2020  
•  The halfspace Airy stat process, with Dan Betea and Alessandra Occelli,
arXiv:2012.10337 (2020).
We study the multipoint distribution of stationary halfspace last passage percolation with exponentially weighted times. We derive both finitesize and asymptotic results for this distribution. In the latter case we observe a new oneparameter process we call halfspace Airy stat. It is a oneparameter generalization of the Airy stat process of BaikFerrariPéché, which is recovered far away from the diagonal. All these results extend the onepoint results previously proven by the authors.
 
•  Universality of the geodesic tree in last passage percolation, with Ofer Busani,
arXiv:2008.07844 (2020).
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 linetopoint geodesic that terminates in a cylinder located around the point (N,N), and whose width and length are o(N^{2/3}) and o(N) respectively, agrees in the cylinder, with the stationary geodesic sharing the same end point. In the case of the pointtopoint model where the geodesic starts from the origin, we consider width δN^{2/3}, length up to δ^{3/2}N/(log(δ−1))^{3}, and provide lower and upper bounds for the probability that the geodesics agree in that cylinder.
 
•  Upper tail decay of KPZ models with Brownian initial conditions, with Bálint Vető,
arXiv:2007.13496 (2020).
In this paper we consider the limiting distribution of KPZ growth models with random but not stationary initial conditions introduced in [ChhitaFerrariSpohn 2018]. The onepoint 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 [MeersonSchmidt 2017].
 
•  The Preisach graph and longest increasing subsequences, with with Muhittin Mungan and M. Mert Terzi,
arXiv:2004.03138 (2020).
The Preisach graph is a directed graph associated with a permutation ρ. We give an explicit bijection between the vertices and increasing subsequences of ρ, with the property that its length equals the degree of nesting of the vertex inside a hierarchy of cycles and subcycles. As a consequence, the nesting degree of the Preisach graph equals the length of the longest increasing subsequence.
 
•  Shock fluctuations in TASEP under a variety of time scalings, with Alexey Bufetov,
arXiv:2003.12414 (2020).
We consider the totally asymmetric simple exclusion process (TASEP) with two different initial conditions with shock discontinuities, made by block of fully packed particles. Initially a second class particle is at the left of a shock discontinuity. Using multicolored TASEP we derive an exact formulas for the distribution of the second class particle and colored height functions. These are given in terms of the height function at different positions of a single TASEP configuration. We study the limiting distributions of second class particles (and colored height functions). The result depends on how the width blocks of particles scale with the observation time; we study a variety of such scalings.
 
2019  
•  Statistics of TASEP with three merging characteristics, with Peter Nejjar,
J. Stat. Phys. 180 (2020), 398–413.
In this paper we consider the totally asymmetric simple exclusion process, with nonrandom 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 TracyWidom distribution functions. We work directly in TASEP without relying on the connection to last passage percolation.
 
•  Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains, with Bálint Vető,
arXiv:1909.10840 (2019). To appear in Ann. Appl. Probab.
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 hardedge tacnode process which was first discovered in the framework of nonintersecting Brownian bridges. The proof relies on a random walk representation of the correlation kernel of the nonintersecting line ensemble which corresponds to a random tiling.
 
•  Stationary halfspace last passage percolation, with Dan Betea and Alessandra Occelli,
Commun. Math. Phys. 377 (2020), 421467.
In this paper we study stationary last passage percolation (LPP) with exponential weights and in halfspace geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new twoparameter family of distributions: one parameter for the strength of the diagonal bounding the halfspace (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 oneparameter family giving the BaikRains distributions for fullspace geometry. We finally show that far enough away from the characteristic line, our distributions indeed converge to the BaikRains family. We derive our results using a related inhomogeneous integrable model having Pfaffian correlations, together with careful analytic continuation, and steepest descent analysis.
 
2018  
•  Timetime covariance for last passage percolation with generic initial profile, with Alessandra Occelli,
Math. Phys. Anal. Geom. (2019), 22:1.
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.
 
2017  
•  Finite GUE distribution with cutoff at a shock,
J. Stat. Phys. 172 (2018), 505521.
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 spacetime correlations, without employing the mapping to the last passage percolation. We then consider a special case, where the asymptotic distribution is a cutoff 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.
 
•  Limit law of a second class particle in TASEP with nonrandom initial condition, with Peter Nejjar and Promit Ghosal,
Ann. Inst. Henri Poincaré Probab. Statist. 55 (2019), 12031225.
We consider the totally asymmetric simple exclusion process (TASEP) with nonrandom initial condition having density λ on Z_{−} and ρ on Z_{+}, and a second class particle initially at the origin. For λ < ρ there is a shock and the second class particle moves with speed 1 − ρ  λ . 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.
 
