Patrik L. Ferrari, Herbert Spohn and Thomas Weiss Scaling Limit for Brownian Motions with One-sided Collisions Ann. Appl. Probab., 25(3): 1349-1382 2015 http://dx.doi.org/10.1214/14-AAP1025
Abstract: We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütz-type formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, i.e., particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the particle positions are described by the Airy\(_1\) process.
Patrik L. Ferrari and Balint Vető Tracy-Widom asymptotics for q-TASEP Ann. Inst. H. Poincaré Probab. Statist., 51(4): 1465-1485 2015 http://dx.doi.org/10.1214/14-AIHP614
Abstract: We consider the q-TASEP, that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on \(\mathbb{Z}\) for \(q \in [0,1)\) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of \(q\)-TASEP at time \(\tau\) are of order \(\tau^{1/3}\) and asymptotically distributed as the GUE Tracy-Widom distribution.
Patrik L. Ferrari, Herbert Spohn and Thomas Weiss Brownian motions with one-sided collisions: the stationary case Electronic Journal of Probability, 20(Art. 69): 1-41 2015 http://dx.doi.org/10.1214/EJP.v20-4177
Abstract: We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the large time limit. This leads to a new universal cross-over process.
Patrik L. Ferrari and Peter Nejjar Shock fluctuations in flat TASEP under critical scaling J. Stat. Phys., 160(4): 985-1004 2015 http://arxiv.org/abs/1408.4850
Abstract: We consider TASEP with two types of particles starting at every second site. Particles to the left of the origin have jump rate $1$, while particles to the right have jump rate $\alpha$. When $\alpha<1$ there is a formation of a shock where the density jumps to $(1-\alpha)/2$. For $\alpha<1$ fixed, the statistics of the associated height functions around the shock is asymptotically (as time $t\to\infty$) a maximum of two independent random variables as shown in [arXiv:1306.3336]. In this paper we consider the critical scaling when $1-\alpha=a t^{-1/3}$, where $t\gg 1$ is the observation time. In that case the decoupling does not occur anymore. We determine the limiting distributions of the shock and numerically study its convergence as a function of $a$. We see that the convergence to product $F_{\rm GOE}^2$ occurs quite rapidly as $a$ increases. The critical scaling is analogue to the one used in the last passage percolation to obtain the BBP transition processes.
2014
Gerard Barkema, Patrik L. Ferrari, Joel L. Lebowitz and Herbert Spohn KPZ universality class and the anchored Toom interface Phys. Rev. E, 90(Art. 042116) 2014 http://dx.doi.org/10.1103/PhysRevE.90.042116
Abstract: We revisit the anchored Toom interface and use KPZ scaling theory to argue that the interface fluctuations are governed by the Airy1 process with the role of space and time interchanged. There is no free parameter. The predictions are numerically well confirmed for space-time statistics in the stationary state. In particular the spatial fluctuations of the interface are given by the GOE edge distribution of Tracy and Widom.
Alexei Borodin, Ivan Corwin, Patrik L. Ferrari and Balint Vető Height fluctuations for the stationary KPZ equation Math. Phys. Anal. Geom., 18(1, Art. 20): 1-95 2014 http://arxiv.org/abs/1407.6977
Abstract: We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data $\mathcal{H}(0,X)=B(X)$, for $B(X)$ a two-sided standard Brownian motion) and show that as time $T$ goes to infinity, the fluctuations of the height function $\mathcal{H}(T,X)$ grow like $T^{1/3}$ and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula for Whittaker processes. We relate this to a polymer model which mixes the semi-discrete and log-gamma random polymers. A special case of this model has a limit to the KPZ equation with initial data given by a two-sided Brownian motion with drift $β$ to the left of the origin and $b$ to the right of the origin. The Fredholm determinant has a limit for $β>b$, and the case where $β=b$ (corresponding to the stationary initial data) follows from an analytic continuation argument.
Patrik L. Ferrari and René Frings Perturbed GUE Minor Process and Warren’s Process with Drifts J. Stat. Phys., 154(1): 356-377 2014 http://dx.doi.org/10.1007/s10955-013-0887-5
Abstract: We consider the minor process of (Hermitian) matrix diffusions with constant diagonal drifts. At any given time, this process is determinantal and we provide an explicit expression for its correlation kernel. This is a measure on the Gelfand–Tsetlin pattern that also appears in a generalization of Warren’s process (Electron. J. Probab. 12:573–590, 2007), in which Brownian motions have level-dependent drifts. Finally, we show that this process arises in a diffusion scaling limit from an interacting particle system in the anisotropic KPZ class in 2+1 dimensions introduced in Borodin and Ferrari (Commun. Math. Phys., 2008). Our results generalize the known results for the zero drift situation.
