is a non-random quantity. We want to investigate if the inequality $$\limsup _{N\rightarrow \infty } (Z_N)^{1/N} \le \lim _{N\rightarrow \infty } {\mathbb E} [Z_N]^{1/N}$$ obtained with the Borel-Cantelli Lemma is strict or not. In other words, we want to know if the quenched and annealed versions of the connective constant are equal. On a heuristic level, this indicates whether or not localization of the trajectories occurs. We prove that when $$d$$ is sufficiently large there exists $$p^{(2)}_c>p_c$$ such that the inequality is strict for $$p\in (p_c,p^{(2)}_c)$$ . [ABSTRACT FROM AUTHOR]

We estimate the variance of the value function for a random optimal control problem. The value function is the solution w of a Hamilton-Jacobi equation with random Hamiltonian H( p, x, ω)= K( p)− V( x/ ϵ, ω) in dimension d≥2. It is known that homogenization occurs as ϵ→0, but little is known about the statistical fluctuations of w. Our main result shows that the variance of the solution w is bounded by O( ϵ/|log ϵ|). The proof relies on a modified Poincaré inequality of Talagrand. [ABSTRACT FROM AUTHOR]

In this paper we revisit the parabolic approximation for wave propagation in random media by taking into account backscattering. We obtain a system of transport equations for the moments of the components of reflection and transmission operators. In the regime in which forward scattering is strong and backward scattering is weak, we obtain closed form expressions for physically relevant quantities related to the reflected wave, such as the beam width, the spectral width and the mean spatial power profile. In particular, we analyze the enhanced backscattering phenomenon, that is, we show that the mean power reflected from an incident quasi-plane wave has a maximum in the backscattered direction. This enhancement can be observed in a small cone around the backscattered direction and we compute the enhancement factor as well as the shape of the enhanced backscattering cone. [ABSTRACT FROM AUTHOR]

The stationary states of boundary driven zero-range processes in random media with quenched disorder are examined, and the motion of a tagged particle is analyzed. For symmetric transition rates, also known as the random barrier model, the stationary state is found to be trivial in absence of boundary drive. Out of equilibrium, two further cases are distinguished according to the tail of the disorder distribution. For strong disorder, the fugacity profiles are found to be governed by the paths of normalized α-stable subordinators. The expectations of integrated functions of the tagged particle position are calculated for three types of routes. [ABSTRACT FROM AUTHOR]

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W n ( S), and rooted self-avoiding polygons P n ( S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P n ( S), and W n( S) for an arbitrary point S on the lattice. These are used to compute the averages $$\langle P_{n}(S) \rangle$$ , $$\langle W_{n}(S) \rangle$$ , $$\langle \log P_{n}(S) \rangle$$ and $$\langle \log W_{n}(S) \rangle$$ over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent $$\nu$$ are the same for the annealed and quenched averages. However, $$\langle \log P_{n}(S) \rangle \simeq n \log \mu + (\alpha_q - 2)\log n$$ , and $$\langle \log W_{n}(S) \rangle \simeq n \log \mu + (\gamma_q-1) log{n}$$ , where the exponents $$\alpha_q$$ and $$\gamma_q$$ , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives $$\alpha_q \simeq 0.72837 \pm 0.00001;$$ and $$\gamma_q \simeq 1.37501 \pm 0.00003$$ , to be compared with the known annealed values $$\alpha_a = 0.73421$$ and $$\gamma_q = 1.37522$$ . [ABSTRACT FROM AUTHOR]

In this paper the transport of quantum particles in time-dependent random media is studied. In the white noise limit, a quantum model for collisions is obtained. At the level of Wigner equation, this limit is described by a linear Wigner-Boltzmann equation. [ABSTRACT FROM AUTHOR]

This paper deals with the transmission of a soliton in a random medium described by a randomly perturbed Korteweg–de Vries equation. Different kinds of perturbations are addressed, depending on their specific time or position dependences, with or without damping. We derive effective evolution equations for the soliton parameter by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus. Original results are derived that are very different compared to a randomly perturbed Nonlinear Schrödinger equation. First the emission of a soliton gas is proved to be a very general feature. Second some perturbations are shown to involve a speeding-up of the soliton, instead of the decay that is usually observed in random media. [ABSTRACT FROM AUTHOR]

