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Kay Joerg Wiese

Publications and source records attributed to Kay Joerg Wiese.

At least 37 records · Page 2Linked to original sources
arXiv2020

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 non-linear drift. The latter appears naturally when applying non-linear variable transformations. Via a perturbative expansion in $ε= H-1/2$, 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 $N_{\rm eff}=2^{28}\approx 2.7\times 10^{8}$ points. The agreement between theory and simulations is excellent, and by far exceeds in precision what can be obtained by scaling alone.

cond-mat.stat-mech↗
arXiv2020

Fractal dimension of critical curves in the $O(n)$-symmetric $ϕ^4$-model and crossover exponent at 6-loop order: Loop-erased random walks, self-avoiding walks, Ising, XY and Heisenberg models

We calculate the fractal dimension $d_{\rm f}$ of critical curves in the $O(n)$ symmetric $(\vec ϕ^2)^2$-theory in $d=4-\varepsilon$ 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 $d_{\rm f}^{\rm tot}$ of all lines, i.e. backbone plus the surrounding loops, identical to $d_{\rm f}^{\rm tot} = 1/ν$. The combination $ϕ_{\rm c}= d_{\rm f}/d_{\rm f}^{\rm tot} = νd_{\rm f}$ is the crossover exponent, describing a system with mass anisotropy. Introducing a novel 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.

cond-mat.stat-mech↗
arXiv2019

Depinning transition of charge-density waves: mapping onto $O(n)$ symmetric $ϕ^4$ theory with $n\to -2$ and loop-erased random walks

Driven periodic elastic systems such as charge-density waves (CDWs) pinned by impurities show a non-trivial, 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 $n\to -2$. We demonstrate that both theories yield identical results to 4-loop order and give both a perturbative and a non-perturbative 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 (SAWs) and LERWs, can both be mapped onto $ϕ^4$ theory taken, with formally $n=0$ and $n\to -2$ 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, $z(d=3)= 1.6243 \pm 0.001$, in excellent agreement with the estimate $z = 1.624 00 \pm 0.00005$ of numerical simulations.

cond-mat.dis-nn↗
arXiv2019

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 $N_{\rm eff}$, 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 $P_{\rm error}^{\rm tot}=10^{-6}$. This leads to significant improvements in both memory and speed. For $H=0.33$ and $N_{\rm eff}=2^{32}$, we need $5\,000$ times less CPU time and $10\, 000$ times less memory than the classical Davies Harte algorithm. The gain grows for $H=0.25$ and $N_{\rm eff} = 2^{42}$ to $3\cdot 10^{5}$ for CPU and $10^6$ for memory. We estimate our algorithmic complexity as ${\cal C}^{\rm ABSec}(N_{\rm eff}) = {\cal O}\left(\left( \ln N_{\rm eff}\right)^{3}\right)$, to be compared to Davies Harte which has complexity ${\cal C}^{\rm DH}(N) = {\cal O}\left(N \ln N \right)$. Decreasing $P_{\rm error}^{\rm tot}$ results in a small increase in complexity, proportional to $\ln (1/P_{\rm error}^{\rm tot})$. Our current implementation is limited to the values of $N_{\rm eff}$ given above, due to a loss of floating-point precision. The algorithm can be adapted to other extreme events and arbitrary Gaussian processes. It enables one to numerically validate theoretical predictions that were hitherto inaccessible.

cond-mat.stat-mech↗
arXiv2019

Distribution of velocities in an avalanche, and related quantities: Theory and numerical verification

We study several probability distributions relevant to the avalanche dynamics of elastic interfaces driven on a random substrate: The distribution of size, duration, lateral extension or area, as well as velocities. Results from the functional renormalization group and scaling relations involving two independent exponents, roughness $ζ$, and dynamics $z$, are confronted to high-precision numerical simulations of an elastic line with short-range elasticity, i.e. of internal dimension $d=1$. The latter are based on a novel stochastic algorithm which generates its disorder on the fly. Its precision grows linearly in the time-discretization step, and it is parallelizable. Our results show good agreement between theory and numerics, both for the critical exponents as for the scaling functions. In particular, the prediction ${\sf a} = 2 - \frac{2}{d+ ζ- z}$ for the velocity exponent is confirmed with good accuracy.