•  Speed and fluctuations for some driven dimer models, with Sunil Chhita and Fabio Toninelli,
Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), 489532.
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.
 
•  Universality of the GOE TracyWidom distribution for TASEP with arbitrary particle density, with Alessandra Occelli,
Electron. J. Probab. 23 (2018), no. 51, 124.
We consider TASEP in continuous time with nonrandom initial conditions and arbitrary fixed density of particles. We show GOE TracyWidom universality of the onepoint fluctuations of the associated height function. The result phrased in last passage percolation language is the universality for the pointtoline problem where the line has an arbitrary slope.
 
2016  
•  Anisotropic (2+1)d growth and Gaussian limits of qWhittaker processes, with Alexei Borodin and Ivan Corwin,
Probab. Theory Relat. Fields 172 (2018), 245321.
We consider a discrete model for anisotropic (2+1)dimensional growth of an interface height function. Owing to a connection with qWhittaker 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 spacetime limit to the (2+1)dimensional additive stochastic heat equation (or EdwardsWilkinson equation) along characteristic directions. In particular, the bulk height function converges to the Gaussian free field which evolves according to this stochastic PDE.
 
•  Limit distributions for KPZ growth models with spatially homogeneous random initial conditions, with Sunil Chhita and Herbert Spohn,
Ann. Appl. Probab. 28 (2018), 15731603.
For stationary KPZ growth in 1+1 dimensions the height fluctuations are governed by the BaikRains 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 oneparameter 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 sixvertex model at is conical point.
 
•  The hardedge tacnode process for Brownian motion, with Bálint Vető,
Electron. J. Probab. 22 (2017), no. 79, 132.
We consider N nonintersecting 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 oneparameter family of processes which depends on the tuning of the threshold position on the natural fluctuation scale. We also discuss the relation to the sixvertex model and the Aztec diamond on restricted domains.
 
•  Fluctuations of the competition interface in presence of shocks, with Peter Nejjar,
ALEA, Lat. Am. J. Probab. Math. Stat. 14 (2017), 299325.
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), 12351254] 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 onepoint result of [Probab. Theory Relat. Fields 61 (2015), 61109].
 
•  On time correlations for KPZ growth in one dimension, with Herbert Spohn,
SIGMA 12 (2016), 074.
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 timeintegrated 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.
 
2015  
•  A combinatorial identity for the speed of growth in an anisotropic KPZ model, with Sunil Chhita,
Ann. Inst. H. Poincaré Comb. Phys. Interact. 4 (2017), 453477.
The speed of growth for a particular stochastic growth model introduced by Borodin and Ferrari in [Comm. Math. Phys. 325 (2014), 603684], which belongs to the KPZ anisotropic universality class, was computed using multitime 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.
 
•  Random tilings and Markov chains for interlacing particles, with Alexei Borodin,
Markov Processes Relat. Fields 24 (2018), no. 3.
We explain the relation between certain random tiling models and interacting
particle systems belonging to the anisotropic KPZ (KardarParisiZhang)
universality class in 2+1dimensions. The link between these two a
priori disjoint sets of models is a consequence of the presence of shuffling
algorithms that generate random tilings under consideration. To see the
precise connection, we represent both a random tiling and the corresponding
particle system through a set of nonintersecting lines, whose dynamics is
induced by the shuffling algorithm or the particle dynamics. The resulting class
of measures on line ensembles also fits into the framework of the Schur
processes.
 
•  Brownian motions with onesided collisions: the stationary case, with Herbert Spohn and Thomas Weiss,
Electron. J. Probab. 20 (2015), 141.
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 spacetime stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finitedimensional 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 crossover process.
 
2014  
•  Shock fluctuations in flat TASEP under critical scaling, with Peter Nejjar,
J. Stat. Phys. 60 (2015), 9851004.
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 α. When α<1 there is a formation of a shock where the density jumps to (1α)/2. For α<1 fixed, the statistics of the associated height functions around the shock is asymptotically (as time t→∞) a maximum of two independent random variables as shown in [Ferrari,Nejjar'14]. In this paper we consider the critical scaling when 1α=a t^{1/3}, where t→∞ 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_{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.
 
•  Height fluctuations for the stationary KPZ equation, with Alexei Borodin, Ivan Corwin, and Bálint Vető,
Math. Phys. Anal. Geom. (2015), 18:20.
We compute the onepoint probability distribution for the stationary KPZ equation (i.e.\ initial data h(0,X)=B(X), for B(X) a twosided standard Brownian motion) and show that as time T goes to infinity, the fluctuations of the height function 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 semidiscrete and loggamma random polymers. A special case of this model has a limit to the KPZ equation with initial data given by a twosided 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.  
•  KPZ universality class and the anchored Toom interface, with Gerard Barkema, Joel Lebowitz and Herbert Spohn,
Phys. Rev. E 90 (2014), 042116.
We revisit the anchored Toom interface and use KPZ scaling theory to argue that the interface fluctuations are governed by the Airy_{1} process with the role of space and time interchanged. There is no free parameter. The predictions are numerically well confirmed for spacetime statistics in the stationary state. In particular the spatial fluctuations of the interface are given by the GOE edge distribution of Tracy and Widom.
 
