Gaussian Random Potential Research Articles

Non-integrability and chaos for natural Hamiltonian systems with a random potential

Consider the ensemble of Gaussian random potentials {VL(q)}L=1∞ on the d-dimensional torus where, essentially, VL(q) is a real-valued trigonometric polynomial of degree L whose coefficients are independent standard normal variables. Our main result ensures that, with a probability tending to 1 as L→∞, the dynamical system associated with the natural Hamiltonian function defined by this random potential, HL:=12|p|2+VL(q), exhibits a number of chaotic regions which coexist with a positive-volume set of invariant tori. In particular, these systems are typically neither integrable with non-degenerate first integrals nor ergodic. An analogous result for random natural Hamiltonian systems defined on the cotangent bundle of an arbitrary compact Riemannian manifold is presented too.

Seven Études on dynamical Keldysh model

We present a comprehensive pedagogical discussion of a family of models describing the propagation of a single particle in a multicomponent non-Markovian Gaussian random field. We report some exact results for single-particle Green’s functions, self-energy, vertex part and T-matrix. These results are based on a closed form solution of the Dyson equation combined with the Ward identity. Analytical properties of the solution are discussed. Further we describe the combinatorics of the Feynman diagrams for the Green’s function and the skeleton diagrams for the self-energy and vertex, using recurrence relations between the Taylor expansion coefficients of the self-energy. Asymptotically exact equations for the number of skeleton diagrams in the limit of large NN are derived. Finally, we consider possible realizations of a multicomponent Gaussian random potential in quantum transport via complex quantum dot experiments.

Invariance principles for random walks in random environment on trees

In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge with respect to the Gromov-Hausdorff-vague topology, and a certain uniform recurrence condition is satisfied. Such a theorem finds particularly nice applications if the resistance metric measure space is a metric measure tree. To illustrate this, we state functional limit theorems in old and new examples of suitably rescaled random walks in random environment on trees. First, we take a critical Galton-Watson tree conditioned on its total progeny and a non-lattice branching random walk on $\mathbb{R}^d$ indexed by it. Then, conditional on that, we consider a biased random walk on the range of the preceding. Here, by non-lattice we mean that distinct branches of the tree do not intersect once mapped in $\mathbb{R}^d$. This excludes the possibility that the random walk on the range may jump from one branch to the other without returning to the recent common ancestor. We prove, after introducing the bias parameter $\beta^{n^{-1/4}}$, for some $\beta>1$, that the biased random walk on the range of a large critical non-lattice branching random walk converges to a Brownian motion in a random Gaussian potential on Aldous' continuum random tree (CRT). Our second new result introduces the scaling limit of the edge-reinforced random walk on a size-conditioned Galton-Watson tree with finite variance as a Brownian motion in a random Gaussian potential on the CRT with a drift proportional to the distance to the root.

The distribution of vacua in random landscape potentials

Landscape cosmology posits the existence of a convoluted, multidimensional, scalar potential—the “landscape”—with vast numbers of metastable minima. Random matrices and random functions in many dimensions provide toy models of the landscape, allowing the exploration of conceptual issues associated with these scenarios. We compute the relative number and slopes of minima as a function of the vacuum energy Λ in an N-dimensional Gaussian random potential, quantifying the associated probability density, p(Λ). After normalisations p(Λ) depends only on the dimensionality N and a single free parameter γ, which is related to the power spectrum of the random function. For a Gaussian landscape with a Gaussian power spectrum, the fraction of positive minima shrinks super-exponentially with N; at N=100, p(Λ>0) ≈ 10−1197. Likewise, typical eigenvalues of the Hessian matrices reveal that the flattest approaches to typical minima grow flatter with N, while the ratio of the slopes of the two flattest directions grows with N. We discuss the implications of these results for both swampland and conventional anthropic constraints on landscape cosmologies. In particular, for parameter values when positive minima are extremely rare, the flattest approaches to minima where Λ ≈ 0 are much flatter than for typical minima, increasingly the viability of quintessence solutions.

