Random Landscape Research Articles

Anderson Localization of Walking Droplets

Understanding the ability of particles to maneuver through disordered environments is a central problem in innumerable settings, from active matter and biology to electronics. Macroscopic particles ultimately exhibit diffusive motion when their energy exceeds the characteristic potential barrier of the random landscape. In stark contrast, wave-particle duality causes electrons in disordered media to come to rest even when the potential is weak—a remarkable phenomenon known as Anderson localization. Here, we present a hydrodynamic active system with wave-particle features, a millimetric droplet self-guided by its own wave field over a submerged random topography, whose dynamics exhibits localized statistics analogous to those of electronic systems. Consideration of an ensemble of particle trajectories reveals a suppression of diffusion when the guiding wave field extends over the disordered topography. We rationalize mechanistically the emergent statistics by virtue of the wave-mediated resonant coupling between the droplet and topography, which produces an attractive wave potential about the localization region. This hydrodynamic analog, which demonstrates how a classical particle may localize like a wave, suggests new directions for future research in various areas, including active matter, wave localization, many-body localization, and topological matter. Published by the American Physical Society 2024

A chiral active particle on two-dimensional random landscapes: ergodic uncertain diffusion and non-ergodic subdiffusion

A chiral active particle on two-dimensional random landscapes: ergodic uncertain diffusion and non-ergodic subdiffusion

Quantum exploration of high-dimensional canyon landscapes

Canyon landscapes in high dimension can be described as manifolds of small, but extensive dimension, immersed in a higher dimensional ambient space and characterized by a zero potential energy on the manifold. Here we consider the problem of a quantum particle exploring a prototype of a high-dimensional random canyon landscape. We characterize the thermal partitionfunction and show that around the point where the classical phase space has a satisfiability transition so that zero potential energy canyons disappear, moderate quantum fluctuations have a deleterious effect: they induce glassy phasesat temperature where classical thermal fluctuations alone would thermalize the system. Surprisingly we show that even when, classically, diffusion is expected to be unbounded in space, the interplay between quantum fluctuations and the randomness of the canyon landscape conspire to have a confining effect.

Observation and all-optical manipulation of replica symmetry breaking dynamics in a multi-Stokes-involved Brillouin random fiber laser photonic system

In this paper, we propose and demonstrate an all-optical control of RSB transition in a multi-wavelength Brillouin random fiber laser (MWBRFL). Multi-order Stokes light components can be subsequently generated by increasing the power of the Erbium-doped fiber amplifier (EDFA) inside the MWBRFL, providing additional disorder as well as multiple Stokes-involved interplay. It essentially allows diversified laser mode landscapes with adjustable average mode lifetime and random mode density of the 1st order Stokes, which benefits the switching between replica symmetry breaking (RSB) and replica symmetry (RS) states in an optically controlled manner. Results show that the average mode lifetime of the 1st order Stokes component gradually decreases from 250.0 ms to 1.2 ms as high orders from the 2nd to the 5th of Stokes components are activated. Meanwhile, the order parameter q of the 1st order Stokes random lasing emission presents distinct statistical distributions within the selective sub-window under various EDFA optical powers. Consequently, all-optical dynamical control of the 1st Stokes random laser mode landscapes with adjustable average mode lifetime turns out to be attainable, facilitating the RSB transition under an appropriate observation time window. These findings open a new avenue for exploring the underlying physical mechanisms behind the occurrence of the RSB phenomenon in photonic complex systems.

Evolutionary accessibility of random and structured fitness landscapes

Biological evolution can be conceptualized as a search process in the space of gene sequences guided by the fitness landscape, a mapping that assigns a measure of reproductive value to each genotype. Here, we discuss probabilistic models of fitness landscapes with a focus on their evolutionary accessibility, where a path in a fitness landscape is said to be accessible if the fitness values encountered along the path increase monotonically. For uncorrelated (random) landscapes with independent and identically distributed fitness values, the probability of existence of accessible paths between genotypes at a distance linear in the sequence length L becomes nonzero at a nontrivial threshold value of the fitness difference between the initial and final genotypes, which can be explicitly computed for large classes of genotype graphs. The behaviour of uncorrelated random landscapes is contrasted with landscape models that display additional, biologically motivated structural features. In particular, landscapes defined by a tradeoff between adaptation to environmental extremes have been found to display a combinatorially large number of accessible paths to all local fitness maxima. We show that this property is characteristic of a broad class of models that satisfy a certain global constraint, and provide further examples from this class.

Avalanches and Extreme Value Statistics of a Mesoscale Moving Contact Line.