2013  
•  TracyWidom asymptotics for qTASEP, with Bálint Vető,
Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), 14651485.
We consider the qTASEP, that is a qdeformation of the totally asymmetric simple exclusion process (TASEP) on 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 qTASEP at time t are of order t^{1/3} and asymptotically distributed as the GUE TracyWidom distribution.
 
•  Coupled KardarParisiZhang equations in one dimension, with Tomohiro Sasamoto and Herbert Spohn,
J. Stat. Phys. 153 (2013), 377399.
Over the past years our understanding of the scaling properties of the solutions to the onedimensional 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 multicomponent driven stochastic lattice gases. To check the predictions of the theory, we perform Monte Carlo simulations of the twocomponent 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 nonuniversal coefficients.
 
•  Scaling limit for Brownian motions with onesided collisions, with Herbert Spohn and Thomas Weiss,
Ann. Appl. Probab. 25 (2015), 13491382.
We consider Brownian motions with onesided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütztype 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.
 
•  Anomalous shock fluctuations in TASEP and last passage percolation models, with Peter Nejjar,
Probab. Theory Relat. Fields 161 (2015), 61109.
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.
 
2012  
•  Perturbed GUE Minor Process and Warren’s Process with Drifts, with René Frings,
J. Stat. Phys. 154 (2014), 356377.
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 GelfandTsetlin pattern that also appears in a generalization of Warren's process, in which Brownian motions have leveldependent 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. Our results generalize the known results for the zero drift situation.
 
•  On the spatial persistence for Airy processes, with René Frings,
J. Stat. Mech. (2013), P02001.
In this short paper we derive a formula for the spatial persistence probability of the Airy_{1} and the Airy_{2} processes. We then determine numerically a persistence coefficient for the Airy_{1} process and its dependence on the threshold.
 
•  Convergence of the twopoint function of the stationary TASEP, with Jinho Baik and Sandrine Péché,
In Singular Phenomena and Scaling in Mathematical Models (2014), 91110.
We consider the twopoint function of the totally asymmetric simple exclusion process with stationary initial conditions. The twopoint function can be expressed as the discrete Laplacian of the variance of the associated height function. The limit of the distribution function of the appropriately scaled height function was obtained previously by Ferrari and Spohn. In this paper we show that the convergence can be improved to the convergence of moments. This implies the convergence of the twopoint function in a weak sense along the nearcharacteristic direction as time tends to infinity, thereby confirming the conjecture in the paper of Ferrari and Spohn.
 
•  Free energy fluctuations for directed polymers in random media in 1+1 dimension, with Alexei Borodin and Ivan Corwin,
Comm. Pure Appl. Math. 67(2014), 1129–1214.
We consider two models for directed polymers in spacetime independent random media (the O'ConnellYor semidiscrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time tau, the probability distributions for the free energy fluctuations, when rescaled by t^{1/3}, converges to the GUE TracyWidom distribution.
We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semidiscrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting BaikBen ArousPeche distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm  the KardarParisiZhang equation. The Laplace transform formula we prove can be inverted to give the onepoint probability distribution of the solution to these stochastic PDEs for the class of initial data.  
2011  
•  Noncolliding Brownian bridges and the asymmetric tacnode process, with Bálint Vető,
Electron. J. Probab. 17 (2012), no. 44, 117.
We consider noncolliding Brownian bridges starting from two points and re turning to the same position. These positions are chosen such that, in the limit of large number of bridges, the two families of bridges just touch each other forming a tacnode. We obtain the limiting process at the tacnode, the (asymmetric) tacnode process. It is a determinantal point process with correlation kernel given by two parameters: (1) the curvature's ratio λ > 0 of the limit shapes of the two families of bridges, (2) a realvalued parameter σ controlling the interaction on the fluctuation scale. This generalizes the result for the symmetric tacnode process (λ = 1 case).
 
•  Finite time corrections in KPZ growth models, with René Frings;
J. Stat. Phys. 144 (2011), 11231150.
We consider some models in the KardarParisiZhang universality class, namely the polynuclear growth model and the totally/partially asymmetric simple exclusion process. For these models, in the limit of large time t, universality of fluctuations has been previously obtained. In this paper we consider the convergence to the limiting distributions and determine the (nonuniversal) first order corrections, which turn out to be a nonrandom shift of order t^{1/3} (of order 1 in microscopic units). Subtracting this deterministic correction, the convergence is then of order t^{2/3}. We also determine the strength of asymmetry in the exclusion process for which the shift is zero. Finally, we discuss to what extend the discreteness of the model has an effect on the fitting functions.
 
