We consider excited random walks (ERWs) on integers with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [KZ08] have shown that when the total expected drift per site, delta, is larger than 1 then ERW is transient to the right and, moreover, for delta>4 under the averaged measure it obeys the Central Limit Theorem. We show that when delta in (2,4] the limiting behavior of an appropriately centered and scaled excited random walk under the averaged measure is described by a strictly stable law with parameter delta/2. Our method also extends the results obtained by Basdevant and Singh [BS08b] for delta in (1,2] under the non-negativity assumption to the setting which allows both positive and negative cookies.Comment: 27 page

We consider the asymptotic almost sure behavior of the solution of the equationwhere {Y x : x ∈ Z d } is a field of independent Lévy processes and is the discrete Laplacian.

We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett [2], who showed that for arbitrarily small infection parameter λ, the survival time of the process is larger than a stretched exponential function of the number of vertices, n. We obtain sharp bounds for the typical density of infected sites in the graph, as λ is kept fixed and n tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.

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Let E x be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on Z 2 is a Markov chain X(t) whose transition rates are given by w xy = ν exp(−βE x ) if x, y are neighbours in Z 2 . We study the behaviour of two correlation functions: P[X(t w + t) = X(t w )] and P X(t ) = X(t w )∀t ∈ [t w , t w + t] . We prove the (sub)aging behaviour of these functions when β > 1.