Patrik Ferrari

In this thesis we consider two models, the first belonging to non-equilibrium and the second one to equilibrium statistical mechanics. The two models are connected the mathematical methods used to their analysis.

The first model analyzed is the polynuclear growth model (PNG) in one dimension, which belongs to the KPZ (Kardar-Parisi-Zhang) universality class. For growth processes, when the growth time t is large, the statistical properties of the surface are expected to depend only on qualitative properties of the dynamics and on symmetries, but not on the details of the models. In the case of the PNG, for large growth time t the surface height fluctuations scale as t^1/3 and the spatial correlation length as t^2/3. For boundary conditions inducing a droplet shaped surface, it was shown by Prhofer and Spohn that the statistics of the surface is described by the Airy process. This result was obtained by extending the surface line to a multi-layer model. In this thesis we consider the space-translation invariant case and determine the limit point process of the multi-layer model at fixed position. The process coincides with the edge scaling of eigenvalues of the Gaussian orthogonal ensemble (GOE) of random matrices.

The second model we study is the 3D-Ising corner at zero temperature. The corner of the crystal is composed by three facets (flat pieces) and a rounded piece interpolating between them. We analyze the border line between the rounded and a flat piece. When the corner defect size is large, say of linear length L, the fluctuations of the border line are of order L^1/3 and the spatial correlation length scales as L^2/3. We prove that the (properly rescaled) border line is well described by the Airy process. This is also the case for the terrace-ledge-kink (TLK) model, a simple model used to describe surfaces close to the high symmetry ones. We expect that the Airy process describes the border of the facets in the class of surface models with short range interactions.

Although the two models describe physically very different systems, the mathematical methods employed for their investigation are similar. Both models can be mapped into some non-intersecting line ensembles, which can also be viewed as trajectories of fermions. One can associate some point processes to the line ensembles. For the 3D-Ising corner it is an extended determinantal point process, whose kernel converges to the extended Airy kernel. The Airy kernel appears also in the edge scaling of Dyson's Brownian motion for GUE random matrices. The process for the PNG is a Pfaffian point process (at fixed position) and the 2x2 matrix kernel converges to the one of GOE random matrices. In the thesis we also discuss the connection with some other models: the longest increasing subsequence problem, directed polymers, last passage percolation, totally asymmetric exclusion process, random tiling, 3D-Young diagrams, and indirectly, Gaussian ensembles of random matrices.

Backwards geodesics for TASEP were introduced in [J. Stat. Phys. 172 (2018)]. We consider flat initial conditions and show that under proper scaling its end-point converges to maximizer argument of the Airy$_2$ process minus a parabola. We generalize its definition to generic non-integrable models including ASEP and speed changed ASEP (call it quasi-geodesics). We numerically verify that its end-point is universal, where the scaling coefficients are analytically computed through the KPZ scaling theory.

Integrable spin chains with a continuous non-Abelian symmetry, such as the one-dimensional isotropic Heisenberg model, show superdiffusive transport with little theoretical understanding. Although recent studies reported a surprising connection to the Kardar-Parisi-Zhang (KPZ) universality class in that case, this view was most recently questioned by discrepancies in full counting statistics. Here, by combining extensive numerical simulations of the Landau-Lifshitz one-dimensional magnet, with a framework developed by exact studies of the KPZ class, we characterize various two-point quantities that remain hitherto unexplored in spin chains, and find full agreement with KPZ scaling laws. This establishes the partial emergence of the KPZ class in isotropic spin chains. Moreover, we reveal that the KPZ scaling laws are intact in the presence of an energy current, under the appropriate Galilean boost required by the propagation of spacetime correlation.

We consider the totally asymmetric simple exclusion process on $\Z$ with step initial condition and with the presence of a rightward-moving wall that prevents the particles from jumping. This model was first studied in [Borodin-Bufetov-Ferrari'21]. We extend their work by determining the limiting distribution of a tagged particle in the case where the wall has influence on its fluctuations in neighbourhoods of multiple macroscopic times.