2010  
•  Nonintersecting random walks in the neighborhood of a symmetric tacnode, with Mark Adler and Pierre Van Moerbeke,
Ann. Probab. 41 (2013), 2599–2647.
Consider a continuous time random walk in Z with independent and exponentially distributed jumps ±1. The model in this paper consists in an infinite number of such random walks starting from the complement of {m,m+1,...,m1,m} at time t, returning to the same starting positions at time t, and conditioned not to intersect. This yields a determinantal process, whose gap probabilities are given by the Fredholm determinant of a kernel. Thus this model consists of two groups of random walks, which are contained into two ellipses which, with the choice m=2t to leading order, just touch: so we have a tacnode. We determine the new limit extended kernel under the scaling m=2t+σt^{1/3}, where parameter σ controls the strength of interaction between the two groups of random walkers.
 
•  On the partial connection between random matrices and interacting particle systems, with René Frings;
J. Stat. Phys. 141 (2010), 613637.
In the last decade there has been increasing interest in the fields of random matrices, interacting particle systems, stochastic growth models, and the connections between these areas. For instance, several objects appearing in the limit of large matrices arise also in the long time limit for interacting particles and growth models. Examples of these are the famous TracyWidom distribution functions and the Airy_{2} process.
The link is however sometimes fragile. For example, the connection between the eigenvalues in the Gaussian Orthogonal Ensembles (GOE) and growth on a flat substrate is restricted to onepoint distribution, and the connection breaks down if we consider the joint distributions. In this paper we first discuss known relations between random matrices and the asymmetric exclusion process (and a 2+1dimensional extension). Then, we show that the correlation functions of the eigenvalues of the matrix minors for β=2 Dyson's Brownian motion have, when restricted to increasing times and decreasing matrix dimensions, the same correlation kernel as in the 2+1dimensional interacting particle system under diffusion scaling limit. Finally, we analyze the analogous question for a diffusion on (complex) sample covariance matrices.  
•  Limit processes for TASEP with shocks and rarefaction fans, with Ivan Corwin and Sandrine Péché,
J. Stat. Phys. 140 (2010), 232267.
We consider the totally asymmetric simple exclusion process (TASEP) with twosided Bernoulli initial condition, i.e., with left density ρ_{} and right density ρ_{+}. We consider the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of ρ_{±}. We characterize the large time scaling limit of the fluctuations as a function of the densities ρ_{±} and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole spacetime, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open.
On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with twosided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices.  
•  Universality of slow decorrelation in KPZ growth, with Ivan Corwin and Sandrine Péché,
Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), 134150.
There has been much success in describing the limiting spatial fluctuations of growth models in the KardarParisiZhang (KPZ) universality class. A proper rescaling of time should introduce a nontrivial temporal dimension to these limiting fluctuations. In onedimension, the KPZ class has the dynamical scaling exponent z=3/2, that means one should find a universal spacetime limiting process under the scaling of time as t T, space like t^{2/3}X and fluctuations like t^{1/3} as t goes to infinity.
In this paper we provide evidence for this belief. We prove that under certain hypotheses, growth models display temporal slow decorrelation. That is to say that in the scalings above, the limiting spatial process for times t T and t T+t^{ν} are identical, for any ν<1. The hypotheses are known to be satisfied for certain last passage percolation models, the polynuclear growth model, and the totally / partially asymmetric simple exclusion process. Using slow decorrelation we may extend known fluctuation limit results to spacetime regions where correlation functions are unknown. The approach we develop requires the minimal expected hypotheses for slow decorrelation to hold and provides a simple and intuitive proof which applied to a wide variety of models.  
2009  
•  Limit process of stationary TASEP near the characteristic line, with Jinho Baik and Sandrine Péché,
Comm. Pure Appl. Math 63 (2010), 10171070.
The totally asymmetric simple exclusion process (TASEP) on Z with the Bernoulliρ measure as initial conditions, 0<ρ<1, is stationary. It is known that along the characteristic line, the current fluctuates as of order t^{1/3}. The limiting distribution has also been obtained explicitly. In this paper we determine the limiting multipoint distribution of the current fluctuations moving away from the characteristics by the order t^{2/3}. The main tool is the analysis of a related directed last percolation model. We also discuss the process limit in tandem queues in equilibrium.
 
•  Maximum of Dyson Brownian motion and noncolliding systems with a boundary, with Alexei Borodin, Michael Prähofer, Tomohiro Sasamoto and Jon Warren,
Electron. Comm. Probab. 14 (2009), 486494.
We prove an equalityinlaw relating the maximum of GUE Dyson's Brownian motion and the noncolliding systems with a wall. This generalizes the well known relation between the maximum of a Brownian motion and a reflected Brownian motion.
 