We report direct atomic force microscopy measurements of pinning-depinning dynamics of a circular moving contact line (CL) over the rough surface of a micron-sized vertical hanging glass fiber, which intersects a liquid-air interface. The measured capillary force acting on the CL exhibits sawtoothlike fluctuations, with a linear accumulation of force of slope k (stick) followed by a sharp release of force δf, which is proportional to the CL slip length. From a thorough analysis of a large volume of the stick-slip events, we find that the local maximal force F_{c} needed for CL depinning follows the extreme value statistics and the measured δf follows the avalanche dynamics with a power law distribution in good agreement with the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model. The experiment provides an accurate statistical description of the CL dynamics at mesoscale, which has important implications to a common class of problems involving stick-slip motion in a random defect or roughness landscape.

Transient replica symmetry breaking in Brillouin random fiber lasers

Replica symmetry breaking (RSB), as a featured phase transition between paramagnetic and spin glass state in magnetic systems, has been predicted and validated among random laser-based complex systems, which involves numerous random modes interplayed via gain competition and exhibits disorder-induced frustration for glass behavior. However, the dynamics of RSB phase transition involving micro-state evolution of a photonic complex system have never been well investigated. Here, we report experimental evidence of transient RSB in a Brillouin random fiber laser (BRFL)-based photonic system through high-resolution unveiling of random laser mode landscape based on heterodyne technique. Thanks to the prolonged lifetime of activated random modes in BRFLs, an elaborated mapping of time-dependent statistics of the Parisi overlap parameter in both time and frequency domains was timely resolved, attributing to a compelling analogy between the transient RSB dynamics and the random mode evolution. These findings highlight that BRFL-based systems with the flexible harness of a customized photonic complex platform allow a superb opportunity for time-resolved transient RSB observation, opening new avenues in exploring fundamentals and application of complex systems and nonlinear phenomena.

Overlaps between eigenvectors of spiked, correlated random matrices: From matrix principal component analysis to random Gaussian landscapes.

We consider pairs of Gaussian orthogonal ensemble matrices which are correlated with each other and subject to additive and multiplicative rank-one perturbations. We focus on the regime of parameters in which the finite-rank perturbations generate outliers in the spectrum of the matrices. We investigate the statistical correlation (i.e., the typical overlap) between the eigenvectors associated to the outlier eigenvalues of each matrix in the pair, as well as the typical overlap between the outlier eigenvector of one matrix with the eigenvectors in the bulk of the spectrum of the other matrix. We discuss implications of these results for the signal recovery problem for spiked matrices, as well as for problems of high-dimensional random landscapes.

Non-self-averaging of current in a totally asymmetric simple exclusion process with quenched disorder.

We investigate the current properties in the totally asymmetric simple exclusion process (TASEP) on a quenched random energy landscape. In low- and high-density regimes, the properties are characterized by single-particle dynamics. In the intermediate one, the current becomes constant and is maximized. Based on the renewal theory, we derive accurate results for the maximum current. The maximum current significantly depends on a disorder realization, i.e., non-self-averaging (SA). We demonstrate that the disorder average of the maximum current decreases with the system size, and the sample-to-sample fluctuations of the maximum current exceed those of current in the low- and high-density regimes. We find a significant difference between single-particle dynamics and the TASEP. In particular, the non-SA behavior of the maximum current is always observed, whereas the transition from non-SA to SA for current in single-particle dynamics exists.

Sample-to-sample fluctuations of transport coefficients in the totally asymmetric simple exclusion process with quenched disorder.

We consider the totally asymmetric simple exclusion processes on quenched random energy landscapes. We show that the current and the diffusion coefficient differ from those for homogeneous environments. Using the mean-field approximation, we analytically obtain the site density when the particle density is low or high. As a result, the current and the diffusion coefficient are described by the dilute limit of particles or holes, respectively. However, in the intermediate regime, due to the many-body effect, the current and the diffusion coefficient differ from those for single-particle dynamics. The current is almost constant and becomes the maximal value in the intermediate regime. Moreover, the diffusion coefficient decreases with the particle density in the intermediate regime. We obtain analytical expressions for the maximal current and the diffusion coefficient based on the renewal theory. The deepest energy depth plays a central role in determining the maximal current and the diffusion coefficient. As a result, the maximal current and the diffusion coefficient depend crucially on the disorder, i.e., non-self-averaging. Based on the extreme value theory, we find that sample-to-sample fluctuations of the maximal current and diffusion coefficient are characterized by the Weibull distribution. We show that the disorder averages of the maximal current and the diffusion coefficient converge to zero as the system size is increased and quantify the degree of the non-self-averaging effect for the maximal current and the diffusion coefficient.

