Depinning in the quenched Kardar-Parisi-Zhang class. I. Mappings, simulations, and algorithm.

Depinning of elastic systems advancing on disordered media can usually be described by the quenched Edwards-Wilkinson equation (qEW). However, additional ingredients such as anharmonicity and forces that cannot be derived from a potential energy may generate a different scaling behavior at depinning. The most experimentally relevant is the Kardar-Parisi-Zhang (KPZ) term, proportional to the square of the slope at each site, which drives the critical behavior into the so-called quenched KPZ (qKPZ) universality class. We study this universality class both numerically and analytically: by using exact mappings we show that at least for d=1,2 this class encompasses not only the qKPZ equation itself, but also anharmonic depinning and a well-known class of cellular automata introduced by Tang and Leschhorn. We develop scaling arguments for all critical exponents, including size and duration of avalanches. The scale is set by the confining potential strength m^{2}. This allows us to estimate numerically these exponents as well as the m-dependent effective force correlator Δ(w), and its correlation length ρ:=Δ(0)/|Δ^{'}(0)|. Finally, we present an algorithm to numerically estimate the effective (m-dependent) elasticity c, and the effective KPZ nonlinearity λ. This allows us to define a dimensionless universal KPZ amplitude A:=ρλ/c, which takes the value A=1.10(2) in all systems considered in d=1. This proves that qKPZ is the effective field theory for all these models. Our work paves the way for a deeper understanding of depinning in the qKPZ class, and in particular, for the construction of a field theory that we describe in a companion paper.

Depinning in the quenched Kardar-Parisi-Zhang class. II. Field theory.

There are two main universality classes for depinning of elastic interfaces in disordered media: quenched Edwards-Wilkinson (qEW) and quenched Kardar-Parisi-Zhang (qKPZ). The first class is relevant as long as the elastic force between two neighboring sites on the interface is purely harmonic and invariant under tilting. The second class applies when the elasticity is nonlinear or the surface grows preferentially in its normal direction. It encompasses fluid imbibition, the Tang-Leschorn cellular automaton of 1992 (TL92), depinning with anharmonic elasticity (aDep), and qKPZ. While the field theory is well developed for qEW, there is no consistent theory for qKPZ. The aim of this paper is to construct this field theory within the functional renormalization group (FRG) framework, based on large-scale numerical simulations in dimensions d=1, 2, and 3, presented in a companion paper [Mukerjee et al., Phys. Rev. E 107, 054136 (2023)10.1103/PhysRevE.107.054136]. In order to measure the effective force correlator and coupling constants, the driving force is derived from a confining potential with curvature m^{2}. We show, that contrary to common belief, this is allowed in the presence of a KPZ term. The ensuing field theory becomes massive and can no longer be Cole-Hopf transformed. In exchange, it possesses an IR attractive stable fixed point at a finite KPZ nonlinearity λ. Since there is neither elasticity nor a KPZ term in dimension d=0, qEW and qKPZ merge there. As a result, the two universality classes are distinguished by terms linear in d. This allows us to build a consistent field theory in dimension d=1, which loses some of its predictive powers in higher dimensions.

Experimental Test of Sinai's Model in DNA Unzipping.

The experimental measurement of correlation functions and critical exponents in disordered systems is key to testing renormalization group (RG) predictions. We mechanically unzip single DNA hairpins with optical tweezers, an experimental realization of the diffusive motion of a particle in a one-dimensional random force field, known as the Sinai model. We measure the unzipping forces F_{w} as a function of the trap position w in equilibrium and calculate the force-force correlator Δ_{m}(w), its amplitude, and correlation length, finding agreement with theoretical predictions. We study the universal scaling properties since the effective trap stiffness m^{2} decreases upon unzipping. Fluctuations of the position of the base pair at the unzipping junction u scales as u∼m^{-ζ}, with a roughness exponent ζ=1.34±0.06, in agreement with the analytical prediction ζ=4/3. Our study provides a single-molecule test of the functional RG approach for disordered elastic systems in equilibrium.

Force Correlations in Disordered Magnets.

We present a proof of principle for the validity of the functional renormalization group, by measuring the force correlations in Barkhausen-noise experiments. Our samples are soft ferromagnets in two distinct universality classes, differing in the range of spin interactions, and the effects of eddy currents. We show that the force correlations have a universal form predicted by the functional renormalization group, distinct for short-range and long-range elasticity, and mostly independent of eddy currents. In all cases correlations grow linearly at small distances, as in mean-field models, but in contrast to the latter are bounded at large distances. As a consequence, avalanches are anti-correlated. We derive bounds for these anticorrelations, which are saturated in the experiments, showing that the multiple domain walls in our samples effectively behave as a single wall.

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles.

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Functionals of fractional Brownian motion and the three arcsine laws.

Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent H∈(0,1), generalizing standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical applications as a standard reference point for nonequilibrium dynamics. We describe a perturbation expansion allowing us to evaluate many nontrivial observables analytically: We generalize the celebrated three arcsine laws of standard Brownian motion. The functionals are: (i) the fraction of time the process remains positive, (ii) the time when the process last visits the origin, and (iii) the time when it achieves its maximum (or minimum). We derive expressions for the probability of these three functionals as an expansion in ɛ=H-1/2, up to second order. We find that the three probabilities are different, except for H=1/2, where they coincide. Our results are confirmed to high precision by numerical simulations.

Mean-field theories for depinning and their experimental signatures.

Mean-field theory is an approximation replacing an extended system by a few variables. For depinning of elastic manifolds, these are the position u of its center of mass and the statistics of the forces F(u). There are two proposals how to model the latter: as a random walk (ABBM model), or as uncorrelated forces at integer u (discretized particle model, DPM). While for many experiments the ABBM model (in the literature misleadingly equated with mean-field theory) makes quantitatively correct predictions for the distributions of velocities, or avalanche size and duration, the microscopic disorder force-force correlations cannot grow linearly, and thus unboundedly as a random walk, with distance. Even the effective (renormalized) disorder forces which do so at small distances are bounded at large distances. To describe both regimes, we model forces as an Ornstein-Uhlenbeck process. The latter has the statistics of a random walk at small scales, and is uncorrelated at large scales. By connecting to results in both limits, we solve the model largely analytically, allowing us to describe in all regimes the distributions of velocity, avalanche size, and duration. To establish experimental signatures of this transition, we study the response function, and the correlation function of position u, velocity u[over ̇], and forces F under slow driving with velocity v>0. While at v=0 force or position correlations have a cusp at the origin and then decay at least exponentially fast to zero, this cusp is rounded at a finite driving velocity. We give a detailed analytic analysis for this rounding by velocity, which allows us, given experimental data, to extract the timescale of the response function, and to reconstruct the force-force correlator at v=0. The latter is the central object of the field theory, and as such contains detailed information about the universality class in question. We test our predictions by careful numerical simulations extending over up to ten orders in magnitude.

Extreme events for fractional Brownian motion with drift: Theory and numerical validation.

We study the first-passage time, the distribution of the maximum, and the absorption probability of fractional Brownian motion of Hurst parameter H with both a linear and a nonlinear drift. The latter appears naturally when applying nonlinear variable transformations. Via a perturbative expansion in ɛ=H-1/2, we give the first-order corrections to the classical result for Brownian motion analytically. Using a recently introduced adaptive-bisection algorithm, which is much more efficient than the standard Davies-Harte algorithm, we test our predictions for the first-passage time on grids of effective sizes up to N_{eff}=2^{28}≈2.7×10^{8} points. The agreement between theory and simulations is excellent, and by far exceeds in precision what can be obtained by scaling alone.

Sampling first-passage times of fractional Brownian motion using adaptive bisections.

We present an algorithm to efficiently sample first-passage times for fractional Brownian motion. To increase the resolution, an initial coarse lattice is successively refined close to the target, by adding exactly sampled midpoints, where the probability that they reach the target is non-negligible. Compared to a path of N equally spaced points, the algorithm achieves the same numerical accuracy N_{eff}, while sampling only a small fraction of all points. Though this induces a statistical error, the latter is bounded for each bridge, allowing us to bound the total error rate by a number of our choice, say P_{error}^{tot}=10^{-6}. This leads to significant improvements in both memory and speed. For H=0.33 and N_{eff}=2^{32}, we need 5000 times less CPU time and 10000 times less memory than the classical Davies-Harte algorithm. The gain grows for H=0.25 and N_{eff}=2^{42} to 3×10^{5} for CPU and 10^{6} for memory. We estimate our algorithmic complexity as C^{ABSec}(N_{eff})=O[(lnN_{eff})^{3}], to be compared to Davies-Harte, which has complexity C^{DH}(N)=O(NlnN). Decreasing P_{error}^{tot} results in a small increase in complexity, proportional to ln(1/P_{error}^{tot}). Our current implementation is limited to the values of N_{eff} given above, due to a loss of floating-point precision. Our algorithm can be adapted to other extreme events and arbitrary Gaussian processes. It enables one to numerically validate theoretical predictions that were hitherto inaccessible.

