We prove a Central Limit Theorem for the empirical measure in the one-dimensional Totally Asymmetric Zero-Range Process in the hyperbolic scaling $N$, starting from the equilibrium measure $ν_ρ$. We also show that when taking the direction of the characteristics, the limit density fluctuation field does not evolve in time until $N^{4/3}$, which implies the current across the characteristics to vanish in this longer time scale.
We study equilibrium fluctuations for a class of totally asymmetric zero-range type interacting particle systems. As a main result, we show that density fluctuation of our process converges to the stationary energy solution of the stochastic Burgers equation. As a special case, microscopic system we consider here is related to $q$-totally asymmetric simple exclusion processes ($q$-TASEPs) and our scaling limit corresponds to letting the quantum parameter $q$ to be one.
We estimate up to universal constants tails of symmetric and totally asymmetric 1-dimensional $α$-stable distributions in terms of functions of the parameters of these distributions. In particular, for values of $α$ close to $2$ we specify where exactly the tail changes from being Gaussian and starts to behave like in the Pareto distribution
These are the lecture notes for the mini-course at the PCMI graduate summer school in 2017. These notes are based on the article by Matetski, Quastel and Remenik arXiv:1701.00018 and give a self-contained exposition of construction of the KPZ fixed point.
It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space is covered a.s., or the covered volume fraction is strictly less than one. The aim of this work is to demonstrate that there is a third possibility in a Boolean model with totally asymmetric grains on a half-line: a.s. there is no unbounded component, but the covered volume fraction is one. An explicit condition is given characterizing the existence of an unbounded occupied component.
We prove a weak law of large numbers for a tagged particle in a totally asymmetric exclusion process on the one-dimensional lattice. The particles are allowed to take long jumps but not pass each other. The object of the paper is to illustrate a special technique for proving such theorems. The method uses a coupling that mimics the Hopf-Lax formula from the theory of viscosity solutions of Hamilton-Jacobi equations.
Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the integrated current $ \mathsf{h}(t,x) $ under the hyperbolic scaling of space and time by $ N $, i.e., $ \mathsf{h}_N(t,ξ) := \frac{1}{N}\mathsf{h}(Nt,Nξ) $. As hinted by the asymmetry in the upper- and lower-tail large deviations of the exponential Last Passage Percolation, the TASEP exhibits two types of deviations. One type of deviations occur with probability $ \exp(-O(N)) $, referred to as speed-$ N $; while the other with probability $ \ex…
We give a combinatorial description of the stationary measure for a totally asymmetric exclusion process (TASEP) with second class particles, on either Z or on the cycle Z_N. The measure is the image by a simple operation of the uniform measure on some larger finite state space. This reveals a combinatorial structure at work behind several results on the TASEP with second class particles.
We study aspects of the hydrodynamics of one-dimensional totally asymmetric K-exclusion, building on the hydrodynamic limit of Seppalainen (1999). We prove that the weak solution chosen by the particle system is the unique one with maximal current past any fixed location. A uniqueness result is needed because we can prove neither differentiability nor strict concavity of the flux function, so we cannot use the Lax-Oleinik formula or jump conditions to define entropy solutions. Next we prove laws of large numbers for a second class particle in K-exclusion. The macroscopic trajectories of second…
We describe the hydrodynamic behavior of the $k$-step exclusion process. Since the flux appearing in the hydrodynamic equation for this particle system is neither convex nor concave, the set of possible solutions include in addition to entropic shocks and continuous solutions those with contact discontinuities. We finish with a limit theorem for the tagged particle.
We prove that the variance of the current across a characteristic is of order t^{2/3} in a stationary constant rate totally asymmetric zero range process, and that the diffusivity has order t^{1/3}. This is a step towards proving universality of this scaling behavior in the class of one-dimensional interacting systems with one conserved quantity and concave hydrodynamic flux. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle. We build on the methods developed by Balazs-Seppalainen for asymmetric simple exclusion. However, some modifications we…
The time-integrated current of the TASEP has non-Gaussian fluctuations of order $t^{1/3}$. The recently discovered connection to random matrices and the Painlevé II Riemann-Hilbert problem provides a technique through which we obtain the probability distribution of the current fluctuations, in particular their dependence on initial conditions, and the stationary two-point function. Some open problems are explained.