cond-mat.dis-nn↗
arXiv2019

Field Theories for Loop-Erased Random Walks

Self-avoiding walks (SAWs) and loop-erased random walks (LERWs) are two ensembles of random paths with numerous applications in mathematics, statistical physics and quantum field theory. While SAWs are described by the $n \to 0$ limit of $ϕ^4$-theory with $O(n)$-symmetry, LERWs have no obvious field-theoretic description. We analyse two candidates for a field theory of LERWs, and discover a connection between the corresponding and a priori unrelated theories. The first such candidate is the $O(n)$-symmetric $ϕ^4$ theory at $n=-2$ whose link to LERWs was known in two dimensions due to conformal field theory. Here it is established in arbitrary dimension via a perturbation expansion in the coupling constant. The second candidate is a field theory for charge-density waves pinned by quenched disorder, whose relation to LERWs had been conjectured earlier using analogies with Abelian sandpiles. We explicitly show that both theories yield identical results to 4-loop order and give both a perturbative and a non-perturbative proof of their equivalence. This allows us to compute the fractal dimension of LERWs to order $ε^5$ where $ε=4-d$. In particular, in $d=3$ our theory gives $z_{\rm LERW}(d=3)= 1.6243 \pm 0.001$, in excellent agreement with the estimate $z = 1.624 00 \pm 0.00005$ of numerical simulations.

cond-mat.stat-mech↗
arXiv2019

Span observables - "When is a foraging rabbit no longer hungry?"

Be $X_t$ a random walk. We study its span $S$, i.e. the size of the domain visited up to time $t$. We want to know the probability that $S$ reaches $1$ for the first time, as well as the density of the span given $t$. Analytical results are presented, and checked against numerical simulations. We then generalize this to include drift, and one or two reflecting boundaries. We also derive the joint probability of the maximum and minimum of a process. Our results are based on the diffusion propagator with reflecting or absorbing boundaries, for which a set of useful formulas is derived.

cond-mat.stat-mech↗
arXiv2019

First passage in an interval for fractional Brownian motion

Be $X_t$ a random process starting at $x \in [0,1]$ with absorbing boundary conditions at both ends of the interval. Denote $P_1(x)$ the probability to first exit at the upper boundary. For Brownian motion, $P_1(x)=x$, equivalent to $P_1'(x)=1$. For fractional Brownian motion with Hurst exponent $H$, we establish that $P_1'(x) = {\cal N} [x(1-x)]^{\frac1H -2} e^{ε{\cal F}(x)+ {\cal O}(ε^2)}$, where $ε=H-\frac12$. The function ${\cal F}(x)$ is analytic, and well approximated by its Taylor expansion, ${\cal F}(x)\simeq 16 (C-1) (x-1/2)^2 +{\cal O}(x-1/2)^4$, 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, then generalized to fBm. 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 to go to the large systems mentioned above.

cond-mat.stat-mech↗
arXiv2018

Field Theory of Disordered Elastic Interfaces at 3-Loop Order: The $β$-Function

We calculate the effective action for disordered elastic manifolds in the ground state (equilibrium) up to 3-loop order. This yields the renormalization-group $β$-function to third order in $ε=4-d$, in an expansion in the dimension $d$ around the upper critical dimension $d=4$. The calculations are performed using exact RG, and several other techniques, which allow us to resolve consistently the problems associated with the cusp of the renormalized disorder.

cond-mat.dis-nn↗
arXiv2017

The 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 $\ell$ in space, i.e. only a part of the interface of size $\ell$ moves during an avalanche. Here we study their spatial shape $\left< S(x)\right>_{\ell}$ given $\ell$, as well as its fluctuations encoded in the second cumulant $\left< S^{2}(x)\right>_{\ell}^{\rm c}$. 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, its fluctuations, including all amplitudes.

cond-mat.dis-nn↗
arXiv2016

Dynamical Selection of Critical Exponents

In renormalized field theories there are in general one or few fixed points which 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 function of a non-renormalizing 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 ${\cal V}(ϕ)$, as in standard $ϕ^4$ theory, but where the potential $\cal V$ at large $ϕ$ goes to zero, as e.g. the attraction by a defect.

cond-mat.stat-mech↗
arXiv2015

The Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory

Fractional Brownian motion is a non-Markovian Gaussian process $X_t$, indexed by the Hurst exponent $H$. It generalises standard Brownian motion (corresponding to $H=1/2$). We study the probability distribution of the maximum $m$ of the process and the time $t_{\rm max}$ at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting $H=1/2 + \varepsilon$. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of $H$ test these analytical predictions and show excellent agreement, even for large $\varepsilon$.

cond-mat.stat-mech↗
arXiv2013

Collective excitations in a large-d model for graphene

We consider a model of Dirac fermions coupled to flexural phonons to describe a graphene sheet fluctuating in dimension $2+d$. We derive the self-consistent screening equations for the quantum problem, exact in the limit of large $d$. We first treat the membrane alone, and work out the quantum to classical, and harmonic to anharmonic crossover. For the coupled electron-membrane problem we calculate the dressed two-particle propagators of the elastic and electron interactions and find that it exhibits a collective mode which becomes unstable at some wave-vector $q_{\rm c}$ for large enough coupling $g$. The saddle point analysis, exact at large $d$, indicates that this instability corresponds to spontaneous and simultaneous appearance of gaussian curvature and electron puddles. The relevance to ripples in graphene is discussed.

