Timo Seppäläinen

Hydrodynamics and Platoon Formation for a Totally Asymmetric Exclusion Model with Particlewise Disorder

We consider a one-dimensional totally asymmetric exclusion model with quenched random jump rates associated with the particles, and an equivalent interface growth process on the square lattice. We obtain rigorous limit theorems for the shape of the interface, the motion of a tagged particle, and the macroscopic density profile on the hydrodynamic scale. The theorems are valid under almost every realization of the disordered rates. Under suitable conditions on the distribution of jump rates the model displays a disorder-dominated low-density phase where spatial inhomogeneities develop below the hydrodynamic resolution. The macroscopic signature of the phase transition is a density discontinuity at the front of the rarefaction wave moving out of an initial step-function profile. Numerical simulations of the density fluctuations ahead of the front suggest slow convergence to the predictions of a deterministic particle model on the real line, which contains only random velocities but no temporal noise.

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 allows 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.

Large Deviations from the Almost Everywhere Central Limit Theorem

We prove large deviation principles for the almost everywhere central limit theorem, assuming that the i.i.d. summands have finite moments of all orders. The level 3 rate function is a specific entropy relative to Wiener measure and the level 2 rate the Donsker-Varadhan entropy of the Ornstein-Uhlenbeck process. In particular, the rate functions are independent of the particular distribution of the i.i.d. process under study. We deduce these results from a large deviation theory for Brownian motion via Skorokhods representation of random walk as Brownian motion evaluated at random times. The results for Brownian motion come from the well-known large deviation theory of the Ornstein-Uhlenbeck process, by a mapping between the two processes.

Large deviation principles for Euclidean functionals and other nearly additive processes

Abstract. We prove a large deviation principle for a process indexed by cubes of the multidimensional integer lattice or Euclidean space, under approximate additivity and regularity hypotheses. The rate function is the convex dual of the limiting logarithmic moment generating function. In some applications the rate function can be expressed in terms of relative entropy. The general result applies to processes in Euclidean combinatorial optimization, statistical mechanics, and computational geometry. Examples include the length of the minimal tour (the traveling salesman problem), the length of the minimal matching graph, the length of the minimal spanning tree, the length of the k-nearest neighbors graph, and the free energy of a short-range spin glass model.

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.Abstract We prove a hydrodynamic limit for ballistic deposition on a multidimensional integer 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.

A microscopic mechanism for the porous medium equation

The porous medium equation on a d-dimensional torus is obtained as a hydrodynamic scaling limit, with the usual diffusion scaling, of the empirical measures of a sequence of reversible Markov jump processes on approximating periodic lattices. Each process can be viewed as a randomly interacting configuration of sticks (or energies, etc.). The configuration evolves through exchanges of stick portions that occur between nearest neighbours through a zero-range pressure mechanism, with conservation of total sticklength.

Parametric Multiple Sequence Alignment and Phylogeny Construction

Bounds are given on the size of the parameter-space decomposition induced by multiple sequence alignment problems where phylogenetic information may be given or inferred. It is shown that many of the usual formulations of these problems fall within the same integer parametric framework, implying that the number of distinct optima obtained as the parameters are varied across their ranges is polynomially bounded in the length and number of sequences.

Bounds for parametric sequence comparison

We consider the problem of computing a global alignment between two or more sequences subject to varying mismatch and indel penalties. We prove a tight 3(n/2/spl pi/)/sup 2/3/+O(n/sup 1/3/logn) bound on the worst-case number of distinct optimum alignments for two sequences of length n as the parameters are varied. This refines a O(n/sup 2/3/) upper bound by D. Gusfield et al. (1994). Our lower bound requires an unbounded alphabet. For strings over a binary alphabet, we prove a /spl Omega/(n/sup 1/2/) lower bound. For the parametric global alignment of k/spl ges/2 sequences under sum-of-pairs scoring, we prove a 3((k/2)n/2/spl pi/)/sup 2/3/+O(k/sup 2/3/n/sup 1/3/logn) upper bound on the number of distinct optimality regions and a /spl Omega/(n/sup 2/3/) lower bound. Based on experimental evidence, we conjecture that for two random sequences, the number of optimality regions is approximately /spl radic/n with high probability.

A class of stochastic evolutions that scale to the porous medium equation

A class of reversible Markov jump processes on a periodic lattice is described and a result about their scaling behavior stated: Under diffusion scaling, the empirical measure converges to a solution of the porous medium equation on thed-dimensional torus. The process can be viewed as a randomly interacting configuration of sticks that evolves through exchanges of stick pieces between nearest neighbors through a zero-range pressure mechanism, with conservation of total stick length.

Bounds for least relative vacancy in a simple mosaic process

Let