We consider the Airy₁ process, which is the limit process in KPZ growth models with flat and non-random initial conditions. We study the persistence probability, namely the probability that the process stays below a given threshold c for a time span of length L. This is expected to decay as e^{-k(c) L}. We determine an analytic expression for k(c) for all c≥3/2 starting with the continuum statistics formula for the persistence probability. As the formula is analytic only for c>0, we determine an analytic continuation of k(c) and numerically verify the validity for c<0 as well.

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 ρ to the right of it and lambda λ<ρ. 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.

The Ferrari-Spohn diffusion process arises as limit process for the 2D Ising model as well as random walks with area penalty. Motivated by the 3D Ising model, we consider M such diffusions conditioned not to intersect. We show that the top process converges to the Airy₂ process as M goes to infinity. We then explain the relation with the 3D Ising model and present some conjectures about it.

We study the decay of the covariance of the Airy₁ process, A₁, a stationary stochastic process on R that arises as a universal scaling limit in the Kardar-Parisi-Zhang (KPZ) universality class. We show that the decay is super-exponential and determine the leading order term in the exponent by showing that Cov(A₁(0),A₁(u))= exp(-(4/3+o(1))u³) as u goes to infinity. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of Airy₁ to planar exponential last passage percolation and several new results on the geometry of point-to-line geodesics in the latter model which are of independent interest; while the lower bound is primarily analytic, using the Fredholm determinant expressions for the two point function of the Airy₁ process together with the FKG inequality.

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.

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 equal to the probability that the Airy₂ process is below a barrier function 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.

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.

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 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 point-to-point model where the geodesic starts from the origin, we consider width δN^2/3, length up to δ^3/2N/(log(δ−1))³, and provide lower and upper bounds for the probability that the geodesics agree in that cylinder.

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].

The Preisach graph is a directed graph associated with a permutation ρ. We give an explicit bijection between its vertices and increasing subsequences of ρ with the property that the length of a subsequence equals to the degree of nesting of the corresponding vertex inside a hierarchy of cycles and sub-cycles of the graph. As a consequence, the nesting degree of the Preisach graph equals the length of the longest increasing subsequence.

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.

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.

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 Airy₂ 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.

In this paper we study stationary last passage percolation (LPP) with exponential weights and 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 new 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 Baik--Rains distributions for full-space geometry. We finally show that far enough away from the characteristic line, our distributions indeed converge to the Baik-Rains family. We derive our results using a related inhomogeneous integrable model having Pfaffian correlations, together with careful analytic continuation, and steepest descent analysis.

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.

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.

We consider the totally asymmetric simple exclusion process (TASEP) with non-random 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.

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.

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.

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.

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.

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.

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].

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.

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.

We explain the relation between certain random tiling models and interacting particle systems belonging to the anisotropic KPZ (Kardar-Parisi-Zhang) universality class in 2+1-dimensions. 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 non-intersecting 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.

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.

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 (1-α)/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 1-α=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_GOE² 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.

We compute the one-point probability distribution for the stationary KPZ equation (i.e.\ initial data 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 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.

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.

We consider the q-TASEP, that is a q-deformation 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 q-TASEP at time t are of order t^1/3 and asymptotically distributed as the GUE Tracy-Widom distribution.

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.

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 Airy₁ process.

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 Airy₁ 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.

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, 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. Our results generalize the known results for the zero drift situation.

In this short paper we derive a formula for the spatial persistence probability of the Airy₁ and the Airy₂ processes. We then determine numerically a persistence coefficient for the Airy₁ process and its dependence on the threshold.

We consider the two-point function of the totally asymmetric simple exclusion process with stationary initial conditions. The two-point 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 two-point function in a weak sense along the near-characteristic direction as time tends to infinity, thereby confirming the conjecture in the paper of Ferrari and Spohn.

We consider two models for directed polymers in space-time independent random media (the O'Connell-Yor semi-discrete 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 Tracy-Widom distribution.

