Preprints - IAM Publications


	2018 Sergio Conti, Matteo Focardi and Flaviana Iurlano Which special functions of bounded deformation have bounded variation Proc. Roy. Soc. Edinb. A, 148: 33-50 2018 10.1017/S030821051700004X Abstract: Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions which are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that $SBD^p$ functions are approximately continuous $H^n-1$-a.e. away from the jump set. On the negative side, we construct a function which is $BD$ but not in BV and has distributional strain consisting only of a jump part, and one which has a distributional strain consisting of only a Cantor part.

	Matthias Erbar and Max Fathi Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature J. Funct. Anal., 274(11): 3056--3089 2018 10.1016/j.jfa.2018.03.011

	M. Furlan and M. Gubinelli Weak universality for a class of 3d stochastic reaction-diffusion models Probability Theory and Related Fields 2018 10.1007/s00440-018-0849-6

	2017 S. Chhita, P.L. Ferrari and F.L. Toninelli Speed and fluctuations for some driven dimer models preprint: arXiv:1705.07641 2017 https://arxiv.org/abs/1705.07641 Abstract: We consider driven dimer models on the square and honeycomb graphs, starting from a stationary Gibbs measure. Each model can be thought of as a two dimensional stochastic growth model of an interface, belonging to the anisotropic KPZ universality class. We use a combinatorial approach to determine the speed of growth and show logarithmic growth in time of the variance of the height function fluctuations.

	Sergio Conti, Matteo Focardi and Flaviana Iurlano Integral representation for functionals defined on $SBD^p$ in dimension two Arch. Ration. Mech. Anal., 223(3): 1337--1374 2017 10.1007/s00205-016-1059-y

	P.L. Ferrari and A. Occelli Universality of the GOE Tracy-Widom distribution for TASEP with arbitrary particle density preprint: arXiv:1704.01291 2017 https://arxiv.org/abs/1704.01291 Abstract: We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles. We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased in last passage percolation language is the universality for the point-to-line problem where the line has an arbitrary slope.

	P.L. Ferrari, P. Ghosal and P. Nejjar Limit law of a second class particle in TASEP with non-random initial condition preprint: arXiv:1710.02323 2017 https://arxiv.org/abs/1710.02323 Abstract: We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition having density $\rho$ on $\mathbb{Z}_-$ and $\lambda$ on $\mathbb{Z}_+$, and a second class particle initially at the origin. For $\rho<\lambda$, there is a shock and the second class particle moves with speed $1-\lambda-\rho$. For large time $t$, we show that the position of the second class particle fluctuates on a $t^{1/3}$ scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time $t$.

	P.L. Ferrari Finite GUE distribution with cut-off at a shock preprint: arXiv:1712.00102 2017 https://arxiv.org/abs/1712.00102 Abstract: We consider the totally asymmetric simple exclusion process with initial conditions generating a shock. The fluctuations of particle positions are asymptotically governed by the randomness around the two characteristic lines joining at the shock. We describe this in terms of space-time correlations, without employing the mapping to the last passage percolation. We then consider a special case, where the asymptotic distribution is a cut-off of the distribution of the largest eigenvalue of a finite GUE matrix. Finally we discuss the strength of the probabilistic and physically motivated approach and compare it with the mathematical difficulties of a direct computation.

	2016 A. Borodin, I. Corwin and P.L. Ferrari Anisotropic (2+1)d growth and Gaussian limits of q-Whittaker processes arXiv:1612.00321 2016 https://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.

	S. Chhita, P.L. Ferrari and H. Spohn Limit distributions for KPZ growth models with spatially homogeneous random initial conditions preprint, arXiv:1611.06690 2016 http://arxiv.org/abs/1611.06690 Abstract: For stationary KPZ growth in 1+1 dimensions the height fluctuations are governed by the Baik-Rains distribution. Using the totally asymmetric single step growth model, alias TASEP, we investigate height fluctuations for a general class of spatially homogeneous random initial conditions. We prove that for TASEP there is a one-parameter family of limit distributions, labeled by the roughness of the initial conditions. The distributions are defined through a variational formula. We use Monte Carlo simulations to obtain their numerical plots. Also discussed is the connection to the six-vertex model at is conical point.