•  Two speed TASEP, with Alexei Borodin and Tomohiro Sasamoto,
J. Stat. Phys. 137 (2009), 936977.
We consider the TASEP on Z with two blocks of particles having different jump rates. We study the large time behavior of particles' positions. It depends both on the jump rates and the region we focus on, and we determine the complete process diagram. In particular, we discover a new transition process in the region where the influence of the random and deterministic parts of the initial condition interact.
Slow particles may create a shock, where the particle density is discontinuous and the distribution of a particle's position is asymptotically singular. We determine the diffusion coefficient of the shock without using second class particles. We also analyze the case where particles are effectively blocked by a wall moving with speed equal to their intrinsic jump rate.  
2008  
•  Airy processes with wanderers and new universality classes, with Mark Adler and Pierre Van Moerbeke,
Ann. Probab. 38 (2010), 714769.
Consider n+m nonintersecting Brownian bridges, with n of them leaving from 0 at time t=1 and returning to 0 at time t=1, while the m remaining ones (wanderers) go from m points a_i to m points b_i. First we keep m fixed and we scale a_i,b_i appropriately with n. In the largen limit we obtain a new Airy process with wanderers, in the neighborhood of (2n)^{1/2}, the approximate location of the rightmost particle in the absence of wanderers. This new process is governed by an Airytype kernel, with a rational perturbation.
Letting the number m of wanderers tend to infinity as well, leads to two Pearcey processes about two cusps, a closing and an opening cusp, the location of the tips being related by an elliptic curve. Upon tuning the starting and target points, one can let the two tips of the cusps grow very close; this leads to a new process, which we conjecture to be governed by a kernel, represented as a double integral involving the exponential of a quintic polynomial in the integration variables.  
•  Anisotropic growth of random surfaces in 2+1 dimensions: fluctuations and covariance structure, with Alexei Borodin,
J. Stat. Mech. (2009), P02009.
In the linked paper we studied an interacting particle system which can be also interpreted as a stochastic growth model. This model belongs to the anisotropic KPZ class in 2+1 dimensions. In this paper we present the results that are relevant from the perspective of stochastic growth models, in particular: (a) the surface fluctuations are asymptotically Gaussian on a (ln(t))^{1/2} scale and (b) the correlation structure of the surface is asymptotically given by the massless field.
 
•  The Airy_{1} process is not the limit of the largest eigenvalue in GOE matrix diffusion, with Folkmar Bornemann and Michael Prähofer,
J. Stat. Phys. 133 (2008), 405415.
Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1 process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion.
 
•  Slow decorrelations in KPZ growth, J. Stat. Mech. (2008), P07022.
Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1 process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion.
 
•  Anisotropic growth of random surfaces in 2+1 dimensions, with Alexei Borodin,
Comm. Math. Phys. 325 (2014), 603684.
We construct a family of stochastic growth models in 2+1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1+1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models.
The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece. (2) The onepoint fluctuations of the height function in the curved part are asymptotically normal with variance of order ln(t) for time t>>1. (3) There is a map of the (2+1)dimensional spacetime to the upper halfplane H such that on spacelike submanifolds the multipoint fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on H.  
2007  
•  Large time asymptotics of growth models on spacelike paths II: PNG and parallel TASEP, with Alexei Borodin and Tomohiro Sasamoto,
Comm. Math. Phys. 283 (2008), 417449.
We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy1 process. This result holds more generally for the observation points located along any spacelike path in the spacetime plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update.
 
•  Large time asymptotics of growth models on spacelike paths I: PushASEP, with Alexei Borodin,
Electron. J. Probab. 13 (2008), 13801418.
We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way.
We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any spacelike path are described by the Airy_{1} and Airy_{2} processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.  
•  Transition between Airy_{1} and Airy_{2} processes and TASEP fluctuations, with Alexei Borodin and Tomohiro Sasamoto,
Comm. Pure Appl. Math. 61 (2008), 16031629.
We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. For large time t, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy_{1} and Airy_{2} processes, whose onepoint distributions are the GOE and GUE TracyWidom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its onepoint distribution is a new interpolation between GOE and GUE edge distributions.
 
2006  
•  Fluctuations in the discrete TASEP with periodic initial configurations and the Airy_{1} process, with Alexei Borodin and Michael Prähofer,
Int. Math. Res. Papers 2007, rpm002 (2007).
We consider the totally asymmetric simple exclusion process (TASEP) in discrete time with sequential update. The joint distribution of the positions of selected particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. We focus on periodic initial conditions where particles occupy Z, d>=2. In the proper large time scaling limit, the fluctuations of particle positions are described by the Airy_{1} process. Interpreted as a growth model, this confirms universality of fluctuations with flat initial conditions for a discrete set of slopes.
 