We study a totally asymmetric simple exclusion process where jumps happen at rate one, except at the origin where the rate is lower. We prove a hydrodynamic scaling limit to a macroscopic profile described by a variational formula. The limit is valid for all values of the slow rate. The only assumption required is that a law of large numbers holds for the initial particle distribution. This includes also deterministic initial configurations. The hydrodynamic description contains as an unknown parameter the macroscopic rate at the origin, which is strictly larger than the microscopic slow rate.…
Let $\bb T_L = \bb Z/L \bb Z$ be the one-dimensional torus with $L$ points. For $α>0$, let $g: \bb N\to \bb R_+$ be given by $g(0)=0$, $g(1)=1$, $g(k) = [k/(k-1)]^α$, $k\ge 2$. Consider the totally asymmetric zero range process on $\bb T_L$ in which a particle jumps from a site $x$, occupied by $k$ particles, to the site $x+1$ at rate $g(k)$. Let $N$ stand for the total number of particles. In the stationary state, if $α>1$, as $N\uparrow\infty$, all particles but a finite number accumulate on one single site. We show in this article that in the time scale $N^{1+α}$ the site which concentrates…
We consider the one-dimensional totally asymmetric zero-range process starting from a step decreasing profile leading in the hydrodynamic limit to the rarefaction fan of the associate hydrodynamic equation. Under that initial condition, we show that the weighted sum of joint probabilities for second class particles sharing the same site, is convergent and we compute its limit. We derive the Law of Large Numbers for the position of a second class particle initially at the origin under the initial state in which all positive sites are empty and all negative sites are occupied and also for a slig…
We present a class of random cellular automata with multiple invariant measures which are all non-Gibbsian. The automata have configuration space {0,1}^{Z^d}, with d > 1, and they are noisy versions of automata with the "eroder property". The noise is totally asymmetric in the sense that it allows random flippings of "0" into "1" but not the converse. We prove that all invariant measures assign to the event "a sphere with a large radius L is filled with ones" a probability μ_L that is too large for the measure to be Gibbsian. For example, for the NEC automaton -ln(μ_L) ~ L while for any Gibbs …
We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous-Diaconis-Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under…
We study the evolution of a small perturbation of the equilibrium of a totally asymmetric one-dimensional interacting system. The model we take as example is Hammersley's process as seen from a tagged particle, which can be viewed as a process of interacting positive-valued stick heights on the sites of Z. It is known that under Euler scaling (space and time scale n) the empirical stick profile obeys the Burgers equation. We refine this result in two ways: If the process starts close enough to equilibrium, then over times n^νfor 1\leν<3, and up to errors that vanish in hydrodynamic scale, the …
In this paper we study the probability distribution of the position of a tagged particle in the $q$-deformed Totally Asymmetric Zero Range Process ($q$-TAZRP) with site dependent jumping rates. For a finite particle system, it is derived from the transition probability previously obtained by Wang and Waugh. We also provide the probability distribution formula for a tagged particle in the $q$-TAZRP with the so-called step initial condition in which infinitely many particles occupy one single site and all other sites are unoccupied. For the $q$-TAZRP with step initial condition, we provide a Fre…
We consider totally asymmetric simple exclusion processes with n types of particle and holes ($n$-TASEPs) on $\mathbb {Z}$ and on the cycle $\mathbb {Z}_N$. Angel recently gave an elegant construction of the stationary measures for the 2-TASEP, based on a pair of independent product measures. We show that Angel's construction can be interpreted in terms of the operation of a discrete-time $M/M/1$ queueing server; the two product measures correspond to the arrival and service processes of the queue. We extend this construction to represent the stationary measures of an n-TASEP in terms of a sys…
This paper is a further investigation of the problem studied in \cite{xue2020hydrodynamics}, where the authors proved a law of large numbers for the empirical measure of the weakly asymmetric normalized binary contact path process on $\mathbb{Z}^d,\, d \geq 3$, and then conjectured that a central limit theorem should hold under a non-equilibrium initial condition. We prove that the aforesaid conjecture is true when the dimension $d$ of the underlying lattice and the infection rate $λ$ of the process are sufficiently large.
The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli ρmeasure 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 …
We consider the asymmetric zero range process in dimensions $d \geq 2$. Assume the initial density profile is a perturbation of the constant density, which has order $N^{-α}$, $α\in (0,1)$, and is constant along the drift direction. Here, $N$ is the scaling parameter. We show that under some constraints on the jump rate of the zero range process, the perturbed quantity macroscopically obeys the heat equation under diffusive scaling.
We study the large deviation functional of the current for the Weakly Asymmetric Simple Exclusion Process in contact with two reservoirs.We compare this functional in the large drift limit to the one of the Totally Asymmetric Simple Exclusion Process, in particular to the Jensen-Varadhan functional. Conjectures for generalizing the Jensen-Varadhan functional to open systems are also stated.