Accessibility percolation on Cartesian power graphs

A fitness landscape is a mapping from a space of discrete genotypes to the real numbers. A path in a fitness landscape is a sequence of genotypes connected by single mutational steps. Such a path is said to be accessible if the fitness values of the genotypes encountered along the path increase monotonically. We study accessible paths on random fitness landscapes of the House-of-Cards type, on which fitness values are independent, identically and continuously distributed random variables. The genotype space is taken to be a Cartesian power graph {mathcal {A}^L}, where L is the number of genetic loci and the allele graph mathcal {A} encodes the possible allelic states and mutational transitions on one locus. The probability of existence of accessible paths between two genotypes at a distance linear in L displays a transition from 0 to a positive value at a threshold beta _text {c} for the fitness difference between the initial and final genotype. We derive a lower bound on beta _text {c} for general mathcal {A} and show that this bound is tight for a large class of allele graphs. Our results generalize previous results for accessibility percolation on the biallelic hypercube, and compare favorably to published numerical results for multiallelic Hamming graphs.

Exponential growth of random determinants beyond invariance

We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with covariance profiles, Wigner matrices and covariance matrices with subexponential tails, Erd\H{o}s-R\'enyi and $d$-regular graphs for any polynomial sparsity parameter, and non-mean-field random matrix models, such as random band matrices for any polynomial bandwidth. The proof builds on recent tools, including the theory of the Matrix Dyson Equation as developed in [Ajanki, Erd\H{o}s, Kr\"uger 2019]. We use these asymptotics as an important input to identify the complexity of classes of Gaussian random landscapes in our companion papers [Ben Arous, Bourgade, McKenna 2021; McKenna 2021].

Statistical properties of inflationary saddles in Gaussian random landscapes

Random, multifield functions can set generic expectations for landscape-style cosmologies. We consider the inflationary implications of a landscape defined by a Gaussian random function, which is perhaps the simplest such scenario. Many key properties of this landscape, including the distribution of saddles as a function of height in the potential, depend only on its dimensionality, N, and a single parameter, γ, which is set by the power spectrum of the random function. We show that for saddles with a single downhill direction the negative mass term grows smaller relative to the average mass as N increases, a result with potential implications for the η-problem in landscape scenarios. For some power spectra, Planck-scale saddles have η ∼ 1 and eternal, topological inflation would be common in these scenarios. Lower-lying saddles typically have large η, but the fraction of these saddles which would support inflation is computable, allowing us to identify which scenarios can deliver a universe that resembles ours. Finally, by drawing inferences about the relative viability of different multiverse proposals we also illustrate ways in which quantitative analyses of multiverse scenarios are feasible.

Heterogeneity and Memory Effect in the Sluggish Dynamics of Vacancy Defects in Colloidal Disordered Crystals and Their Implications to High-Entropy Alloys.

Vacancy dynamics of high-density 2D colloidal crystals with a polydispersity in particle size are studied experimentally. Heterogeneity in vacancy dynamics is observed. Inert vacancies that hardly hop to other lattice sites and active vacancies that hop frequently between different lattice sites are found within the same samples. The vacancies show high probabilities of first hopping from one lattice site to another neighboring lattice site, then staying at the new site for some time, and later hopping back to the original site in the next hop. This back-returning hop probability increases monotonically with the increase in packing fraction, up to 83%. This memory effect makes the active vacancies diffuse sluggishly or even get trapped in local regions. Strain-induced vacancy motion on a distorted lattice is also observed. New glassy properties in the disordered crystals are discovered, including the dynamical heterogeneity, the presence of cooperative rearranging regions, memory effect, etc. Similarities between the colloidal disordered crystals and the high-entropy alloys (HEAs) are also discussed. Molecular dynamics simulations further support the experimental observations. These results help to understand the microscopic origin of the sluggish dynamics in materials with ordered structures but in random energy landscapes, such as high-entropyalloys.

Superposition of random plane waves in high spatial dimensions: Random matrix approach to landscape complexity

Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in RN obtained by superimposing M > N plane waves of random wavevectors and amplitudes and further restricted by a uniform parabolic confinement in all directions. For this landscape, we show how to compute the “annealed complexity,” controlling the asymptotic growth rate of the mean number of stationary points as N → ∞ at fixed ratio α = M/N > 1. The framework of this computation requires us to study spectral properties of N × N matrices W = KTKT, where T is a diagonal matrix with M mean zero independent and identically distributed (i.i.d.) real normally distributed entries, and all MN entries of K are also i.i.d. real normal random variables. We suggest to call the latter Gaussian Marchenko–Pastur ensemble as such matrices appeared in the seminal 1967 paper by those authors. We compute the associated mean spectral density and evaluate some moments and correlation functions involving products of characteristic polynomials for such matrices.

Hessian characterization of the pinning landscape in a type-II superconductor

Supercurrent transport requires vortices in superconductors to be pinned by material defects. Vortices are trapped in minima of the pinning potential landscape, and one has to determine what fraction of this landscape comes with a positive curvature. Such a curvature analysis is done by studying the Hessian matrix of the landscape. The authors find the stable area fraction to be unexpectedly low (yellow color in the figure). In particular, for Gaussian random landscapes it is $(3\ensuremath{-}\sqrt{3})/6\ensuremath{\approx}21%$. The results offer new hints for optimizing pinning landscapes. Their transdisciplinary application to natural landscapes exhibits a surprising universality.