•  Fluctuation properties of the TASEP with periodic initial configuration, with Alexei Borodin, Michael Prähofer and Tomohiro Sasamoto,
J. Stat. Phys. 129 (2007), 10551080.
We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us and here we provide a selfcontained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.
 
•  Domino tilings and the sixvertex model at its free fermion point, with Herbert Spohn,
J. Phys. A: Math. Gen. 39 (2006), 1029710306.
At the freefermion point, the sixvertex model with domain wall boundary conditions (DWBC) can be related to the Aztec diamond, a domino tiling problem. We study the mapping on the level of complete statistics for general domains and boundary conditions. This is obtained by associating to both models a set of nonintersecting lines in the LindströmGesselViennot (LGV) scheme. One of the consequence for DWBC is that the boundaries of the ordered phases are described by the Airy process in the thermodynamic limit.
 
2005  
•  Scaling Limit for the SpaceTime Covariance of the Stationary Totally Asymmetric Simple Exclusion Process, with Herbert Spohn, Comm. Math. Phys. 265 (2006), 144.
The totally asymmetric simple exclusion process (TASEP) on the onedimensional lattice with the Bernoulli ρ measure as initial conditions, 0<ρ<1, is stationary in space and time. Let N_{t}(j) be the number of particles which have crossed the bond from j to j+1 during the time span [0,t]. For j=(12ρ)t+2w(ρ(1ρ))^{1/3} t^{2/3} we prove that the fluctuations of N_{t}(j) for large t are of order t^{1/3} and we determine the limiting distribution function F_{w}(s), which is a generalization of the GUE TracyWidom distribution. The family F_{w}(s) of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a RiemannHilbert problem. In our work we arrive at F_{w}(s) through the asymptotics of a Fredholm determinant. F_{w}(s) is simply related to the scaling function for the spacetime covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle.
 
•  A determinantal formula for the GOE TracyWidom distribution, with Herbert Spohn,
J. Phys. A, Math. Gen. 38 (2005), L557L561.
We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the RobinsonSchenstedKnuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of nonintersecting lines, the top one being the PNG height. The statistics of the lines is translation invariant and at a fixed position the lines define a point process. We prove that for large times the edge of this point process, suitably scaled, has a limit. This limit is a Pfaffian point process and identical to the one obtained from the edge scaling of Gaussian orthogonal ensemble (GOE) of random matrices. Our results give further insight to the universality structure within the KPZ class of 1+1 dimensional growth models.
 
2004  
•  Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues,
Comm. Math. Phys. 252 (2004), 77109.
We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the RobinsonSchenstedKnuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of nonintersecting lines, the top one being the PNG height. The statistics of the lines is translation invariant and at a fixed position the lines define a point process. We prove that for large times the edge of this point process, suitably scaled, has a limit. This limit is a Pfaffian point process and identical to the one obtained from the edge scaling of Gaussian orthogonal ensemble (GOE) of random matrices. Our results give further insight to the universality structure within the KPZ class of 1+1 dimensional growth models.
 
2003  
•  Constrained Brownian motion: fluctuations away from circular and parabolic barriers, with Herbert Spohn,
Ann. Probab. 33 (2005), 13021325.
Motivated by the polynuclear growth model, we consider a Brownian bridge b(t) with b(T)=b(T)=0 conditioned to stay above the semicircle c_T(t)=sqrt{T^2t^2}. In the limit of large T, the fluctuation scale of b(t)c_T(t) is T^{1/3} and its timecorrelation scale is T^{2/3}. We prove that, in the sense of weak convergence of path measures, the conditioned Brownian bridge, when properly rescaled, converges to a stationary diffusion process with a drift explicitly given in terms of Airy functions. The dependence on the reference point t=tau T, tau ∈ (1,1), is only through the second derivative of c_T(t) at t=tau T. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height T^gamma, gamma>1/2. The fluctuation scale is then T^{(2gamma)/3}. More general conditioning shapes are briefly discussed.
 
•  Fluctuations of an Atomic Ledge Bordering a Crystalline Facet, with Michael Prähofer and Herbert Spohn,
Phys. Rev. E, Rapid Communications 69 (2004), 035102(R).
When a high symmetry facet joins the rounded part of a crystal, the step line density vanishes as sqrt(r) with r denoting the distance from the facet edge. This means that the ledge bordering the facet has a lot of space to meander as caused by thermal activation. We investigate the statistical properties of the border ledge fluctuations. In the scaling regime they turn out to be nonGaussian and related to the edge statistics of GUE random matrices.
 