We study asymmetric zero-range processes on Z with nearest-neighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. We prove quenched strong local equilibrium at subcritical and critical hydrodynamic densities, and dynamic local loss of mass at supercritical hydrodynamic densities. Our results do not assume starting from local Gibbs states. As byproducts of these results, we prove convergence of the process from given initial configurations with an asymptotic density of particles to the left of the origin. In p…
We study the large space and time scale behavior of a totally asymmetric, nearest-neighbor exclusion process in one dimension with random jump rates attached to the particles. When slow particles are sufficiently rare the system has a phase transition. At low densities there are no equilibrium distributions, and on the hydrodynamic scale the initial profile is transported rigidly. We elaborate this situation further by finding the correct order of the correction from the hydrodynamic limit, together with distributional bounds averaged over the disorder. We consider two settings, a macroscopica…
Consider a tagged particle in zero-range dynamics on the integer lattice in dimension d with rate g whose finite-range jump probabilities p possess a drift. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all dimensions and at most order t in d=1 and d larger or equal to 3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and when the rate g(k) increases and g(k)/k decreases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian…
We prove the law of large numbers and invariance principles for the tagged particle in the asymmetric exclusion process with long jumps when the process starts from its equilibrium measure.
This contribution aims at presenting and generalizing a recent work of Hernandez, Jara and Valentim [DOI:10.1016/j.spa.2016.06.026]. We consider the weakly asymmetric version of the so-called discrete Atlas model, which has been introduced there. Precisely, we look at some equilibrium fluctuation field of a weakly asymmetric zero-range process which evolves on a discrete half-line, with a source of particles at the origin. We prove that its macroscopic evolution is governed by a stochastic heat equation with Neumann or Robin boundary conditions, depending on the range of the parameters of the …
This paper develops a new methodology for studying continuous-time Nash equilibrium in a financial market with asymmetrically informed agents. This approach allows us to lift the restriction of risk neutrality imposed on market makers by the current literature. It turns out that, when the market makers are risk averse, the optimal strategies of the agents are solutions of a forward-backward system of partial and stochastic differential equations. In particular, the price set by the market makers solves a nonstandard "quadratic" backward stochastic differential equation. The main result of the …
We prove distributional convergence for a family of random processes on $\mathbb{Z}$, which we call asymmetric cooperative motions. The model generalizes the "totally asymmetric hipster random walk" introduced in [Addario-Berry, Cairns, Devroye, Kerriou and Mitchell, 2020]. We present a novel approach based on connecting a temporal recurrence relation satisfied by the cumulative distribution functions of the process to the theory of finite difference schemes for Hamilton-Jacobi equations [Crandall and Lyons, 1984]. We also point out some surprising lattice effects that can persist in the distr…
In this paper, we obtain the transition probability formulas for the Asymmetric Simple Exclusion Process (ASEP) and the $q$-deformed Totally Asymmetric Zero Range Process ($q$-TAZRP) on the ring by applying the coordinate Bethe ansatz. We also compute the distribution function for a tagged particle with general initial condition.
We study in detail and explicitly solve the version of Kyle's model introduced in a specific case in \cite{BB}, where the trading horizon is given by an exponentially distributed random time. The first part of the paper is devoted to the analysis of time-homogeneous equilibria using tools from the theory of one-dimensional diffusions. It turns out that such an equilibrium is only possible if the final payoff is Bernoulli distributed as in \cite{BB}. We show in the second part that the signal of the market makers use in the general case is a time-changed version of the one that they would have …
We consider finite-range asymmetric exclusion processes on $\mathbb Z$ with non-zero drift. The diffusivity $D(t)$ is expected to be of ${\mathcal O}(t^{1/3})$. We prove that $D(t)\ge Ct^{1/3}$ in the weak (Tauberian) sense that $\int_0^\infty e^{-λt}tD(t)dt \ge Cλ^{-7/3}$ as $λ\to 0$. The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn. In the nearest neighbor case, we show further that $tD(t)$ is mono…
We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$ started from step initial data and obtain the exact Lyapunov exponents for $H_0(t)$, the integrated current of ASEP. As a corollary, we derive an explicit formula for the upper-tail large deviation rate function for $-H_0(t)$. Our result matches with the rate function for the integrated current of the totally asymmetric simple exclusion process (TASEP) obtained in [Johansson 00](arXiv:math/9903134).
We examine the entropy of non-equilibrium stationary states of boundary driven totally asymmetric simple exclusion processes. As a consequence, we obtain that the Gibbs-Shannon entropy of the non equilibrium stationary state converges to the Gibbs-Shannon entropy of the local equilibrium state. Moreover, we prove that its fluctuations are Gaussian, except when the mean displacement of particles produced by the bulk dynamics agrees with the particle flux induced by the density reservoirs in the maximal phase regime.
The asymmetric exclusion process is an idealised stochastic model of transport, whose exact solution has given important insight into a general theory of nonequilibrium statistical physics. In this work, we consider a totally asymmetric exclusion process with multiple species of particles on a one-dimensional lattice in contact with reservoirs. We derive the exact nonequilibrium phase diagram for the system in the long time limit. We find two new phenomena in certain regions of the phase diagram: when the density of a species becomes zero throughout the system, and dynamical localisation when …