Irene Fonseca, Aldo Pratelli and Barbara Zwicknagl Shapes of Epitaxially Grown Quantum Dots Archive for Rational Mechanics and Analysis, 214(2): 359-401 2014 http://dx.doi.org/10.1007/s00205-014-0767-4
Fabian Franzelin, Patrick Diehl and Dirk Pflüger Non-intrusive Uncertainty Quantification with Sparse Grids for Multivariate Peridynamic Simulations In M. Griebel and M. A. Schweitzer, editor, Meshfree Methods for Partial Differential Equations VII, Volume 100 of Lecture Notes in Computational Science and Engineering
Chapter 7, page 115-143.
Publisher: Springer
2014 http://dx.doi.org/10.1007/978-3-319-06898-5_7
2013
Benjamin Berkels, Tom Fletcher, Behrend Heeren, Martin Rumpf and Benedikt Wirth Discrete geodesic regression in shape space In Anders Heyden, Fredrik Kahl, Carl Olsson, Magnus Oskarsson, Xue-Cheng Tai, editor, Energy Minimization Methods in Computer Vision and Pattern Recognition, Volume 8081 of Lecture Notes in Computer Science
page 108-122.
Publisher: Springer International
2013 http://dx.doi.org/10.1007/978-3-642-40395-8_9
Abstract: A new approach for the effective computation of geodesic re- gression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity mea- sures between given two- or three-dimensional input shapes and corre- sponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is de- duced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging.
Patrik L. Ferrari, Tomohiro Sasamoto and Herbert Spohn Coupled Kardar-Parisi-Zhang Equations in One Dimension J. Stat. Phys., 153(3): 377-399 2013 http://dx.doi.org/10.1007/s10955-013-0842-5
Abstract: Over the past years our understanding of the scaling properties of the solutions to the one-dimensional KPZ equation has advanced considerably, both theoretically and experimentally. In our contribution we export these insights to the case of coupled KPZ equations in one dimension. We establish equivalence with nonlinear fluctuating hydrodynamics for multi-component driven stochastic lattice gases. To check the predictions of the theory, we perform Monte Carlo simulations of the two-component AHR model. Its steady state is computed using the matrix product ansatz. Thereby all coefficients appearing in the coupled KPZ equations are deduced from the microscopic model. Time correlations in the steady state are simulated and we confirm not only the scaling exponent, but also the scaling function and the non-universal coefficients.
0
A. Borodin, I. Corwin and P.L. Ferrari Anisotropic (2+1)d growth and Gaussian limits of q-Whittaker processes preprint, arXiv:1612.00321 0 http://arxiv.org/abs/1612.00321
Abstract: We consider a discrete model for anisotropic (2+1)-dimensional growth of an interface height function. Owing to a connection with q-Whittaker functions, this system enjoys many explicit integral formulas. By considering certain Gaussian stochastic differential equation limits of the model we are able to prove a space-time limit to the (2+1)-dimensional additive stochastic heat equation (or Edwards-Wilkinson equation) along characteristic directions. In particular, the bulk height function converges to the Gaussian free field which evolves according to this stochastic PDE.
A. Borodin, A. Bufetov and P.L. Ferrari TASEP with a moving wall preprint: arXiv:2111.02530 0 http://arxiv.org/abs/2111.02530
Abstract: We consider a totally asymmetric simple exclusion on Z with the step initial condition, under the additional restriction that the first particle cannot cross a deterministally moving wall. We prove that such a wall may induce asymptotic fluctuation distributions of particle positions of the form P(supÏâR{Airy2(Ï)âg(Ï)}â¤S) with arbitrary barrier functions g. This is the same class of distributions that arises as one-point asymptotic fluctuations of TASEPs with arbitrary initial conditions. Examples include Tracy-Widom GOE and GUE distributions, as well as a crossover between them, all arising from various particles behind a linearly moving wall. We also prove that if the right-most particle is second class, and a linearly moving wall is shock-inducing, then the asymptotic distribution of the position of the second class particle is a mixture of the uniform distribution on a segment and the atomic measure at its right end.
P.L. Ferrari and P. Nejjar The second class particle process at shocks preprint: arXiv:2309.09570 0 https://arxiv.org/abs/2309.09570
Abstract: We consider the totally asymmetric simple exclusion process (TASEP) starting with a shock discontinuity at the origin, with asymptotic densities λ to the left of the origin and Ï to the right of it and λ<Ï. We find an exact identity for the distribution of a second class particle starting at the origin. Then we determine the limiting joint distributions of the second class particle. Bypassing the last passage percolation model, we work directly in TASEP, allowing us to extend previous one-point distribution results via a more direct and shorter ansatz.
P.L. Ferrari and A. Occelli Time-time covariance for last passage percolation in half-space preprint:arXiv:2204.06782 0 https://arxiv.org/abs/2204.06782
Abstract: This article studies several properties of the half-space last passage percolation, in particular the two-time covariance. We show that, when the two end-points are at small macroscopic distance, then the first order correction to the covariance for the point-to-point model is the same as the one of the stationary model. In order to obtain the result, we first derive comparison inequalities of the last passage increments for different models. This is used to prove tightness of the point-to-point process as well as localization of the geodesics. Unlike for the full-space case, for half-space we have to overcome the difficulty that the point-to-point model in half-space with generic start and end points is not known.