2002  
•  Step fluctuations for a faceted crystal, with Herbert Spohn,
J. Stat. Phys. 113 (2003), 146.
A statistical mechanics model for a faceted crystal is the 3D Ising model at zero temperature. It is assumed that in one octant all sites are occupied by atoms, the remaining ones being empty. Allowed atom configurations are such that they can be obtained from the filled octant through successive removals of atoms with breaking of precisely three bonds. If V denotes the number of atoms removed, then the grand canonical Boltzmann weight is q^V, q<1. In the rounded piece it is given by a determinantal process based on the discrete sinekernel. Exactly at the facet edge, the steps have more space to meander. Their statistics is again determinantal, but this time based on the Airykernel. In particular, the border step is well approximated by the Airy process, which has been obtained previously in the context of growth models. Our results are based on the asymptotic analysis for spacetime inhomogeneous transfer matrices.
 
•  Information Loss in Coarse Graining of Polymer Configurations via Contact Matrices, with Joel L. Lebowitz,
J. Phys. A: Math. Gen. 36 (2003), 57195730; Appendix with the complete mathematical proof of Proposition 4.2, see here.
Contact matrices provide a coarse grained description of the configuration omega of a linear chain (polymer or random walk) on Z^n: C_{ij}(omega)=1 when the distance between the position of the ith and jth step are less than or equal to some distance "a" and C_{ij}(omega)=0 otherwise. We consider models in which polymers of length N have weights corresponding to simple and selfavoiding random walks, SRW and SAW, with "a" the minimal permissible distance. We prove that to leading order in N, the number of matrices equals the number of walks for SRW, but not for SAW. The coarse grained Shannon entropies for SRW agree with the fine grained ones for n<=2, but differs for n>=3.
 
•  Last Branching in Directed Last Passage Percolation, with Herbert Spohn,
Markov Process and Related Fields 9 (2003), 323339.
The 1+1 dimensional directed polymers in a Poissonian random environment is studied. For two polymers of maximal length with the same origin and distinct end points we establish that the point of last branching is governed by the exponent for the transversal fluctuations of a single polymer. We also investigate the density of branches.

•  Reflected Brownian motions in the KPZ universality class, Springer Brief, with Herbert Spohn and Thomas Weiss available here  
•  Random Growth Models, with Herbert Spohn (2010).
One chapter of the The Oxford Handbook of Random Matrix Theory.
The link between a particular class of growth processes and random matrices was established in the now famous 1999 article of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation. During the past ten years, this connection has been worked out in detail and led to an improved understanding of the large scale properties of onedimensional growth models. The reader will find a commented list of references at the end. Our objective is to provide an introduction highlighting random matrices. From the outset it should be emphasized that this connection is fragile. Only certain aspects, and only for specific models, the growth process can be reexpressed in terms of partition functions also appearing in random matrix theory.
 
•  Dimers and orthogonal polynomials: connections with random matrices, Extended lecture notes of the minicourse at IHP (57 Oct. 2009). Published as part of Dimer Models and Random Tilings,
Panoramas et Synthèses 45 (2015).
B. de Tilière, P. Ferrari, edited by C. Boutillier, N. Enriquez.
Extended lecture notes of the minicourse at IHP (57 Oct. 2009).
In these lecture notes we present some connections between random matrices, the asymmetric exclusion process, random tilings. These three apparently unrelated objects have (sometimes) a similar mathematical structure, an interlacing structure, and the correlation functions are given in terms of a kernel. In the basic examples, the kernel is expressed in terms of orthogonal polynomials. 
•  From interacting particle systems to random matrices, Contribution to StatPhys24 special issue,
J. Stat. Mech. (2010), P10016.
In this contribution we consider stochastic growth models in the KardarParisiZhang universality class in 1+1 dimension. We discuss the large time distribution and processes and their dependence on the class on initial condition. This means that the scaling exponents do not uniquely determine the large time surface statistics, but one has to further divide into subclasses.
Some of the fluctuation laws were first discovered in random matrix models. Moreover, the limit process for curved limit shape turned out to show up in a dynamical version of hermitian random matrices, but this analogy does not extend to the case of symmetric matrices. Therefore the connections between growth models and random matrices is only partial.
 
•  The universal Airy_{1} and Airy_{2} processes in the Totally Asymmetric Simple Exclusion Process, In Proceeding ''Integrable Systems and Random Matrices: In Honor of Percy Deift'',
Contemporary Mathematics 458 (2008), 321332.
In the totally asymmetric simple exclusion process (TASEP) two processes arise in the large time limit: the Airy_{1} and Airy_{2} processes. The Airy_{2} process is an universal limit process occurring also in other models: in a stochastic growth model on 1+1dimensions, 2d last passage percolation, equilibrium crystals, and in random matrix diffusion. The Airy_{1} and Airy_{2} processes are defined and discussed in the context of the TASEP. We also explain a geometric representation of the TASEP from which the connection to growth models and directed last passage percolation is immediate.
 