Improved Field Theoretical Approach to Noninteracting Brownian Particles in a Quenched Random Potential

We construct a dynamical field theory for noninteracting Brownian particles in the presence of a quenched Gaussian random potential. The main variable for the field theory is the density fluctuation which measures the difference between the local density and its average value. The average density is spatially inhomogeneous for given realization of the random potential. It becomes uniform only after averaged over the disorder configurations. We develop the diagrammatic perturbation theory for the density correlation function and calculate the zero-frequency component of the response function exactly by summing all the diagrams contributing to it. From this exact result and the fluctuation dissipation relation, which holds in an equilibrium dynamics, we find that the connected density correlation function always decays to zero in the long-time limit for all values of disorder strength implying that the system always remains ergodic. This nonperturbative calculation relies on the simple diagrammatic structure of the present field theoretical scheme. We compare in detail our diagrammatic perturbation theory with the one used in a recent paper [B.\ Kim, M.\ Fuchs and V.\ Krakoviack, J.\ Stat.\ Mech.\ (2020) 023301], which uses the density fluctuation around the uniform average, and discuss the difference in the diagrammatic structures of the two formulations.

Dynamics of a noninteracting colloidal fluid in a quenched Gaussian random potential: a time-reversal-symmetry-preserving field-theoretic approach

We develop a field-theoretic perturbation method preserving the fluctuation–dissipation relation (FDR) for the dynamics of the density fluctuations of a noninteracting colloidal gas plunged in a quenched Gaussian random field. It is based on an expansion about the Brownian noninteracting gas and can be considered and justified as a low-disorder or high-temperature expansion. The first-order bare theory yields the same memory integral as the mode-coupling theory (MCT) developed for (ideal) fluids in random environments, apart from the bare nature of the correlation functions involved. It predicts an ergodic dynamical behavior for the relaxation of the density fluctuations, in which the memory kernels and correlation functions develop long-time algebraic tails. An FDR-consistent renormalized theory is also constructed from the bare theory. It is shown to display a dynamic ergodic–nonergodic transition similar to the one predicted by the MCT at the level of the density fluctuations, but, at variance with the MCT, the transition does not fully carry over to the self-diffusion, which always reaches normal diffusive behavior at long time, in agreement with known rigorous results.

Magneto-conductivity of tilted type-I Weyl semimetals with different types of impurities

We theoretically study the longitudinal magneto-conductivity of tilted type-I Weyl semimetals with three different types of impurities via a two-node model with opposite chirality and tilt vector. The Landau level solutions suggest that the Fermi velocity of the zeroth Landau level is dressed by the tilt and the angle between the magnetic field and the tilt direction. At low temperature and only considering the zeroth Landau level, we find that the magneto-conductivity σzz depends strongly on the tilt-dressed Fermi velocity and thus shows a strong angle dependence. This angle dependence of conductivity will be enhanced by the increase of the degree of tilt. Besides, the conductivity is very sensitive to impurities. σzz decreases with increasing impurity density and scattering strength, and it has different magnetic field dependence for different scattering potentials. For short-range delta scattering, it is independent of B. But for long-range random Gaussian potential and screened Coulomb potential, it may show a linear and quasi-quadratic dependence on B, respectively. These findings help us have a better understanding of the magneto-transport properties of the tilted type-I Weyl semimetals.

Wegner estimate for discrete Schrödinger operators with Gaussian random potentials

Abstract We prove a Wegner estimate for discrete Schrödinger operators with a potential given by a Gaussian random process. The only assumption is that the covariance function decays exponentially; no monotonicity assumption is required. This improves earlier results where abstract conditions on the conditional distribution, compactly supported and non-negative, or compactly supported covariance functions with positive mean are considered.

A Spin Glass Model for Reconstructing Nonlinearly Encrypted Signals Corrupted by Noise