Fractal dimension of critical curves in the O(n)-symmetric ϕ

We calculate the fractal dimension d_{f} of critical curves in the O(n)-symmetric (ϕ[over ⃗]^{2})^{2} theory in d=4-ε dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at n=-2, self-avoiding walks (n=0), Ising lines (n=1), and XY lines (n=2), in agreement with numerical simulations. It can be compared to the fractal dimension d_{f}^{tot} of all lines, i.e., backbone plus the surrounding loops, identical to d_{f}^{tot}=1/ν. The combination ϕ_{c}=d_{f}/d_{f}^{tot}=νd_{f} is the crossover exponent, describing a system with mass anisotropy. Introducing a self-consistent resummation procedure and combining it with analytic results in d=2 allows us to give improved estimates in d=3 for all relevant exponents at 6-loop order.

Depinning Transition of Charge-Density Waves: Mapping onto O(n) Symmetric ϕ

Driven periodic elastic systems such as charge-density waves (CDWs) pinned by impurities show a nontrivial, glassy dynamical critical behavior. Their proper theoretical description requires the functional renormalization group. We show that their critical behavior close to the depinning transition is related to a much simpler model, O(n) symmetric ϕ^{4} theory in the unusual limit of n→-2. We demonstrate that both theories yield identical results to four-loop order and give both a perturbative and a nonperturbative proof of their equivalence. As we show, both theories can be used to describe loop-erased random walks (LERWs), the trace of a random walk where loops are erased as soon as they are formed. Remarkably, two famous models of non-self-intersecting random walks, self-avoiding walks and LERWs, can both be mapped onto ϕ^{4} theory, taken with formally n=0 and n→-2 components. This mapping allows us to compute the dynamic critical exponent of CDWs at the depinning transition and the fractal dimension of LERWs in d=3 with unprecedented accuracy, z(d=3)=1.6243±0.001, in excellent agreement with the estimate z=1.62400±0.00005 of numerical simulations.

Behavior of random RNA secondary structures near the glass transition.

RNA forms elaborate secondary structures through intramolecular base pairing. These structures perform critical biological functions within each cell. Due to the availability of a polynomial algorithm to calculate the partition function over these structures, they are also a suitable system for the statistical physics of disordered systems. In this model, below the denaturation temperature, random RNA secondary structures exist in one of two phases: a strongly disordered, low-temperature glass phase and a weakly disordered, high-temperature molten phase. The probability of two bases to pair decays with their distance with an exponent 3/2 in the molten phase and about 4/3 in the glass phase. Inspired by previous results from a renormalized field theory of the glass transition separating the two phases, we numerically study this transition. We introduce distinct order parameters for each phase that both vanish at the critical point. We finally explore the driving mechanism behind this transition.

First passage in an interval for fractional Brownian motion.

Let X_{t} be a random process starting at x∈[0,1] with absorbing boundary conditions at both ends of the interval. Denote by P_{1}(x) the probability to first exit at the upper boundary. For Brownian motion, P_{1}(x)=x, which is equivalent to P_{1}^{'}(x)=1. For fractional Brownian motion with Hurst exponent H, we establish that P_{1}^{'}(x)=N[x(1-x)]^{1/H-2}e^{εF(x)+O(ε^{2})}, where ε=H-1/2. The function F(x) is analytic and well approximated by its Taylor expansion F(x)≃16(C-1)(x-1/2)^{2}+O(x-1/2)^{4}, where C=0.915... is the Catalan constant. A similar result holds for moments of the exit time starting at x. We then consider the span of X_{t}, i.e., the size of the (compact) domain visited up to time t. For Brownian motion, we derive an analytic expression for the probability that the span reaches 1 for the first time and then generalize it to fractional Brownian motion. Using large-scale numerical simulations with system sizes up to N=2^{24} and a broad range of H, we confirm our analytic results. There are important finite-discretization corrections which we quantify. They are most severe for small H, necessitating going to the large systems mentioned above.

Generalized Arcsine Laws for Fractional Brownian Motion.

The three arcsine laws for Brownian motion are a cornerstone of extreme-value statistics. For a Brownian B_{t} starting from the origin, and evolving during time T, one considers the following three observables: (i) the duration t_{+} the process is positive, (ii) the time t_{last} the process last visits the origin, and (iii) the time t_{max} when it achieves its maximum (or minimum). All three observables have the same cumulative probability distribution expressed as an arcsine function, thus the name arcsine laws. We show how these laws change for fractional Brownian motion X_{t}, a non-Markovian Gaussian process indexed by the Hurst exponent H. It generalizes standard Brownian motion (i.e., H=1/2). We obtain the three probabilities using a perturbative expansion in ϵ=H-1/2. While all three probabilities are different, this distinction can only be made at second order in ϵ. Our results are confirmed to high precision by extensive numerical simulations.

Spatial shape of avalanches.