Glass-like transition described by toppling of stability hierarchy* *Dedicated to the memory of Fritz Haake.

Building on the work of Fyodorov (2004) and Fyodorov and Nadal (2012) we examine the critical behaviour of population of saddles with fixed instability index k in high dimensional random energy landscapes. Such landscapes consist of a parabolic confining potential and a random part in N ≫ 1 dimensions. When the relative strength m of the parabolic part is decreasing below a critical value m c, the random energy landscapes exhibit a glass-like transition from a simple phase with very few critical points to a complex phase with the energy surface having exponentially many critical points. We obtain the annealed probability distribution of the instability index k by working out the mean size of the population of saddles with index k relative to the mean size of the entire population of critical points and observe toppling of stability hierarchy which accompanies the underlying glass-like transition. In the transition region m = m c + δN −1/2 the typical instability index scales as k = κN 1/4 and the toppling mechanism affects whole instability index distribution, in particular the most probable value of κ changes from κ = 0 in the simple phase (δ > 0) to a non-zero value κ max ∝ (−δ)3/2 in the complex phase (δ < 0). We also show that a similar phenomenon is observed in random landscapes with an additional fixed energy constraint and in the p-spin spherical model.

Interfacial interactions of rough spherical surfaces with random topographies

A mathematical model was created based on surface element integral (SEI) and XDLVO theory for assessing the interfacial interaction between rough spherical surfaces with random topographies, which were generated following a modified two-variable Weierstrass-Mandelbrot (WM) function. The surface construction and interfacial interaction equations used in this study facilitated the generation of randomly rough spherical surfaces, which assisted in improving the prediction accuracy of particle interactions for natural colloidal particles. This modeling study presented discussions on the interaction of rough surfaces having different asperity heights, asperity positions, random landscape, and roughness in colloidal systems. We observed that the asperity number and ratio were primary parameters for influencing the interfacial interaction between spherical surfaces. The arrangement and randomness in the position of asperities on the surface had negligible effects on the interfacial interaction. The elevated asperity height, as a result of increased fractal roughness or relative fractal roughness on spherical surfaces, could hamper the interfacial energy between surfaces. However, increasing the fractal dimension and relative fractal dimension generated smoother surfaces and thus elevated the interfacial energy developed between surfaces. The most impactful parameter of surface morphologies in altering interfacial energy was fractal dimension as it could control the asperity height and asperity number simultaneously. The largest primary maximum was predicted (216 kT) when the fractal dimension was 2.43, which represented the strongest stability of particles in a suspension.

Charge transport in the spatially correlated exponential random energy landscape: effect of the nonpositive correlation function.

Charge transport in amorphous semiconductors having spatially correlated exponential density of states (DOS) has been considered for the arbitrary behavior of the correlation function of random energies. Average carrier velocity is exactly calculated for the quasi-equilibrium (nondispersive) transport regime. For the symmetric exponential DOS with exponential tails for low and high energies and nonpositive correlation function the temperature of the transition to the dispersive transport regime depends on correlation properties and becomes greater than the traditional estimation based on the DOS decay energy kT = U0. Another new feature of the transport in the landscape having nonpositive correlation function is the decrease of the mobility with the field in the low field region and development of the universal mobility field dependence for stronger fields.

Efficient approximation of branching random walk Gibbs measures

Disordered systems such as spin glasses have been used extensively as models for high-dimensional random landscapes and studied from the perspective of optimization algorithms. In a recent paper by L. Addario-Berry and the second author, the continuous random energy model (CREM) was proposed as a simple toy model to study the efficiency of such algorithms. The following question was raised in that paper: what is the threshold $\beta_G$, at which sampling (approximately) from the Gibbs measure at inverse temperature $\beta$ becomes algorithmically hard? This paper is a first step towards answering this question. We consider the branching random walk, a time-homogeneous version of the continuous random energy model. We show that a simple greedy search on a renormalized tree yields a linear-time algorithm which approximately samples from the Gibbs measure, for every $\beta < \beta_c$, the (static) critical point. More precisely, we show that for every $\varepsilon>0$, there exists such an algorithm such that the specific relative entropy between the law sampled by the algorithm and the Gibbs measure of inverse temperature $\beta$ is less than $\varepsilon$ with high probability. In the supercritical regime $\beta > \beta_c$, we provide the following hardness result. Under a mild regularity condition, for every $\delta > 0$, there exists $z>0$ such that the running time of any given algorithm approximating the Gibbs measure stochastically dominates a geometric random variable with parameter $e^{-z\sqrt{N}}$ on an event with probability at least $1-\delta$.