•  Onedimensional stochastic growth and Gaussian ensembles of random matrices, with Michael Prähofer; In proceedings of ''Inhomogeneous Random Systems 2005'',
Markov Processes Relat. Fields 12 (2006), 203234.
In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of random matrices. This connection can be explained via point processes associated to the PNG model and the random matrices ensemble by an extension to the multilayer PNG and multimatrix models, respectively. We also explain other models which are equivalent to the PNG model: directed polymers, the longest increasing subsequence problem, Young tableaux, a directed percolation model, kinkantikink gas, and Hammersley process.

•  Timetime covariance for last passage percolation with generic initial profile, Oberwolfach 2019.  
•  Limit distributions for KPZ growth models with spatially homogeneous random initial conditions, in Oberwolfach Report 13 (2016), 3031–3086.  
•  Height fluctuations for the stationary KPZ equation, in Oberwolfach Report 28 (2014), 15601563  
•  Anomalous shock fluctuations in TASEP, in Oberwolfach Report 10 (2013), 30593062  
•  Interacting particle systems and random matrices, in Oberwolfach Report 7 (2010), 2915–2918  
•  Around the universality of the Airy_{1} process, in Oberwolfach Report 4 (2007), 24842486  
•  The Airy_{1} and Airy_{2} processes in the TASEP, in Oberwolfach Report 3 (2006), 24732476  
•  Stochastic growth in one dimension and Gaussian multimatrix models, with Michael Prähofer and Herbert Spohn; In proceedings of the
''14th International Congress on Mathematical Physics'' (ICMP 2003), World Scientific (J.C. Zambrini ed.) (2006), 404411.
We discuss the spacetime determinantal random field which arises for the PNG model in one dimension and resembles the one for Dyson's Brownian motion. The information of interest for growth processes is carried by the edge statistics of the random field and therefore their universal scaling is related to the edge properties of Gaussian multimatrix models.

•  Mark Adler, Brandeis University, USA 
•  Jinho Baik, University of Michigan, USA 
•  Gerard Barkema, Universiteit Utrecht, The Netherlands 
•  Alexei Borodin, MIT, USA 
•  Alexey Bufetov, Bonn University, Germany 
•  Ofer Busani, Briston Univesity, UK 
•  Folkmar Bornemann, TUMunich, Germany 
•  Sunil Chhita, Durham University, UK 
•  Ivan Corwin, Columbia University,USA 
•  René Frings, Bonn University, Germany 
•  Promit Ghosal, Columbia University,USA 
•  Joel L. Lebowitz, Rutgers University, USA 
•  Peter Nejjar, Bonn University, Germany 
•  Alessandra Occelli, Bonn University, Germany 
•  Sandrine Péché, Université Paris Diderot, France 
•  Michael Prähofer, TUMunich, Germany 
•  Tomohiro Sasamoto, Tokyo Institute of Technology, Japan 
•  Herbert Spohn, TUMunich, Germany 
•  Fabio Toninelli, Lyon University, France 
•  Pierre Van Moerbeke, Université Catholique de Louvain, Belgium 
•  Bálint Vető, Hungarian Academy of Sciences, Hungary 
•  Jon Warren, Warwick University, England 
•  Thomas Weiss, TUMunich, Germany 
•  Random matrices and determinantal processes,
contributo del ''Colloquio di matematica'', Note di Matematica e Fisica, CERFIM 12 (2003), 6783.
The aim of this work is to explain some connections between random matrices and determinantal processes. First we consider the eigenvalue distributions of the classical Gaussian random matrices ensembles. Of particular interest is the distribution of their largest eigenvalue in the limit of large matrices. For the Gaussian Unitary Ensemble, GUE, it is known as GUE TracyWidom distribution and appears in a lot of different models in combinatorics, growth models, equilibrium statistical mechanics, and in noncolliding random walks or Brownian particles. Secondly we introduce the determinantal processes, which are point processes which npoint correlation functions are given by a determinants of a kernel of an integral operator. It turns out that the eigenvalue distribution of the GUE random matrices is a determinantal process which kernel has a particular structure. This is the reason why the GUE TracyWidom distribution appears in a lot of models which are not related with random matrices.
 
•  Dall'informatica alla geometria ortogonale passando per l'algebra universale, con R. Moresi, Note di Matematica e Fisica, CERFIM 11 (2000), 115140  
•  Contact Matrices for Random Walks,diploma thesis supervised by Joel L. Lebowitz  
•  Notes basées sur le cours du Prof. Amrein's (Unige): Théorie des groupes pour la physique  
•  Project d'ingénieur: a computer program for the computation of lattices  
•  Project STS sur les nombres complexes 