Stability of the stochastic matching model

We introduce and study a new model that we call the {\em matching model}. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be {\em matched}. There is a finite set of classes \maV for the items, and the allowed matchings depend on the classes, according to a {\em matching graph} on \maV. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a {\em matching policy}. When the sequence of classes of the arriving items is i.i.d., the sequence of buffer-contents is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is non-bipartite

Uniform and Bernoulli measures on the boundary of trace monoids

Trace monoids and heaps of pieces appear in various contexts in combinatorics. They also constitute a model used in computer science to describe the executions of asynchronous systems. The design of a natural probabilistic layer on top of the model has been a long standing challenge. The difficulty comes from the presence of commuting pieces and from the absence of a global clock. In this paper, we introduce and study the class of Bernoulli probability measures that we claim to be the simplest adequate probability measures on infinite traces. For this, we strongly rely on the theory of trace combinatorics with the M\"obius polynomial in the key role. These new measures provide a theoretical foundation for the probabilistic study of concurrent systems.Comment: 34 pages, 5 figures, 27 reference

Randomly growing braid on three strands and the manta ray

Consider the braid group $B_3=$ and the nearest neighbor random walk defined by a probability $\nu$ with support $\{a,a^{-1},b,b^{-1}\}$. The rate of escape of the walk is explicitly expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitly describe the harmonic measure of the induced random walk on $B_3$ quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups.Comment: Published at http://dx.doi.org/10.1214/105051606000000754 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Probabilistic cellular automata and random fields with i.i.d. directions

Let us consider the simplest model of one-dimensional probabilistic cellular automata (PCA). The cells are indexed by the integers, the alphabet is {0, 1}, and all the cells evolve synchronously. The new content of a cell is randomly chosen, independently of the others, according to a distribution depending only on the content of the cell itself and of its right neighbor. There are necessary and sufficient conditions on the four parameters of such a PCA to have a Bernoulli product invariant measure. We study the properties of the random field given by the space-time diagram obtained when iterating the PCA starting from its Bernoulli product invariant measure. It is a non-trivial random field with very weak dependences and nice combinatorial properties. In particular, not only the horizontal lines but also the lines in any other direction consist in i.i.d. random variables. We study extensions of the results to Markovian invariant measures, and to PCA with larger alphabets and neighborhoods