	Sergio Conti, Matteo Focardi and Flaviana Iurlano Phase field approximation of cohesive fracture models Annales de l'Institut Henri Poincar{\'e} / Analyse non lin{\'e}aire, 33: 1033-1067 2016 10.1016/j.anihpc.2015.02.001 Abstract: We obtain a cohesive fracture model as a $\Gamma$-limit of scalar damage models in which the elastic coefficient is computed from the damage variable $v$ through a function $f_k$ of the form $f_k(v)=min\{1,\varepsilon_k^{1/2} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening $s$ at small values of $s$ and has a finite limit as $s\to\infty$. If the function $f$ is allowed to depend on the index $k$, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings.

	Sergio Conti, Matteo Focardi and Flaviana Iurlano Existence of minimizers for the 2d stationary Griffith fracture model C. R. Math. Acad. Sci. Paris, 354(11): 1055--1059 2016 10.1016/j.crma.2016.09.003

	Sergio Conti, Matteo Focardi and Flaviana Iurlano Some recent results on the convergence of damage to fracture Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27(1): 51--60 2016 10.4171/RLM/722

	P.L. Ferrari and P. Nejjar Fluctuations of the competition interface in presence of shocks arXiv:1603.07498 2016 http://arxiv.org/abs/1603.07498 Abstract: We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied by Ferrari and Pimentel in [Ann. Probab. 33 (2005), 1235-1254] for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deterministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of [Probab. Theory Relat. Fields 61 (2015), 61-109].

	P.L. Ferrari and H. Spohn On time correlations for KPZ growth in one dimension preprint: arXiv:1602.00486 2016 http://arxiv.org/abs/1602.00486 Abstract: Time correlations for KPZ growth in 1+1 dimensions are reconsidered. We discuss flat, curved, and stationary initial conditions and are interested in the covariance of the height as a function of time at a fixed point on the substrate. In each case the power laws of the covariance for short and long times are obtained. They are derived from a variational problem involving two independent Airy processes. For stationary initial conditions we derive an exact formula for the stationary covariance with two approaches: (1) the variational problem and (2) deriving the covariance of the time-integrated current at the origin for the corresponding driven lattice gas. In the stationary case we also derive the large time behavior for the covariance of the height gradients.

	P.L. Ferrari and B. Vető The hard-edge tacnode process for Brownian motion preprint, arXiv:1608.00394 2016 https://arxiv.org/abs/1608.00394 Abstract: {We consider N non-intersecting Brownian bridges conditioned to stay below a fixed threshold. We consider a scaling limit where the limit shape is tangential to the threshold. In the large N limit, we determine the limiting distribution of the top Brownian bridge conditioned to stay below a function as well as the limiting correlation kernel of the system. It is a one-parameter family of processes which depends on the tuning of the threshold position on the natural fluctuation scale. We also discuss the relation to the six-vertex model and the Aztec diamond on restricted domains.}

	2015 Alexei Borodin and Patrik L. Ferrari Random tilings and Markov chains for interlacing particles ArXiv e-prints 2015 http://arxiv.org/abs/1506.03910

	Sunil Chhita and Patrik L. Ferrari A combinatorial identity for the speed of growth in an anisotropic KPZ model arXiv e-prints 2015 http://arxiv.org/abs/1508.01665 Abstract: The speed of growth for a particular stochastic growth model introduced by Borodin and Ferrari in [Comm. Math. Phys. 325 (2014), 603-684], which belongs to the KPZ anisotropic universality class, was computed using multi-time correlations. The model was recently generalized by Toninelli in [arXiv:1503.05339] and for this generalization the stationary measure is known but the time correlations are unknown. In this note, we obtain algebraic and combinatorial proofs for the expression of the speed of growth from the prescribed dynamics.

	Max Fathi and Jan Maas Entropic Ricci curvature bounds for discrete interacting systems Ann. Appl. Prob. 2015 http://arxiv.org/abs/1501.00562

	Patrik L. Ferrari and Peter Nejjar Anomalous shock fluctuations in TASEP and last passage percolation models Probab. Theory Related Fields, 161(1): 61-109 2015 http://dx.doi.org/10.1007/s00440-013-0544-6 Abstract: We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time $t$ will have a width of order $t^{1/3}$. We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the Airy$_1$ process. The limiting distribution is a product of two distribution functions, which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics. We show that the result generalizes to generic last passage percolation models.

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