arXiv subjects

Pablo A. Ferrari

Publications and source records attributed to Pablo A. Ferrari.

59 records · Page 4Linked to original sources

arXiv1999

Cesaro mean distribution of group automata starting from measures with summable decay

Consider a finite Abelian group (G,+), with |G|=p^r, p a prime number, and F: G^N -> G^N the cellular automaton given by {F(x)}_n= A x_n + B x_{n+1} for any n in N, where A and B are integers relatively primes to p. We prove that if P is a translation invariant probability measure on G^Z determining a chain with complete connections and summable decay of correlations, then for any w= (w_i:i<0) the Cesaro mean distribution of the time iterates of the automaton with initial distribution P_w --the law P conditioned to w on the left of the origin-- converges to the uniform product measure on G^N. The proof uses a regeneration representation of P.

math.PR↗

arXiv1999

The Asymmetric Simple Exclusion Process with Multiple Shocks

We consider the one dimensional totally asymmetric simple exclusion process with initial product distribution with densities $0 \leq ρ_0 < ρ_1 <...< ρ_n \leq 1$ in $(-\infty,c_1\ve^{-1})$, $[c_1\ve^{-1},c_2ε^{-1}),...,[c_n \ve^{-1}, + \infty)$, respectively. The initial distribution has shocks (discontinuities) at $ε^{-1}c_k$, k=1,...,n and we assume that in the corresponding macroscopic Burgers equation the n shocks meet in $r^*$ at time $t^*$. The microscopic position of the shocks is represented by second class particles whose distribution in the scale $ε^{-1/2}$ is shown to converge to a function of n independent Gaussian random variables representing the fluctuations of these particles ``just before the meeting''. We show that the density field at time $\ve^{-1}t^*$, in the scale $\ve^{-1/2}$ and as seen from $\ve^{-1}r^*$ converges weakly to a random measure with piecewise constant density as $\ve \to 0$; the points of discontinuity depend on these limiting Gaussian variables. As a corollary we show that, as $ε\to 0$, the distribution of the process at site $ε^{-1}r^*+\ve^{-1/2}a$ at time $ε^{-1}t^*$ tends to a non trivial convex combination of the product measures with densities $ρ_k$, the weights of the combination being explicitly computable.

math.PR↗

arXiv1999

Loss network representation of Peierls contours

We present a probabilistic approach for the study of systems with exclusions, in the regime traditionally studied via cluster-expansion methods. In this paper we focus on its application for the gases of Peierls contours found in the study of the Ising model at low temperatures, but most of the results are general. We realize the equilibrium measure as the invariant measure of a loss-network process whose existence is ensured by a subcriticality condition of a dominant branching process. In this regime, the approach yields, besides existence and uniqueness of the measure, properties such as exponential space convergence and mixing, and a central limit theorem. The loss network converges exponentially fast to the equilibrium measure, without metastable traps. This convergence is faster at low temperatures, where it leads to the proof of an asymptotic Poisson distribution of contours. Our results on the mixing properties of the measure are comparable to those obtained with ``duplicated-variables expansion'', used to treat systems with disorder and coupled map lattices. It works in a larger region of validity than usual cluster-expansion formalisms, and it is not tied to the analyticity of the pressure. In fact, it does not lead to any kind of expansion for the latter, and the properties of the equilibrium measure are obtained without resorting to combinatorial or complex analysis techniques.

math.PR↗

arXiv1998

Invariant Measures and Convergence for Cellular Automaton 184 and Related Processes

For a class of one-dimensional cellular automata, we review and complete the characterization of the invariant measures (in particular, all invariant phase separation measures), the rate of convergence to equilibrium, and the derivation of the hydrodynamic limit. The most widely known representatives of this class of automata are: Automaton 184 from the classification of S. Wolfram, an annihilating particle system and a surface growth model.

math.PR↗

arXiv1998

Measures on contour, polymer or animal models. A probabilistic approach

We present a new approach to study measures on ensembles of contours, polymers or other objects interacting by some sort of exclusion condition. For concreteness we develop it here for the case of Peierls contours. Unlike existing methods, which are based on cluster-expansion formalisms and/or complex analysis, our method is strictly probabilistic and hence can be applied even in the absence of analyticity properties. It involves a Harris graphical construction of a loss network for which the measure of interest is invariant. The existence of the process and its mixing properties depend on the absence of infinite clusters for a dual (backwards) oriented percolation process which we dominate by a multitype branching process. Within the region of subcriticality of this branching process the approach yields: (i) exponential convergence to the equilibrium (=contour) measures, (ii) standard clustering and finite-effect properties of the contour measure, (iii) a particularly strong form of the central limit theorem, and (iv) a Poisson approximation for the distribution of contours at low temperature.

math.PR↗