Preprints - IAM Publications


	2020 D. Betea, P.L. Ferrari and A. Occelli The half-space Airy stat process preprint: arXiv:2012.10337 2020 https://arxiv.org/abs/2012.10337 Abstract: We study the multipoint distribution of stationary half-space last passage percolation with exponentially weighted times. We derive both finite-size and asymptotic results for this distribution. In the latter case we observe a new one-parameter process we call half-space Airy stat. It is a one-parameter generalization of the Airy stat process of Baik-Ferrari-Péché, which is recovered far away from the diagonal. All these results extend the one-point results previously proven by the authors.

	Ofer Busani and Patrik L. Ferrari Universality of the geodesic tree in last passage percolation preprint, arXiv:2008.07844 2020 https://arxiv.org/abs/2008.07844 Abstract: In this paper we consider the geodesic tree in exponential last passage percolation. We show that for a large class of initial conditions around the origin, the line-to-point geodesic that terminates in a cylinder of width $o(N^{2/3})$ and length $o(N)$ agrees in the cylinder, with the stationary geodesic sharing the same end point. In the case of the point-to-point model, we consider width $\delta N^{2/3}$ and length up to $\delta^{3/2} N/(\log(\delta^{-1}))^3$ and provide lower and upper bound for the probability that the geodesics agree in that cylinder.

	P.L. Ferrari and B. Vető Upper tail decay of KPZ models with Brownian initial conditions preprint,arXiv:2007.13496 2020 https://arxiv.org/abs/2007.13496 Abstract: In this paper we consider the limiting distribution of KPZ growth models with random but not stationary initial conditions introduced in [Chhita-Ferrari-Spohn 2018]. The one-point distribution of the limit is given in terms of a variational problem. By directly studying it, we deduce the right tail asymptotic of the distribution function. This gives a rigorous proof and extends the results obtained in [Meerson-Schmidt 2017].

	P.L. Ferrari, Muhittin Mungan and M. Mert Terzi The Preisach graph and longest increasing subsequences arXiv:2004.03138 2020 https://arxiv.org/abs/2004.03138 Abstract: The Preisach graph is a directed graph associated with a permutation ÏâSN. We give an explicit bijection between the vertices and increasing subsequences of Ï, with the property that its length equals the degree of nesting of the vertex inside a hierarchy of cycles and sub-cycles. As a consequence, the nesting degree of the Preisach graph equals the length of the longest increasing subsequence.

	2019 D. Betea, P.L. Ferrari and A. Occelli Stationary half-space last passage percolation preprint: arXiv:1905.08582 2019 https://arxiv.org/abs/1905.08582 Abstract: In this paper we study stationary last passage percolation (LPP) in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space (strength of the source at the origin in the equivalent TASEP language) and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the BaikâRains distributions for full-space geometry. The result is obtained by using a related integrable model, having Pfaffian structure, together with careful analytic continuation and steepest descent analysis.

	Anton Bovier, Saeda Marello, P. L. Ferrari and Elena Pulvirenti Metastability for the dilute Curie-Weiss model with Glauber dynamics 2019 https://arxiv.org/abs/1912.10699

	P.L. Ferrari and A. Occelli Time-time covariance for last passage percolation with generic initial profile Math. Phys. Anal. Geom., 22: 1 2019 https://doi.org/10.1007/s11040-018-9300-6 Abstract: We consider time correlation for KPZ growth in 1+1 dimensions in a neighborhood of a characteristics. We prove convergence of the covariance with droplet, flat and stationary initial profile. In particular, this provides a rigorous proof of the exact formula of the covariance for the stationary case obtained in [SIGMA 12 (2016), 074]. Furthermore, we prove the universality of the first order correction when the two observation times are close and provide a rigorous bound of the error term. This result holds also for random initial profiles which are not necessarily stationary.

	P.L. Ferrari and B. Vető Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains preprint: arXiv:1909.10840 2019 https://arxiv.org/abs/1909.10840 Abstract: We consider uniform random domino tilings of the restricted Aztec diamond which is obtained by cutting off an upper triangular part of the Aztec diamond by a horizontal line. The restriction line asymptotically touches the arctic circle that is the limit shape of the north polar region in the unrestricted model. We prove that the rescaled boundary of the north polar region in the restricted domain converges to the Airy$_2$ process conditioned to stay below a parabola with explicit continuous statistics and the finite dimensional distribution kernels. The limit is the hard-edge tacnode process which was first discovered in the framework of non-intersecting Brownian bridges. The proof relies on a random walk representation of the correlation kernel of the non-intersecting line ensemble which corresponds to a random tiling.

	Patrik L. Ferrari and Peter Nejjar Statistics of TASEP with three merging characteristics preprint: arXiv:1910.14083 2019 https://arxiv.org/abs/1910.14083 Abstract: In this paper we consider the totally asymmetric simple exclusion process, with non-random initial condition having three regions of constant densities of particles. From left to right, the densities of the three regions are increasing. Consequently, there are three characteristics which meet, i.e., two shocks merge. We study the particle fluctuations at this merging point and show that they are given by a product of three (properly scaled) GOE Tracy-Widom distribution functions. We work directly in TASEP without relying on the connection to last passage percolation.

	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.

	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.

	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.

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