Vladimir Belitsky

Invariant Measures and Convergence Properties for Cellular Automaton 184 and Related Processes

Our results concern long time limit properties of a deterministic dynamics that is common for a wide class of processes that have been studied so far during at least last two decades. The most widely known process from this class is a cellular automaton that acquired number 184 in the classification of S. Wolfram. This CA 184 is being intensively used to model vehicular traffic. However, our results are mainly derived with help of another process that offers a helpful insight into the studied dynamics, it is a so-called Ballistic Annihilation Model (abbreviated by BA). BA is a model for chemical reaction A+B → inert. In BA, A and B-type particles move in opposite directions with velocities 1 and -1, respectively, and annihilate upon collisions. Certain results concerning BA and CA 184 are also formulated in terms of another process known as a Model of Surface Growth (SG, for short); the surface shape in this process behaves as the integrated profile of particle distribution in CA 184.Our results are as follows. First, we characterize the invariant measures of the dynamics in interest. The bulk of our effort is devoted to the characterization of those of them that are not translation invariant; we call them phase separating invariant measures. In the case of BA, such measures are concentrated on the configurations consisting of two converging infinite blocks of (not necessarily adjacent) particles. In the case of CA 184, a phase separating measure describes the transition from free traffic phase to jammed phase. We also analyze domains of attraction of invariant measures and rates of convergence to them. This analysis then allows us to express the long time limit of particle current in CA 184 as a function of certain characteristics of its initial distribution, when it is translation invariant. This expression has been used in a companion paper (V. Belitsky, J. Krug, E. J. Neves and G. Schütz, A cellular automaton model for two-lane traffic, J. Stat. phys.103(5/6):945--971 (2001)) to show the enhancement of cars’ current caused by the possibility of lane changes in a model of traffic on a two-lane highway that was created by putting two CA 184’s in parallel. Our other results concern hydrodynamic limits of BA and CA 184. We prove that if the integrated profile of initial particle configuration of BA or CA 184 converges, as n → ∞, to some stochastic process W(x), x ∈ R, when being re-scaled by n-1 along x-axis and by cn-1 along y-axis for some sequence cn, then the integrated profile of particle configuration at time n under the same re-scaling, will converge, as time → ∞, to the local moving minimum of the process W(·), that is, to the process Wmin(·) defined by Wmin(x):=min{W(y) :x-1≤ y≤ x+1}. This hydrodynamic limit is then interpreted in terms of the limiting shape of surface in SG.

BALLISTIC ANNIHILATION AND DETERMINISTIC SURFACE GROWTH

A model of deterministic surface growth studied by Krug and Spohn, a model of the annihilating reactionA+B→inert studied by Elskens and Frisch, a one-dimensional three-color cyclic cellular automaton studied by Fisch, and a particular automaton that has the number 184 in the classification of Wolfram can be studied via a cellular automaton with stochastic initial data called ballistic annihilation. This automaton is defined by the following rules: At timet=0, one particle is put at each integer point of ℝ. To each particle, a velocity is assigned in such a way that it may be either +1 or −1 with probabilities 1/2, independent of the velocities of the other particles. As time goes on, each particle moves along ℝ at the velocity assigned to it and annihilates when it collides with another particle. In the present paper we compute the distribution of this automaton for each timet ∈ ℕ. We then use this result to obtain the hydrodynamic limit for the surface profile from the model of deterministic surface growth mentioned above. We also show the relation of this limit process to the process which we call moving local minimum of Brownian motion. The latter is the processBxmin,x ∈ ℝ, defined byBxmin≔min{By;x−1≤y≤x+1} for everyx ∈ ℝ, whereBx,x ∈ ℝ, is the standard Brownian motion withB0=0.

Microscopic structure of shocks and antishocks in the ASEP conditioned on low current

We study the time evolution of the ASEP on a one-dimensional torus with L sites, conditioned on an atypically low current up to a finite time t. For a certain one-parameter family of initial measures with a shock we prove that the shock position performs a biased random walk on the torus and that the measure seen from the shock position remains invariant. We compute explicitly the transition rates of the random walk. For the large scale behavior this result suggests that there is an atypically low current such that the optimal density profile that realizes this current is a hyperbolic tangent with a traveling shock discontinuity. For an atypically low local current across a single bond of the torus we prove that a product measure with a shock at an arbitrary position and an antishock at the conditioned bond remains a convex combination of such measures at all times which implies that the antishock remains microscopically stable under the locally conditioned dynamics. We compute the coefficients of the convex combinations.

A mixture of the exclusion process and the voter model

We consider a one-dimensional nearest-neighbour interacting particle system, which is a mixture of the simple exclusion process and the voter model. The state space is taken to be the countable set of the configurations that have a finite number of particles to the right of the origin and a finite number of empty sites to the left of it. We obtain criteria for the ergodicity and some other properties of this system using the method of Lyapunov functions.

A strong correlation inequality for contact processes and oriented percolation

Let v(A) be the extinction probability for a contact process on a countable set S with initial state A [subset of] S. We prove that for any sets A, B [subset of] S, [nu](A[intersection]B)[nu](A[union or logical sum]B)[greater-or-equal, slanted][nu](A)[nu](B). We also prove an analogous statement for oriented percolation.

Quantum Algebra Symmetry of the ASEP with Second-Class Particles

We consider a two-component asymmetric simple exclusion process (ASEP) on a finite lattice with reflecting boundary conditions. For this process, which is equivalent to the ASEP with second-class particles, we construct the representation matrices of the quantum algebra

Antishocks in the ASEP with open boundaries conditioned on low current

U_q[\mathfrak {gl}(3)]

Microscopic position and structure of a shock in CA 184

Uq[gl(3)] that commute with the generator. As a byproduct we prove reversibility and obtain in explicit form the reversible measure. A review of the algebraic techniques used in the proofs is given.