We define a (symmetric key) encryption of a signal mathbf{s}in {mathbb {R^N}} as a random mapping mathbf{s}mapsto mathbf y =(y_1,ldots ,y_M)^Tin mathbb {R}^M known both to the sender and a recipient. In general the recipients may have access only to images mathbf{y} corrupted by an additive noise of unknown strength. Given the encryption redundancy parameter (ERP) mu =M/Nge 1 and the signal strength parameter R=sqrt{sum _i {s_i^2/N}}, we consider the problem of reconstructing mathbf{s} from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian random interaction potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap p_{infty }in [0,1] between the original signal and its recovered image (known as ’estimate’) as Nrightarrow infty , for a given (’bare’) noise-to-signal ratio (NSR) gamma ge 0. Such an overlap is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full p_{infty } (gamma ) curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in some interval of NSR. We show that encryptions with a nonvanishing linear component permit reconstructions with p_{infty }>0 for any mu >1 and any gamma <infty , with p_{infty }sim gamma ^{-1/2} as gamma rightarrow infty . In contrast, for the case of purely quadratic nonlinearity, for any ERP mu >1 there exists a threshold NSR value gamma _c(mu ) such that p_{infty }=0 for gamma >gamma _c(mu ) making the reconstruction impossible. The behaviour close to the threshold is given by p_{infty }sim (gamma _c-gamma )^{3/4} and is controlled by the replica symmetry breaking mechanism.

Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

By extending the Kac–Rice approach to manifolds of finite internal dimension, we show that the mean numberNtot of all possible equilibria (i.e. force-free configurations, a.k.a. equilibrium points) of an elastic line (directed polymer), confined in a harmonic well and submitted to a quenched random Gaussian potential in dimensiond=1+1, grows exponentiallyNtot∼exp(rL) with its lengthL. The growth rater is found to be directly related to the generalized Lyapunov exponent (GLE) which is a moment-generating function characterizing the large-deviation type fluctuations of the solution to the initial value problem associated with the random Schrödinger operator of the 1D Anderson localization problem. For strong confinement, the rater is small and given by a non-perturbative (instanton, Lifshitz tail-like) contribution to GLE. For weak confinement, the rater is found to be proportional to the inverse Larkin length of the pinning theory. As an application, identifying the depinning with a landscape “topology trivialization” phenomenon, we obtain an upper bound for the depinning thresholdfc, in the presence of an applied force, for elastic lines andd-dimensional manifolds, expressed through the mean modulus of the spectral determinant of the Laplace operators with a random potential. We also discuss the question of counting of stable equilibria. Finally, we extend the method to calculate the asymptotic number of equilibria at fixed energy (elastic, potential and total), and obtain the (annealed) distribution of the energy density over these equilibria (i.e. force-free configurations). Some connections with the Larkin model are also established.

Hessian eigenvalue distribution in a random Gaussian landscape

The energy landscape of multiverse cosmology is often modeled by a multi-dimensional random Gaussian potential. The physical predictions of such models crucially depend on the eigenvalue distribution of the Hessian matrix at potential minima. In particular, the stability of vacua and the dynamics of slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues. The Hessian eigenvalue distribution has been studied earlier, using the saddle point approximation, in the leading order of 1/N expansion, where N is the dimensionality of the landscape. This approximation, however, is insufficient for the small eigenvalue end of the spectrum, where sub-leading terms play a significant role. We extend the saddle point method to account for the sub-leading contributions. We also develop a new approach, where the eigenvalue distribution is found as an equilibrium distribution at the endpoint of a stochastic process (Dyson Brownian motion). The results of the two approaches are consistent in cases where both methods are applicable. We discuss the implications of our results for vacuum stability and slow-roll inflation in the landscape.

Inflation in random landscapes with two energy scales

We investigate inflation in a multi-dimensional landscape with a hierarchy of energy scales, motivated by the string theory, where the energy scale of Kahler moduli is usually assumed to be much lower than that of complex structure moduli and dilaton field. We argue that in such a landscape, the dynamics of slow-roll inflation is governed by the low-energy potential, while the initial condition for inflation are determined by tunneling through high-energy barriers. We then use the scale factor cutoff measure to calculate the probability distribution for the number of inflationary e-folds and the amplitude of density fluctuations Q, assuming that the low-energy landscape is described by a random Gaussian potential with a correlation length much smaller than Mpl. We find that the distribution for Q has a unique shape and a preferred domain, which depends on the parameters of the low-energy landscape. We discuss some observational implications of this distribution and the constraints it imposes on the landscape parameters.

Persistent Sinai-type diffusion in Gaussian random potentials with decaying spatial correlations.

