arXiv subjects

Timo Seppalainen

Publications and source records attributed to Timo Seppalainen.

25 records · Page 2Linked to original sources

arXiv2001

Diffusive fluctuations for one-dimensional totally asymmetric interacting random dynamics

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 hydrodynamic scaling. We look at diffusive fluctuations, by which we mean fluctuations on the scale of the classical central limit theorem. In the scaling limit these fluctuations obey deterministic equations with random initial conditions given by the initial fluctuations. Of particular interest is the effect of macroscopic shocks, which play a dominant role because dynamical noise is suppressed on the scale we are working.

math.PR↗

arXiv2000

Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor K-exclusion processes

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 class particles are characteristics and shocks of the conservation law for the particle density. In particular, we extend to K-exclusion Ferrari's result that the second class particle follows a macroscopic shock in the Riemann solution. The technical novelty of the proofs is a variational representation for the position of a second class particle, in the context of the variational coupling method.

math.PR↗

arXiv2000

Hydrodynamic profiles for the totally asymmetric exclusion process with a slow bond

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. The limit is proved by the variational coupling method.

math.PR↗

arXiv1999

Perturbation of the equilibrium for a totally asymmetric stick process in one dimension

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 dynamics merely translates the initial stick configuration. A time evolution for the perturbation is visible under a particular family of scalings: over times n^ν, 1<ν<3/2, a perturbation of order n^(1-ν) from equilibrium follows the inviscid Burgers equation. The results for the stick model are derived from asymptotic results for tagged particles in Hammersley's process.

math.PR↗

arXiv1999

A variational coupling for a totally asymmetric exclusion process with long jumps but no passing

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.

math.PR↗

arXiv1999

Recent results and open problems on the hydrodynamics of disordered asymmetric exclusion and zero-range processes

This paper summarizes results and some open problems about the large-scale and long-time behavior of asymmetric, disordered exclusion and zero-range processes. These processes have randomly chosen jump rates at the sites of the underlying lattice. The interesting feature is that for suitably distributed random rates there is a phase transition where the process behaves differently at high and low densities. Some of this distinction is visible on the hydrodynamic scale.

math.PR↗

arXiv1999

Strong law of large numbers for the interface in ballistic deposition

We prove a hydrodynamic limit for ballistic deposition on a multidimensional lattice. In this growth model particles rain down at random and stick to the growing cluster at the first point of contact. The theorem is that if the initial random interface converges to a deterministic macroscopic function, then at later times the height of the scaled interface converges to the viscosity solution of a Hamilton-Jacobi equation. The proof idea is to decompose the interface into the shapes that grow from individual seeds of the initial interface. This decomposition converges to a variational formula that defines viscosity solutions of the macrosopic equation. The technical side of the proof involves subadditive methods and large deviation bounds for related first-passage percolation processes.

math.PR↗