In this paper, we study the propagation of acoustic waves in a layered random medium. Using the time-reversed method, we prove the space-time refocusing effect and the stabilization for acoustic signals. This work is a generalization of ref. 4 and 7 in the three-dimensional case. [ABSTRACT FROM AUTHOR]

We show that a signal can propagate in a particular direction through a model random medium regardless of the precise state of the medium. As a prototype, we consider a point particle moving on a one-dimensional lattice whose sites are occupied by scatterers with the following properties: (i) the state of each site is defined by its spin (up or down); (ii) the particle arriving at a site is scattered forward (backward) if the spin is up (down); (iii) the state of the site is modified by the passage of the particle, i.e., the spin of the site where a scattering has taken place, flips (↑⇔↓). We consider one-dimensional and triangular lattices, for which we give a microscopic description of the dynamics, prove the propagation of a particle through the scatterers, and compute analytically its statistical properties. In particular we prove that, in one dimension, the average propagation velocity is 〈 c( q)〉=1/(3−2 q), with q the probability that a site has a spin ↑, and, in the triangular lattice, the average propagation velocity is independent of the scatterers distribution: 〈 c〉=1/8. In both cases, the origin of the propagation is a blocking mechanism, restricting the motion of the particle in the direction opposite to the ultimate propagation direction, and there is a specific reorganization of the spins after the passage of the particle. A detailed mathematical analysis of this phenomenon is, to the best of our knowledge, presented here for the first time. [ABSTRACT FROM AUTHOR]

We study the problem of localization in a disordered one-dimensional nonlinear medium modeled by the nonlinear Schrödinger equation. Devillard and Souillard have shown that almost every time-harmonic solution of this random PDE exhibits localization. We consider the temporal stability of such time-harmonic solutions and derive bounds on the location of any unstable eigenvalues. By direct numerical determination of the eigenvalues we show that these time-harmonic solutions are typically unstable, and find the distribution of eigenvalues in the complex plane. The distributions are distinctly different for focusing and defocusing nonlinearities. We argue further that these instabilities are connected with resonances in a Schrödinger problem, and interpret the earlier numerical simulations of Caputo, Newell, and Shelley, and of Shelley in terms of these instabilities. Finally, in the defocusing case we are able to construct a family of asymptotic solutions which includes the stable limiting time-harmonic state observed in the simulations of Shelley. [ABSTRACT FROM AUTHOR]

We analyze the effective diffusivity of a passive scalar in a two-dimensional, steady, incompressible random flow that has mean zero and a stationary stream function. We show that in the limit of small diffusivity or large Peclet number, with convection dominating, there is substantial enhancement of the effective diffusivity. Our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory, some of which are widely believed to be correct but have not been proved yet. We show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of Isichenko and Kalda that the effective diffusivity behaves like ɛ when the molecular diffusivity ɛ is small, assuming some percolation-theoretic facts. We also analyze the effective diffusivity for a special class of convective flows, random cellular flows, where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows. [ABSTRACT FROM AUTHOR]

We have calculated the self-dynamic structure factor F( k, t) for tagged particle motion in “hopping” Lorentz gases. We find evidence that, even at long times, the probability distribution function for the displacement of the particles is highly non-Gaussian. At very small values of the wave vector this manifests itself as the divergence of the Burnett coefficient (the fourth moment of the distribution never approaching a value characteristic of a Gaussian). At somewhat larger wave vectors we find that F( k, t) decays algebraically, rather than exponentially as one would expect for a Gaussian. The precise form of this power-law decay depends on the nature of the scatterers making up the Lorentz gas. We find different power-law exponents for scatterers which exclude certain sites and scatterers which do not. [ABSTRACT FROM AUTHOR]

We study the conductance of random resistor networks in d≥2 dimensions. It is shown (under some technical assumptions) that if a network exhibits a nonzero conductivity in the infinite-volume limit, then the variance of a finite-volume conductance grows at least like the volume. [ABSTRACT FROM AUTHOR]