Logarithmic or Sinai-type subdiffusion is usually associated with random force disorder and nonstationary potential fluctuations whose root-mean-squared amplitude grows with distance. We show here that extremely persistent, macroscopic logarithmic diffusion also universally emerges at sufficiently low temperatures in stationary Gaussian random potentials with spatially decaying correlations, known to exist in a broad range of physical systems. Combining results from extensive simulations with a scaling approach we elucidate the physical mechanism of this unusual subdiffusion. In particular, we explain why with growing temperature and/or time a first crossover occurs to standard, power-law subdiffusion, with a time-dependent power-law exponent, and then a second crossover occurs to normal diffusion with a disorder-renormalized diffusion coefficient. Interestingly, the initial, nominally ultraslow diffusion turns out to be much faster than the universal de Gennes-Bässler-Zwanzig limit of the renormalized normal diffusion, which realistically cannot be attained at sufficiently low temperatures and/or for strong disorder. The ultraslow diffusion is also shown to be nonergodic and it displays a local bias phenomenon. Our simple scaling theory not only explains our numerical findings but qualitatively also has a predictive character.

Initial conditions for slow-roll inflation in a random Gaussian landscape

In the landscape perspective, our Universe begins with a quantum tunneling from an eternally-inflating parent vacuum, followed by a period of slow-roll inflation. We investigate the tunneling process and calculate the probability distribution for the initial conditions and for the number of e-folds of slow-roll inflation, modeling the landscape by a small-field one-dimensional random Gaussian potential. We find that such a landscape is fully consistent with observations, but the probability for future detection of spatial curvature is rather low, P ∼ 10−3.

Instanton calculus of Lifshitz tails

For noninteracting particles moving in a Gaussian random potential, there exists a disagreement in the literature on the asymptotic expression for the density of states in the tail of the band. We resolve this discrepancy. Further we illuminate the physical facet of instantons appearing in replica and supersymmetric derivations with another derivation employing a Lagrange multiplier field.

Bose–Einstein condensate under Gaussian random potentials

Properties of Bose–Einstein condensates (BEC) are studied in the presence of a Gaussian random potential of arbitrary strength of disorder and with arbitrary correlation length. Using the stochastic mean field approximation, a system having arbitrary strength of interactions is treated at finite temperature. A special case, in which we assume that the second moment of the random potential is Lorentzian, is treated in more detail at zero temperature. The total density of particles consists of condensate density, glassy density and normal density components. Our results are in agreement with those of Huang and Meng (HM). The system is stable below the critical value of the disorder parameter and undergoes a first-order phase transition at the critical value. We find that the critical value of the disorder parameter changes with changing the correlation length of the random potential.

Evaluation of correlation functions and wave-functions of the Gaussian random potentials by numerical shooting method

In this paper, we apply a simple numerical method for evaluating a correlation function of atomic density fluctuation under Gaussian random potentials. Instead of using the 6-points kernel, averaged over disorders, we use the numerical shooting method for solving the Schrodinger equation of this quantum system and directly calculate the correlation function from these solutions. Since our approach does not use complicated formulas, it requires much less computational effort when compared to Green function techniques. Finally, we show results of calculations for setting each of parameters of Gaussian random potentials.

Solitonization of the Anderson localization

We study the affinities between the shape of the bright soliton of the one-dimensional nonlinear Schroedinger equation and that of the disorder induced localization in the presence of a Gaussian random potential. With emphasis on the focusing nonlinearity, we consider the bound states of the nonlinear Schroedinger equation with a random potential; for the state exhibiting the highest degree of localization, we derive explicit expressions for the nonlinear eigenvalue and for the localization length by using perturbation theory and a variational approach following the methods of statistical mechanics of disordered systems. We numerically investigate the linear stability and "superlocalizations". The profile of the disorder averaged Anderson localization is found to obey a nonlocal nonlinear Schroedinger equation

The loss of quantum coherence induced by a Gaussian random potential

We investigate the source of decoherence from a random external potential by the Feynman path integral method. We find that the off-diagonal elements of the density matrix decay with two characteristic time scales, which are separated by the correlation length. The heavy and light mass cases are demonstrated.

High-dimensional Gaussian fields with isotropic increments seen through spin glasses

We study the free energy of a particle in (arbitrary) high-dimensional Gaussian random potentials with isotropic increments. We prove a computable saddle point variational representation in terms of a Parisi-type functional for the free energy in the infinite-dimensional limit. The proofs are based on the techniques developed in the course of the rigorous analysis of the Sherrington-Kirkpatrick model with vector spins.