cond-mat.mes-hall↗
arXiv2013

Non-Gaussian effects and multifractality in the Bragg glass

We study, beyond the Gaussian approximation, the decay of the translational order correlation function for a d-dimensional scalar periodic elastic system in a disordered environment. We develop a method based on functional determinants, equivalent to summing an infinite set of diagrams. We obtain, in dimension d=4-epsilon, the even n-th cumulant of relative displacements as <[u(r)-u(0)]^n>^c = A_n ln r, with A_n = -(ε/3)^n Γ(n-1/2) ζ(2n-3)/π^(1/2), as well as the multifractal dimension x_q of the exponential field e^{q u(r)}. As a corollary, we obtain an analytic expression for a class of n-loop integrals in d=4, which appear in the perturbative determination of Konishi amplitudes, also accessible via AdS/CFT using integrability.

cond-mat.dis-nn↗
arXiv2013

Avalanche dynamics of elastic interfaces

Slowly driven elastic interfaces, such as domain walls in dirty magnets, contact lines, or cracks proceed via intermittent motion, called avalanches. We develop a field-theoretic treatment to calculate, from first principles, the space-time statistics of instantaneous velocities within an avalanche. For elastic interfaces at (or above) their (internal) upper critical dimension d >= d_uc (d_uc = 2, 4 respectively for long-ranged and short-ranged elasticity) we show that the field theory for the center of mass reduces to the motion of a point particle in a random-force landscape, which is itself a random walk (ABBM model). Furthermore, the full spatial dependence of the velocity correlations is described by the Brownian-force model (BFM) where each point of the interface sees an independent Brownian-force landscape. Both ABBM and BFM can be solved exactly in any dimension d (for monotonous driving) by summing tree graphs, equivalent to solving a (non-linear) instanton equation. This tree approximation is the mean-field theory (MFT) for realistic interfaces in short-ranged disorder. Both for the center of mass, and for a given Fourier mode q, we obtain probability distribution functions (PDF's) of the velocity, as well as the avalanche shape and its fluctuations (second shape). Within MFT we find that velocity correlations at non-zero q are asymmetric under time reversal. Next we calculate, beyond MFT, i.e. including loop corrections, the 1-time PDF of the center-of-mass velocity du/dt for dimension d< d_uc. The singularity at small velocity P(du/dt) ~ 1/(du/dt)^a is substantially reduced from a=1 (MFT) to a = 1 - 2/9 (4-d) + ... (short-ranged elasticity) and a = 1 - 4/9 (2-d) + ... (long-ranged elasticity). We show how the dynamical theory recovers the avalanche-size distribution, and how the instanton relates to the response to an infinitesimal step in the force.

cond-mat.dis-nn↗
arXiv2012

Functional renormalization-group approach to decaying turbulence

We reconsider the functional renormalization-group (FRG) approach to decaying Burgers turbulence, and extend it to decaying Navier-Stokes and Surface-Quasi-Geostrophic turbulence. The method is based on a renormalized small-time expansion, equivalent to a loop expansion, and naturally produces a dissipative anomaly and a cascade after a finite time. We explicitly calculate and analyze the one-loop FRG equations in the zero-viscosity limit as a function of the dimension. For Burgers they reproduce the FRG equation obtained in the context of random manifolds, extending previous results of one of us. Breakdown of energy conservation due to shocks and the appearance of a direct energy cascade corresponds to failure of dimensional reduction in the context of disordered systems. For Navier-Stokes in three dimensions, the velocity-velocity correlation function acquires a linear dependence on the distance, zeta_2=1, in the inertial range, instead of Kolmogorov's zeta_2=2/3; however the possibility remains for corrections at two- or higher-loop order. In two dimensions, we obtain a numerical solution which conserves energy and exhibits an inverse cascade, with explicit analytical results both for large and small distances, in agreement with the scaling proposed by Batchelor. In large dimensions, the one-loop FRG equation for Navier-Stokes converges to that of Burgers.

nlin.CD↗
arXiv2011

First-principle derivation of static avalanche-size distribution

We study the energy minimization problem for an elastic interface in a random potential plus a quadratic well. As the position of the well is varied, the ground state undergoes jumps, called shocks or static avalanches. We introduce an efficient and systematic method to compute the statistics of avalanche sizes and manifold displacements. The tree-level calculation, i.e. mean-field limit, is obtained by solving a saddle-point equation. Graphically, it can be interpreted as a the sum of all tree graphs. The 1-loop corrections are computed using results from the functional renormalization group. At the upper critical dimension the shock statistics is described by the Brownian Force model (BFM), the static version of the so-called ABBM model in the non-equilibrium context of depinning. This model can itself be treated exactly in any dimension and its shock statistics is that of a Levy process. Contact is made with classical results in probability theory on the Burgers equation with Brownian initial conditions. In particular we obtain a functional extension of an evolution equation introduced by Carraro and Duchon, which recursively constructs the tree diagrams in the field theory.

cond-mat.dis-nn↗