In disordered elastic systems, driven by displacing a parabolic confining potential adiabatically slowly, all advance of the system is in bursts, termed avalanches. Avalanches have a finite extension in time, which is much smaller than the waiting time between them. Avalanches also have a finite extension ℓ in space, i.e., only a part of the interface of size ℓ moves during an avalanche. Here we study their spatial shape 〈S(x)〉_{ℓ} given ℓ, as well as its fluctuations encoded in the second cumulant 〈S^{2}(x)〉_{ℓ}^{c}. We establish scaling relations governing the behavior close to the boundary. We then give analytic results for the Brownian force model, in which the microscopic disorder for each degree of freedom is a random walk. Finally, we confirm these results with numerical simulations. To do this properly we elucidate the influence of discretization effects, which also confirms the assumptions entering into the scaling ansatz. This allows us to reach the scaling limit already for avalanches of moderate size. We find excellent agreement for the universal shape and its fluctuations, including all amplitudes.

Extreme-value statistics of fractional Brownian motion bridges.

Fractional Brownian motion is a self-affine, non-Markovian, and translationally invariant generalization of Brownian motion, depending on the Hurst exponent H. Here we investigate fractional Brownian motion where both the starting and the end point are zero, commonly referred to as bridge processes. Observables are the time t_{+} the process is positive, the maximum m it achieves, and the time t_{max} when this maximum is taken. Using a perturbative expansion around Brownian motion (H=1/2), we give the first-order result for the probability distribution of these three variables and the joint distribution of m and t_{max}. Our analytical results are tested and found to be in excellent agreement, with extensive numerical simulations for both H>1/2 and H<1/2. This precision is achieved by sampling processes with a free end point and then converting each realization to a bridge process, in generalization to what is usually done for Brownian motion.

Universal correlations between shocks in the ground state of elastic interfaces in disordered media.

The ground state of an elastic interface in a disordered medium undergoes collective jumps upon variation of external parameters. These mesoscopic jumps are called shocks, or static avalanches. Submitting the interface to a parabolic potential centered at w, we study the avalanches which occur as w is varied. We are interested in the correlations between the avalanche sizes S_{1} and S_{2} occurring at positions w_{1} and w_{2}. Using the functional renormalization group (FRG), we show that correlations exist for realistic interface models below their upper critical dimension. Notably, the connected moment 〈S_{1}S_{2}〉^{c} is up to a prefactor exactly the renormalized disorder correlator, itself a function of |w_{2}-w_{1}|. The latter is the universal function at the center of the FRG; hence, correlations between shocks are universal as well. All moments and the full joint probability distribution are computed to first nontrivial order in an ε expansion below the upper critical dimension. To quantify the local nature of the coupling between avalanches, we calculate the correlations of their local jumps. We finally test our predictions against simulations of a particle in random-bond and random-force disorder, with surprisingly good agreement.

Perturbative expansion for the maximum of fractional Brownian motion.

Brownian motion is the only random process which is Gaussian, scale invariant, and Markovian. Dropping the Markovian property, i.e., allowing for memory, one obtains a class of processes called fractional Brownian motion, indexed by the Hurst exponent H. For H=1/2, Brownian motion is recovered. We develop a perturbative approach to treat the nonlocality in time in an expansion in ɛ=H-1/2. This allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics: the maximum m of the process and the time t_{max} at which this maximum is reached, as well as their joint distribution. We test our analytical predictions with extensive numerical simulations for different values of H. They show excellent agreement, even for H far from 1/2.

Distribution of joint local and total size and of extension for avalanches in the Brownian force model.

The Brownian force model is a mean-field model for local velocities during avalanches in elastic interfaces of internal space dimension d, driven in a random medium. It is exactly solvable via a nonlinear differential equation. We study avalanches following a kick, i.e., a step in the driving force. We first recall the calculation of the distributions of the global size (total swept area) and of the local jump size for an arbitrary kick amplitude. We extend this calculation to the joint density of local and global sizes within a single avalanche in the limit of an infinitesimal kick. When the interface is driven by a single point, we find new exponents τ_{0}=5/3 and τ=7/4, depending on whether the force or the displacement is imposed. We show that the extension of a "single avalanche" along one internal direction (i.e., the total length in d=1) is finite, and we calculate its distribution following either a local or a global kick. In all cases, it exhibits a divergence P(ℓ)∼ℓ^{-3} at small ℓ. Most of our results are tested in a numerical simulation in dimension d=1.

Dynamical selection of critical exponents.

In renormalized field theories there are in general one or few fixed points that are accessible by the renormalization-group flow. They can be identified from the fixed-point equations. Exceptionally, an infinite family of fixed points exists, parameterized by a scaling exponent ζ, itself a function of a nonrenormalizing parameter. Here we report a different scenario with an infinite family of fixed points of which seemingly only one is chosen by the renormalization-group flow. This dynamical selection takes place in systems with an attractive interaction V(ϕ), as in standard ϕ^{4} theory, but where the potential V at large ϕ goes to zero, as, e.g., the attraction by a defect.