The probability that a one dimensional excited random walk in stationary ergodic and elliptic cookie environment is transient to the right (left) is either zero or one. This solves a problem posed by Kosygina and Zerner [10]. As an application, a law of large numbers holds in these conditions. RèsumèLa probabilitè q'une marche alèatoire unidimensionnelle excite dans un environnement ergodique et elliptique soit transiente a gauche ou a droite est soit nulle soit un. Ceci rèsout un problème pose par Kosygina et Zerner. Comme application, une loi des grands nombres est valable dans de telles conditions.2000 Mathematics Subject Classification. 60K35, 60K37.

Abstract. In this paper we consider an excited random walk (ERW) on Z in identically piled periodic environment. This is a discrete time process on Z defined by parameters (p 1 , . . . p M ) ∈ [0, 1] M for some positive integer M , where the walker upon the i th visit to z ∈ Z moves to z + 1 with probability p i (mod M ) , and moves to z − 1 with probability 1 − p i (mod M ) . We give an explicit formula in terms of the parameters (p 1 , . . . , p M ) which determines whether the walk is recurrent, transient to the left, or transient to the right. In particular, in the case thatall behaviors are possible, and may depend on the order of the p i . Our framework allows us to reprove some known results on ERW and branching processes with migration with no additional effort.

Abstract. We show that countable increasing unions preserve a large family of well-studied covering properties, which are not necessarily σ-additive. Using this, together with infinite-combinatorial methods and simple forcing theoretic methods, we explain several phenomena, settle problems of

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We study a discrete time self interacting random process on graphs, which we call Greedy Random Walk. The walker is located initially at some vertex. As time evolves, each vertex maintains the set of adjacent edges touching it that have not been crossed yet by the walker. At each step, the walker being at some vertex, picks an adjacent edge among the edges that have not traversed thus far according to some (deterministic or randomized) rule. If all the adjacent edges have already been traversed, then an adjacent edge is chosen uniformly at random. After picking an edge the walk jumps along it to the neighboring vertex. We show that the expected edge cover time of the greedy random walk is linear in the number of edges for certain natural families of graphs. Examples of such graphs include the complete graph, even degree expanders of logarithmic girth, and the hypercube graph. We also show that GRW is transient in $\Z^d$ for all $d \geq 3$