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 dZ, 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.
We study the model of the totally asymmetric exclusion process with generalized update, which compared to the usual totally asymmetric exclusion process, has an additional parameter enhancing clustering of particles. We derive the exact multiparticle distributions of distances travelled by particles on the infinite lattice for two types of initial conditions: step and alternating once. Two different scaling limits of the exact formulas are studied. Under the first scaling associated to Kardar-Parisi-Zhang (KPZ) universality class we prove convergence of joint distributions of the scaled partic…
The Bethe ansatz solution of periodic TASEP is formulated in terms of a ramified covering from a Riemann surface to the sphere. The joint probability distribution of height fluctuations at $n$ distinct times has in particular a relatively simple expression as a function of $n$ variables on the Riemann surface built from exponentials of Abelian integrals, traced over the ramified covering and integrated on $n$ nested contours in the complex plane.
The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multi-time and multi-location distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then ev…
We consider a totally asymmetric simple exclusion on $\mathbb{Z}$ 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 of the form $$ \mathbb{P}\Big(\sup_{τ\in \mathbb{R}}\{\textrm{Airy}_2(τ) -g(τ)\}\leq S\Big)$$ with arbitrary barrier functions $g$. 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 G…
We study the one-dimensional discrete time totally asymmetric simple exclusion process with parallel update rules on a spatially periodic domain. A multi-point space-time joint distribution formula is obtained for general initial conditions. The formula involves contour integrals of Fredholm determinants with kernels acting on certain discrete spaces. For a class of initial conditions satisfying certain technical assumptions, we are able to derive large-time, large-period limit of the joint distribution, under the relaxation time scale $t=O(L^{3/2})$ when the height fluctuations are critically…
We consider the one-dimensional totally asymmetric simple exclusion process with an arbitrary initial condition in a spatially periodic domain, and obtain explicit formulas for the multi-point distributions in the space-time plane. The formulas are given in terms of an integral involving a Fredholm determinant. We then evaluate the large-time, large-period limit in the relaxation time scale, which is the scale such that the system size starts to affect the height fluctuations. The limit is obtained assuming certain conditions on the initial condition, which we show that the step, flat, and ste…
We compute an explicit formula for the integrated density of states of the periodic Airy-Schr{ö}dinger operator on the real line. For this purpose, we study precisely the spectrum of the restriction of the periodic Airy-Schr{ö}dinger operator to a finite number of periods. We prove that all the eigenvalues of this restriction are in the spectral bands of the periodic Airy-Schr{ö}dinger operator and none of them are in its spectral gaps. We count exactly the number of eigenvalues in each of the spectral bands, this number being equal to the number of periods to which the periodic operator is re…
We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems can be thought of as new exactly solvable examples of tandem queues, directed first- or last-passage percolation models, or Robinson-Schensted-Knuth type systems with random input. One of the novel features of the particle systems is the presence of spatial inhomogeneity which can lead to the formation of traffic jams. For systems with special step-like initial data, we find explicit limit shapes, describe hydrodynamic evolution, and ob…
We establish an exact mapping between (i) the equilibrium (imaginary time) dynamics of non-interacting fermions trapped in a harmonic potential at temperature $T=1/β$ and (ii) non-intersecting Ornstein-Uhlenbeck (OU) particles constrained to return to their initial positions after time $β$. Exploiting the determinantal structure of the process we compute the universal correlation functions both in the bulk and at the edge of the trapped Fermi gas. The latter corresponds to the top path of the non-intersecting OU particles, and leads us to introduce and study the time-periodic Airy$_2$ process,…
We show there is a last path at the rough smooth boundary of the two-periodic Aztec diamond with parameter $a\in (0,1)$ that, suitably rescaled, converges to the Airy process, under the condition that $a$ tends to zero as the size of the Aztec diamond tends to infinity at a certain rate. This condition causes the rough region to have a thin, mesoscopic width. We also show that the dimers are described by a discrete Bessel kernel when the width is only of microscopic size.
Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between dominoes decay exponentially. In a previous paper, the authors found that a certain averaging of height function differences at the rough-smooth interface converged to the extended Airy kernel point process. In this paper, we augment the local geometrical picture at this interface by introducing well-defined lattice paths which are closely related to the level li…
We address two pressing questions in the theory of the Korteweg--de Vries (KdV) equation. First, we show the uniqueness of solutions to KdV that are merely bounded, without any further decay, regularity, periodicity, or almost periodicity assumptions. The second question, emphasized by Deift, regards whether almost periodic initial data leads to almost periodic solutions to KdV. Building on the new observation that this is false for the Airy equation, we construct an example of almost periodic initial data whose KdV evolution remains bounded, but fails to be almost periodic at a later time. Ou…
We revisit the periodic Schur process introduced by Borodin in 2007. Our contribution is threefold. First, we provide a new simpler derivation of its correlation functions via the free fermion formalism. In particular, we shall see that the process becomes determinantal by passing to the grand canonical ensemble, which gives a physical explanation to Borodin's "shift-mixing" trick. Second, we consider the edge scaling limit in the simplest nontrivial case, corresponding to a deformation of the poissonized Plancherel measure on partitions. We show that the edge behavior is described, in a certa…