We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semi-discrete 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 Baik-Ben Arous-Peche 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 Kardar-Parisi-Zhang equation. The Laplace transform formula we prove can be inverted to give the one-point probability distribution of the solution to these stochastic PDEs for the class of initial data.

We consider non-colliding 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 real-valued parameter σ controlling the interaction on the fluctuation scale. This generalizes the result for the symmetric tacnode process (λ=1 case).

We consider some models in the Kardar-Parisi-Zhang 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 (non-universal) first order corrections, which turn out to be a non-random 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.

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,...,m-1,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.

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 Tracy-Widom distribution functions and the Airy₂ 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 one-point 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+1-dimensional 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+1-dimensional interacting particle system under diffusion scaling limit. Finally, we analyze the analogous question for a diffusion on (complex) sample covariance matrices.

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided 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 space-time, 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 two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices.

There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar-Parisi-Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent z=3/2, that means one should find a universal space-time limiting process under the scaling of time as t T, space like t^2/3X 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 space-time 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.

The totally asymmetric simple exclusion process (TASEP) on Z with the Bernoulli-ρ 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 multi-point 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.

We prove an equality-in-law relating the maximum of GUE Dyson's Brownian motion and the non-colliding systems with a wall. This generalizes the well known relation between the maximum of a Brownian motion and a reflected Brownian motion.

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.

Consider n+m non-intersecting 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 large-n 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 Airy-type 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.

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.

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.

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.

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 one-point 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 space-time to the upper half-plane H such that on space-like submanifolds the multi-point fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on H.

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 space-like path in the space-time plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update.

We consider a new interacting particle system on the one-dimensional 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 space-like path are described by the Airy₁ and Airy₂ processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.

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 Airy₁ and Airy₂ processes, whose one-point distributions are the GOE and GUE Tracy-Widom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its one-point distribution is a new interpolation between GOE and GUE edge distributions.

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 Airy₁ process. Interpreted as a growth model, this confirms universality of fluctuations with flat initial conditions for a discrete set of slopes.

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 self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.

At the free-fermion point, the six-vertex 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 non-intersecting lines in the Lindström-Gessel-Viennot (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.

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional 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=(1-2ρ)t+2w(ρ(1-ρ))^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 Tracy-Widom 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 Riemann-Hilbert 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 space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle.

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of non-intersecting 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.

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of non-intersecting 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.

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(T²-t²). In the limit of large T, the fluctuation scale of b(t)-c_T(t) is T^1/3 and its time-correlation 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=τT, τ∈(-1,1), is only through the second derivative of c_T(t) at t=τT. We also prove a corresponding result where instead of the semicircle the barrier is a parabola of height T^γ, γ>1/2. The fluctuation scale is then T^(2-γ)/3. More general conditioning shapes are briefly discussed.

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 non-Gaussian and related to the edge statistics of GUE random matrices.

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 sine-kernel. Exactly at the facet edge, the steps have more space to meander. Their statistics is again determinantal, but this time based on the Airy-kernel. 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 space-time inhomogeneous transfer matrices.

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 i-th and j-th 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 self-avoiding 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.

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.

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 one-dimensional 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.

Extended lecture notes of the minicourse at IHP (5-7 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.

In this contribution we consider stochastic growth models in the Kardar-Parisi-Zhang 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.

In the totally asymmetric simple exclusion process (TASEP) two processes arise in the large time limit: the Airy₁ and Airy₂ processes. The Airy₂ process is an universal limit process occurring also in other models: in a stochastic growth model on 1+1-dimensions, 2d last passage percolation, equilibrium crystals, and in random matrix diffusion. The Airy₁ and Airy₂ 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.

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 multi-matrix 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, kink-antikink gas, and Hammersley process.

We discuss the space-time 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 multi-matrix models.

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 Tracy-Widom distribution and appears in a lot of different models in combinatorics, growth models, equilibrium statistical mechanics, and in non-colliding random walks or Brownian particles. Secondly we introduce the determinantal processes, which are point processes which n-point 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 Tracy-Widom distribution appears in a lot of models which are not related with random matrices.